## Abstract

In metamaterial systems, toroidal dipole (TD) plays an important role in determining their optical properties. Here, we proposed an all-dielectric metasurface consisting of two silicon split-ring resonators (SRRs) that can support strong TD resonance. The TD resonance is excited by TD moments both inside the unit cell and between the neighboring unit cells, and can be easily manipulated by altering the gap size or distance of the SRRs, leading to powerful electric and magnetic near-field enhancement. In addition, symmetric unprotected TD bound state in the continuum (TD-BIC) was achieved in closed-ring-resonator (CRR) metasurface, and transformed into leaky resonances with ultrahigh Q factors by adjusting the distance of CRRs. The proposed structure provides a good platform for us to better understand the coupling of SRRs, which is useful for the design and application of TD metasurfaces in biological sensors, nonlinear interactions and other photonic devices.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Toroidal dipole (TD) is created by poloidal currents (currents flowing on the surface of a torus along its meridians), but can be equivalently represented by a set of magnetic dipoles arranged head-to-tail along a loop [1,2]. In 1958, the TD moment was firstly reported by Zel’Dovich to explain parity violation in the weak interaction [3]. However, in classical electrodynamics, the excitations of TD and high-order TD multipoles are easily overlooked as their manifestations are often masked by much stronger electric dipole (ED), magnetic dipole (MD), and other multipoles. In 2007, N. Zheludev et al theoretically proposed TD response in metamaterials (MMs) [4], and then experimentally demonstrated with a periodical three-dimensional (3D) array of metallic split ring resonators (SRRs) in microwave regime in 2010 [5]. By reducing the size and precise fabrication of the structure, TD responses in optical band have been achieved in 3D MMs [6–8]. Since 2D planar MMs or metasurfaces are much easier to fabricate than 3D MMs, the research on TD responses gradually shifts from 3D MMs to metasurfaces [9–16].

Like ED and MD, TD MMs or metasurfaces have potential applications in photonic devices such as polarization conversion, sensing, and resonant transparency [17–19]. In particular, TD is often associated with high quality factor (Q) resonance and plays an important role in ultrahigh sensitive sensing and nonlinear interactions, such as it can excite non-radiative anapole mode [20–22], thus achieving high Q resonance [23–25]. Therefore, the excitation and manipulation of TD by designing effective structure of metasurface has become a hot spot now. In metallic TD metasurface, the contribution of TD to the resonance is relatively weak at beginning compared to other multipoles [26,27], and gradually becomes dominant as the design of structure is improved [28,29]. For dielectric TD metasurface, strong TD response with higher Q-factor can be achieved due to its low material loss [13–16]. So far, most of the works focus on TD resonances excited by TD moment inside the unit cell (called intra-TD). C. Z. Gu and Andrey Sayanskiy studied two TD resonances in MMs [30,31], which are excited by TD moments inside the unit cell and between the neighboring unit cells (called inter-TD), respectively. However, it is rarely reported that one TD resonance is excited and manipulated by both the intra- and inter-TD moments [32].

In addition, recent reports reveal a strong link between high Q TD metasurface and the bound state in the continuum (BIC) [33], which is a localized state with zero linewidth embedded in the continuum [34–38]. Optical BICs offer a unique opportunity to manipulate the light-matter interaction within radiative continuum, due to their ultrahigh-Q properties and the associated giant enhancement of the electromagnetic near-fields [33,39,40]. BIC shows an infinite Q factor and vanishing resonance width. However, in practice, BIC can be realized as a quasi-BIC in the form of a supercavity mode when both Q factor and resonance linewidth become finite at the BIC conditions due to material’s absorption, size effects, and other perturbations [41,42]. Quasi-BIC leaky resonances have been successfully used in ultrasensitive hyperspectral imaging and biodetection [43,44].

In this article, we proposed and numerically studied the TD response and TD-BIC in dual-SRR all-dielectric metasurface in the near infrared spectral region. The simulation results show that the TD resonance is excited by both the intra- and inter-TD moments, and can be easily manipulated by altering the gap size or the distance of SRRs, leading to strong electric and magnetic near-field enhancement. Furthermore, symmetric unprotected TD-BIC was achieved in closed ring resonator (CRR) metasurface. By adjusting the distance of CRRs, we can transform the TD-BIC into leaky resonances with ultrahigh Q factors. Such all-dielectric metasurface can have potential applications in ultrasensitive biosensors, nonlinear devices or other photonic devices.

