## Abstract

All-dielectric resonant metasurfaces are expected to boost Mie resonances with high *Q*-factors and enhanced electromagnetic fields owing to their large mode volumes and low material losses. However, the toroidal dipole (TD) and magnetic dipole (MD) are usually suppressed by other stronger multipoles due to their relatively weak coupling to the incident lights. In this work, the double resonances of TD and MD quasi-bound state in the continuum (quasi-BIC) are excited asymmetrically by breaking the geometric symmetry in an all-dielectric metasurface consisting of arrays of silicon I-bars and silicon Φ-disks, leading to their corresponding enhanced electric field confinements and high *Q*-factors. The sensing performances by these two resonances are investigated as well, achieving refractive index sensitivities of 784.8 nm/RIU and 630 nm/RIU, respectively. This work suggests a route to manipulate strong TD and MD quasi-BIC excitations and facilitates their practical applications such as nonlinear light sources and sensing.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Metasurfaces, as 2D forms of metamaterials, have extraordinary resonant optical responses and have been extensively employed in nanophotonics [1,2]. The surface plasmon resonances can be excited by sub-wavelength metallic metasurfaces, while their *Q*-factors are limited on account of high inherent resistive losses in the metals [3]. Therefore, all-dielectric metasurfaces with small dissipation and high laser damage threshold have been proposed, which can support Mie resonances with high *Q*-factors and low absorption losses [4]. They possess great capabilities of confining and enhancing the near field inside the nanoparticles, which govern the performance of many applications, such as harmonic generation [5,6], Raman scattering [7], and sensing [8,9].

Toroidal dipole (TD) is one of the members of Mie multipoles in all-dielectric metasurfaces, which originates from two inverse circulating currents [10–12]. Compared with electric dipole (ED) and magnetic dipole (MD), TD has weaker capabilities of confining and enhancing the near field [13,14]. Many researches about enhancing TD response have been reported. By manipulating the near-field coupling and the geometry arrangement, the strong and nearly pure TD resonance can be achieved in all-dielectric metasurfaces composed of clusters of subwavelength high-index dielectric cylinders [15,16]. Regulating the asymmetric permittivities of the components in all-dielectric metasurfaces is also an effective way to excite and boost the TD response [17], providing an efficient platform for sensing and optical nonlinearity. However, the electric field enhancements and *Q*-factors of TD resonances in the existing work are still insufficient, which require further explorations in structure design and physical mechanism innovation.

Bound state in the continuum (BIC) can produce extremely high *Q* resonances, which appears firstly in quantum mechanics, and then is reported in other fields such as optics, acoustics, hydrodynamics, and etc. [18–22]. A true BIC only appears in ideal lossless infinite structures or extreme values of parameters, and its resonance linewidth disappears while its *Q*-factor is infinite [23–25]. Normally, the BIC is symmetrically protected since the coupling constants vanish accidentally due to symmetry [26]. Recently it is revealed that symmetry-protected BIC can be transformed into quasi-BIC with a finite yet sharply high *Q* resonance by breaking the symmetry [27–29]. Some structures supporting quasi-BIC have also been reported, such as nanoholes [30,31], circular slots [32], and etc, implying outstanding performances in various applications, such as high *Q* cavity-mode laser [33], dynamical image tuning and display [34], and enhancing the nonlinear interactions between light and matter [35].

In this work, we propose the asymmetric excitations of TD resonance and MD quasi-BIC in an I-Φ shaped all-dielectric metasurface related to the parameter of distance difference. First, double resonances with giant enhancements of electromagnetic fields and extremely high *Q*-factors are excited. Then, by adjusting the distance difference between the I-bar and the neighboring Φ-disks, the physical mechanisms of double resonances are demonstrated. Finally, the sensing performances by these two resonances are investigated as well.

## 2. Structure design and double resonances properties

The proposed all-dielectric metasurface has a square-lattice pattern, as shown in Fig. 1, whose unit cell consists of a silicon I-shaped bar and a silicon Φ-shaped disk surrounded by air. The dispersive refractive indexes of silicon are taken from its experimentally measured data [36]. A *y*-polarized plane wave propagating along the *z*-axis is illuminated normally upon the metasurface. The numerical optical responses of the proposed silicon I-Φ shaped metasurface are simulated by commercial software COMSOL Multiphysics^{TM} based on the finite element method. Periodic boundary conditions are applied in the *x*- and *y*-directions, while perfectly matched layers are used to truncate the infinite space along the *z*-direction. The practical fabrication can be readily operated on silicon film by electron-beam lithography and reactive-ion etching techniques [37,38].

As is well known, the bar-disk tandem structures have been proposed to support Fano resonances [37–5]. However, the excitations of the TD and MD resonances were usually difficult due to their relatively weak coupling to the incident lights. The design consideration of the proposed metasurface lies on the overlap between the anapole and the electric dipole (ED) resonances. The Φ-disk array supports structured anapole modes [39], while the I-bar array excites ED modes. When the geometric parameters are chosen carefully to let them overlap at a same wavelength of 1072 nm (spectrum not shown for simplicity), and the array is arranged by both of the I-bar and Φ-disk, the structured MD dark modes in the Φ-disks would be induced and suppress the structured anapole modes. This is because the dipole axis of the structured MD modes is out-of-plane, and it is easy to be excited by the magnetic field of a co-plane ED mode rather than that of the normally incident wave, which appears to be perpendicular to the axis [37].