## 2. Design and simulation results

#### 2.1 Scattering properties of two SRRs

The schematic of the considered pair of mirror symmetric silicon SRR (refractive index *n* = 3.45) is depicted in the inset of Fig. 1(b). The geometrical parameters of the SRRs are as follows: *l*_{1} = 600 nm, *l*_{2} = 360 nm, *w* = 100 nm, thickness *h* = 200 nm. The two SRRs are separated by a distance of *s* = 100 nm, and the gap size equals to *g*. The surrounding medium is air, and the excitation field is a *y*-polarized plane wave propagating along the *z*-axis. Numerical calculations are implemented by using Comsol Multiphysics simulation software. To better analyze the influence of the gap *g* on TD, we calculated the Cartesian multipole contributions into the scattered powers based on density of the induced current inside the SRRs at three different gap sizes [45–47]. The dominant five scattering powers of multipoles in the wavelength range of 950 ∼ 1600 nm are shown in Fig. 1(a) – 1(c), where *P _{y}*,

*M*,

_{x}*T*,

_{y}*Qe*,

*Qm*are the ED in

*y*-direction, MD in

*x*-direction, TD in

*y*-direction, electric quadrupole, and magnetic quadrupole, respectively.

*I*corresponds to the combined contribution of the destructive interference between the ED and TD. When

_{PT}*g*= 0 nm, the two SRRs structure becomes two CRRs. We can see from Fig. 1(a) that

*P*dominants the scattering powers in the full simulation range, and the second largest scattering power is

_{y}*T*,

_{y}*M*takes the third at short wavelengths. When

_{x}*g*= 100 nm, it is clearly seen in Fig. 1(b) that

*T*increases quickly and becomes the largest contribution in the wavelength range of 1200 ∼ 1350 nm, but

_{y}*P*still occupies the leading role at shorter and longer wavelengths. It is noticed that the scattered powers of

_{y}*P*and

_{y}*T*are crossing with each other at 1200 nm and 1351 nm, but the phase difference between them does not satisfy the condition of the anapole mode. When

_{y}*g*increases further to 160 nm shown in Fig. 1(c), in the wavelength range of 950 ∼ 1100 nm, it is worth mentioning that

*P*and

_{y}*T*not only have almost the same strength, but also the phase difference approaches

_{y}*π*(inset in Fig. 1(c)), leading to nearly zero strength of

*I*. This means that the ED moment and TD moment nearly form destructive interference, resulting in almost cancellation of the far field radiation. In addition, the contributions of

_{PT}*M*,

_{x}*Qe*and

*Qm*are much smaller than that of

*P*and

_{y}*T*. Therefore, it is beneficial to achieve the excitation of non-radiative anapole mode at this relatively broad spectral range.

_{y}#### 2.2 Silicon split-ring metasurface

As shown in Fig. 2(a), the investigated metasurface consists of dual-SRR array deposited on a quartz substrate (*n *= 1.46). The geometric dimensions of the SRRs shown in Fig. 2(b) are the same as that in section 2.1, and the lattice constants of the unit cell are *Λx = Λy *= 900 nm. *s* and *s’* represent the distances of SRRs inside the unit cell and between the neighboring unit cells, respectively, and *s’ = Λx-2×l _{2}-s*. For certain choices of parameter

*s*,

*s’*might become smaller than

*s*, and the strengths of the couplings between the SRRs within and across unit cells can be changed by varying parameter

*s*. The metasurface is illuminated normally by a

*y*-polarized incident wave. A semi-infinite quartz substrate is assumed in simulations, and periodic boundary conditions are applied in both x and y directions, perfectly matched layers (PMLs) are used in the wave propagating direction

*z*.

As a first remark, when the gap of SRRs *g* = 50 nm, the transmission spectrum of the dielectric metasurface in the wavelength range of 1400 ∼ 1500 nm is simulated and given in Fig. 3(a). It is clear to see a resonance with a Q value of 140 located at central wavelength of 1445 nm. The Q factor is defined as the ratio between the central resonant frequency and the bandwidth (FWHM, full width at half maximum). Here, Q value of the resonance was calculated by fitting the transmission spectrum with the following Fano formula [48], [49],

*F*is the Fano parameter, γ and ω

_{0}represent the width and position of the resonance. Then Q value is calculated by ω

_{0}/γ. In this case, the Fano fitted transmission spectrum is denoted with pink dotted line in Fig. 3(a), the fitting parameters are

*F*= 1,

*ω*

_{0 }= 2π × 207.54 THz, γ = 2π × 1.482 THz, respectively.

In order to further access the role of the TD excitation in the observed response, the scattered powers of five major multipoles for the proposed metasurface are calculated using the same way as in Section 2.1 and shown in Fig. 3(b). We can notice that *T _{y}* is the dominating multipole of scattering powers at the resonant dip, and it is approximately 2 times stronger than

*P*, 6 times larger than the magnetic quadrupole

_{y}*Qm*, around 1 order stronger than

*M*and 4 orders stronger than the electric quadrupole

_{x}*Qe*. The corresponding electric and magnetic near-field distributions in the

*x-y*plane and

*x-z*plane at the resonant dip are illustrated in Fig. 3(c) and 3(d), respectively. From Fig. 3(c) we can see that there are clockwise and anticlockwise circular displacement currents around the right and left SRR within the unit cell, respectively. Each of them could produce a magnetic field, the corresponding z-component magnetic field of the right SRR is along + z axis, and -z axis for the left SRR. These induced magnetic field patterns can generate head-to-tail magnetic moment

*m*formed within the unit cell (see in Fig. 3(d)), indicating the existence of the intra-TD moment

_{1}*T*along the

_{1}*y*-direction. Meanwhile, there are also clockwise and anticlockwise circular displacement currents in the neighboring SRRs across the unit cells in

*x*axis, forming head-to-tail magnetic moment

*m*in the neighboring SRRs and showing the inter-TD moment

_{2}*T*along the

_{2}*y*-axis. Therefore, we can obtain that the excitation of the TD resonance is attributed to both the intra- and inter-TD moments.

#### 2.3 Influence of the gap g on TD resonance

To study the resonance properties of the metasurface, the transmission spectra of the metasurface at different gap sizes are calculated and shown in Fig. 4(a). It is revealed that the central wavelength of the TD resonance exhibits a red-shift when the gap *g* decreases from 150 to 0 nm. Figure 4(b) shows the corresponding Q factor of the resonance as a function of *g*. It is clearly seen that Q value of the resonance increases gradually when *g* decreases from 150 nm to 25 nm, then rises dramatically and reaches up to 10830 as *g* decreases from 25 nm to 0 nm. Here we should mention that the two SRRs array becomes CRRs metasurface when *g = *0 nm.

For the purpose of further accessing the role of TD in the enhancement of Q factor of the TD resonance, we calculated the scattering powers of the ED and TD at the resonance for different gap *g*, as shown in Fig. 4(c). It is revealed that *P _{y}* remains substantially stable when

*g*increases from 0 to 150 nm. Contrarily,

*T*is very sensitive to

_{y}*g*and exponentially decays, which is consistent with that of the Q factor shown in Fig. 4(b), indicating that the TD plays a major role in the enhancement of the Q factor of the resonance. We can notice that the scattered powers of

*P*and

_{y}*T*are crossing with each other at

_{y}*g*= 75 nm, but the phase difference between them does not satisfy the condition of the anapole mode. In addition, Fig. 4(d) shows the magnetic near-field distributions in the

*x-z*plane at the resonance dips when

*g*= 0 and 150 nm, respectively. The inter-TD moment is stronger than the intra-TD when

*g*= 0 nm. However, when

*g*= 150 nm, the intra- and inter-TD moments both decrease significantly, and the inter-TD is much weaker than the intra-TD. It indicates that the intra- and inter-TD moments can be simultaneously manipulated through the SRR's gap, and the inter-TD moment is more sensitive to the gap. Furthermore, as

*g*decreases, the TD moment increases, the coupling of SRRs both inside the unit cell and between the neighboring unit cells strengthens. Hence, the incident energy is significantly confined both inside and between the silicon SRRs, the radiation loss is greatly reduced, resulting in the enlargement of Q factor of the resonance. It is mentioned here that the intra-TD is not always the dominant dipole of the TD resonance of the metasurface. As the gap

*g*of SRRs decreases, the TD coupling of the SRRs between the neighboring units becomes strong. Therefore, according to our calculation, when

*g*≤ 30 nm, we can suppress the intra-TD and make the inter-TD be the dominant dipole by increasing the distance

*s*.