To start with, we define *Δg* = *g*_{2} – *g*_{1} as the distance difference between the bar and the neighboring disks. As *Δg* ≠ 0, the symmetry of the structure will be broken. Figure 2(a) gives the transmission spectrum of the proposed metasurface for *Δg* = 200 nm. It can be seen that two prominently sharp dips, Resonance R_{1} and Resonance R_{2}, appear on the almost transparent spectrum in the wavelength range of 1100 nm ∼ 1300 nm, indicating the double resonances are excited. The widths of both resonances are narrow, implying the high *Q*-factors and the strong electromagnetic field confinement. Indeed, one of the fascinating properties of the all-dielectric metasurface is its ability to confine the electric field inside the dielectric nanostructures, which facilitates the practical applications of nonlinear frequency conversion, nanolaser, and etc. Therefore, the average electric field enhancement factor is introduced to quantify the normalized electric field intensity averaged within the silicon nanostructure [40], which is expressed as followed

*E*| is the local electric field inside the silicon nanostructures, |

*E*

_{inc}| is the amplitude of the incident electric field, and

*V*the volume of the silicon nanostructures. Figure 2(b) presents the average enhancement factor as functions of the wavelength. It can be seen that two typical Lorentz line shapes emerge, corresponding to the two Fano-like resonances in Fig. 2(a). In the following, we will investigate the asymmetric excitations of the double resonances by the average enhancement factors at the resonant wavelengths and their corresponding

*Q*-factors, which are extracted from the Lorentz line shapes rather than the Fano-like line shapes in the transmission spectrum in Fig. 2(a).

## 3. Results and discussions

#### 3.1 Dependence on distance difference *Δg*

Figure 3(a) demonstrates the dependence of the transmission spectra on the distance difference *Δg*. It can be seen that the double resonances are symmetric with respect to *Δg* = 0. However, the excitations of these two resonances are asymmetric. With the decreasing of |*Δg*|, Resonance R_{1} has a red shift and the line width is broadened, while Resonance R_{2} has a blue shift and the line width is narrowed. Moreover, it is interesting that R_{2} disappears at *Δg* = 0, as indicated by the white dash circle. For both resonances, their Fano-like line shapes might vary dramatically, but their abilities to confine electromagnetic can be directly manifested on the spectra of average enhancement factor. We thus calculate the spectra of average enhancement factors and extract their corresponding properties.

First of all, the average enhancement factor extracted at the resonant wavelength of Resonance R_{1} and its corresponding *Q*-factor as the functions of *Δg* are given in Fig. 3(b). It can be seen that as |*Δg*| enlarges, the average enhancement factor is increased. The corresponding *Q*-factor is increased synchronously, indicating that the electric field localization comes inherently from the mode resonance. As |*Δg*| reaches 250 nm, the average enhancement factor and the *Q*-factor are as high as 3533 and 7885, which are 7 folds and 9 folds when compared to those of |*Δg*| = 0, respectively. Secondly, in contrast to R_{1}, R_{2} experiences a totally different excitation, as shown in Fig. 3(c). The average enhancement factor and the *Q*-factor both grow exponentially and approach infinity as |*Δg*| decreases. As |*Δg*| reaches 25 nm, they are as high as 7917 and 17684, which are 34 folds and 62 folds when compared to those of |*Δg*| = 250 nm, respectively. These extremely high average enhancement factor and *Q*-factor indicate the great potentials of the proposed all-dielectric metasurface in the applications of nonlinear optics and nanolaser [41,42].

Mathematically, a true bound state in the continuum (BIC) has infinite value of the *Q*-factor, but it is in fact a “dark mode”, which does not manifest itself in the R_{2} transmission spectrum in Fig. 3(a). However, the average enhancement factor and *Q*-factor of R_{2} are not infinite but disappear suddenly as |*Δg*| decreases to be 0, and this can be regarded as the symmetry-protected BIC. Interestingly, the symmetry-protected BIC is easily destroyed by breaking the geometric symmetry to induce leakage, generating the quasi-BIC with radiative loss when |*Δg*| > 0 [43]. To further verify this judgment, we fit the relationship between the *Q*-factor of R_{2} and the asymmetry parameter *α*, as shown in Fig. 3(d). *α* is defined as *Δg*/*g*, and *g* is the distance between the I-bars and the Φ-disks at *Δg* = 0. It can be seen that the evolution of the *Q*-factor of R_{2} is consistent with the theoretical formula between the radiative *Q*-factor *Q*_{rad} and the *α* [26],

The perfect match with the inverse quadratic law of Eq. (2) indicates that R_{2} can be recognized as a quasi-BIC, which is featured to have an extremely high *Q*-factor as *α* approaching zero.

In order to analyze the physical origins of these two resonances, the Cartesian multipole decompositions [44] are performed and those for the distance difference *Δg* = 0, 25 nm, 100 nm and 200 nm are presented in Fig. 4. For Resonance R_{1}, although there are some other dipoles mixing in, the toroidal dipole (TD) contributes more. As *Δg* increases, the intensities of these dipoles change, but TD always dominates, and its line width is getting narrower. Hence, R_{1} can be theoretically attributed to be the TD. On the other hand, for R_{2}, it is a pure MD resonance when *Δg* = 0. However, it is really weak and submerges on the flat curve of other dipole moments, as shown in Fig. 4(a). As the symmetry breaks and *Δg* increases to be 25 nm, this symmetry-protected BIC is transformed into the quasi-BIC, and the MD floats up and surpasses all other moments, as shown in Fig. 4(b). As *Δg* increases further, there are some other dipoles mixing in, and the line width of MD is getting wider. Nevertheless, the MD is the dominated dipole of R_{2}, and its evolution corresponds to the *Q*-factors in Fig. 3(c).

To further verify the origins of these two resonances, Fig. 5 demonstrates the corresponding electromagnetic field distributions. For Resonance R_{1} in Fig. 5(a), when *Δg* = 0, there has two opposite circular currents, revealing the pattern of TD. However, these two currents evenly distribute on two sides of Φ-disk, which can be considered to be two MDs in opposite directions. Therefore, the destructive interference of these two opposite MDs leads to the weakest ability of confinement. When |*Δg*| > 0, as mentioned above, more dipoles are mixed in, and one of the circular currents is destroyed, leaving the other one. As *Δg* increases, the only existing circular current approaches to the center of Φ-disk, resulting in that the electric field is enhanced inside the dielectric structures, which corresponds to the dependence of average enhancement factor and *Q*-factor on *Δg* in Fig. 3(b). On the other hand, for the MD quasi-BIC of R_{2} in Fig. 5(b), when *Δg* = 0, the distributions of both electric and magnetic fields have no any feature. However, as *Δg* slightly increases to be 25 nm, the quasi-BIC is formed and the circular current of MD emerges around the center of Φ-disk, leading to the extremely high localization and enhancement of electric field inside the dielectric structures of Φ-disk. As *Δg* increases further, the circular current moves away from the center of Φ-disk, leading to the rapidly degradation of the electromagnetic field. Together with the evolution in Fig. 5(a), it is exactly the position of circular current with respect to the center of Φ-disk leading to the asymmetric excitations of R_{2} and R_{1}, which have been demonstrated in Fig. 3(a)-(c).

#### 3.2 Refractive index sensing

Besides the ability to confine the electric field inside the dielectric nanostructure, the proposed metasurface also has the opportunity to be applied in refractive index (RI) sensing. In practical operations, the loss and the surface roughness of the metasurfaces will influence the resonances thus the sensing performance. Here, they are not considered for simplicity. Figures 6(a) and (b) give the resonant wavelength shifts as the function of the RI for Resonances R_{1} and R_{2}, respectively. The sensing analyte is set as the infinite surrounding medium. It is interesting to be seen that the wavelength shifts of both resonances are proportional to the variation of RI no matter how *Δg* changes. This stems from the fact that the electric field energies of high *Q* resonances are mainly confined inside the dielectric nanostructure, and rarely distributed in the surrounding medium. Therefore, the RI sensing capabilities depend more on the modes themselves rather than on the electric field hot spot on the surrounding sensing medium [45,46].

Nevertheless, the structured modes supported by the proposed metasurface have high sensing abilities. The refractive index sensitivities (RISs) are as high as 784.8 nm/RIU and 630 nm/RIU for R_{1} and R_{2}, respectively, and they are almost independent on the distance difference *Δg*. The zero wavelength shift for R_{2} under *Δg* = 0 is because there is no resonant mode formed due to the symmetry-protected BIC. The RISs based on all-dielectric metasurfaces have been previously explored in Table 1. Compared with these published works, the proposed metasurface has higher RISs. Together with the ease of fabrication resulting from the independence on *Δg*, the modes supported by our proposed metasurface have great potential for RI sensing application.

## 4. Conclusion

In summary, we have demonstrated that by breaking the geometric symmetry in an all-dielectric metasurface consisting of arrays of silicon I-bars and silicon Φ-disks, TD resonance and MD quasi-BIC are excited asymmetrically by the distance difference *Δg*, realizing giant enhancements of electromagnetic fields and extremely high *Q*-factors. Numerical results show that the average enhancement factor and the *Q*-factor of TD resonance are as high as 3533 and 7885 when *Δg = *250 nm, and those of MD resonance governed by quasi-BIC can be as high as 7917 and 17684 when *Δg = *25 nm. Furthermore, the sensing performances by these two resonances are investigated as well, achieving refractive index sensitivities of 784.8 nm/RIU and 630 nm/RIU, respectively, which are higher than those of previous works based on all-dielectric metasurfaces. Our proposed all-dielectric metasurface broadens the way to efficiently tailor strong TD and MD quasi-BIC excitations, and provides promising prospects for applications in light sources, sensing and many other aspects.

## Funding

National Natural Science Foundation of China (11604276, 61871462); National Key Research and Development Program of China (2018YFC0603503).

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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