#### 2.4 Influence of the lattice constant on TD resonance

To further analyze the influence of the lattice constant on the TD resonance, we calculated the transmission, Q factor and scattering powers of *P _{y}* and

*T*at different lattice periods

_{y}*Λx*, as shown in Fig. 5(a) – 5(c). The resonance wavelength exhibits a monotonous blue shift when

*Λx*decreases from 1000 to 850 nm, and the corresponding Q factor increases from 105 to 175. Meanwhile,

*T*increases slowly and

_{y}*P*remains substantially stable. Since the distance

_{y}*s*is fixed to be 100 nm,

*s’*decreases from 180 to 30 nm when

*Λx*varies, resulting in the increase of inter-TD, but its enhancement factor is smaller compared to the effect of the gap on

*T*. The distributions of magnetic near-field in the

_{y}*x-z*plane at resonance wavelengths when

*Λx*is 850 and 1000 nm are shown in Fig. 5(d), respectively. As

*Λx*decreases from 1000 to 850 nm, we can notice that the strengths of the inter- and intra-TD moments both increase, but the inter-TD moment enhances more significantly.

In brief summary, it is well known that the resonance of dielectric metasurface is based on the coupling between nano particles. Here, we demonstrate that, in a dual-SRR metasurface, a strong TD resonance is excited and attributed to the contributions of both the intra- and inter-TD moments, which are easily manipulated by changing the geometric parameters of the structure. Thus, both the intra- and inter-TD coupling of SRRs inside the unit cell and between the neighboring unit cells were clearly demonstrated, high Q resonance was interpreted. The proposed dual-SRR structure provides a good platform for us to better understand the coupling of SRRs, which is useful for the design and application of such kind toroidal metasurface. It should be mentioned that the definitions of the inter-TD and intra-TD are based on the coupling between the SRRs. Here, we define the couplings of SRR pairs facing each other and facing away from each other as the inter-TD and intra-TD, respectively. Of course, you can define them in reverse way, and the simulated results will not be changed. In addition, when *g = 0 *nm, the metasurface converts to a system of CRRs. In this case, the response will come from two different couplings because of the interaction of CRRs at two different separations *s* and *s’*, which will be studied in the following section.

#### 2.5 TD-BIC in CRR metasurface

In particular, the two mirror symmetric SRRs are transformed into a dual-CRR structure when *g* = 0 nm, hence, the intra- and inter-TD moments are determined only by the distances of the two CRRs inside the unit cell *s* and between the neighboring unit cells *s’*. We calculated the transmission spectra at different *s*, as shown in Fig. 6(a). When *s *= 50 nm (*s’* = 130 nm) and *s *= 130 nm (*s’* = 50 nm), these two *s* parameters correspond to the same CRR metasurface, thus same result of the TD resonance at wavelength of 1553 nm with Q value of 730 can be obtained. Especially, when *s* = *s’* = 90 nm, the TD resonance at wavelength of 1549 nm with extremely narrow bandwidth is clearly observed. The Q factor of the resonance as a function of distance *s* is given in Fig. 6(b). We can see that Q factor of the resonance reaches a peak value of more than 10^{8} when *s* = 90 nm, and decreases dramatically when *s* < 90 nm and *s* > 90 nm, clearly indicating that this is a typical symmetric unprotected TD-BIC [33]. BIC refers to a localized state with zero linewidth that is embedded in the continuum [34–38], and the obtained TD-BIC can be turned into leaky resonances with ultrahigh Q factors when the symmetry of the CRR structure is broken (*s ≠ s’*). The symmetric unprotected BIC refers to which the BIC is very sensitive to the small change of parameter *s*, otherwise, it is called symmetric protected BIC. Although the proposed TD-BIC is sensitive to the lattice constant in the *x* direction, the advantage is that it can be directly excited by a normally incident plane wave, without need for oblique incidence like other BICs to be excited [50], [51]. In addition, our multipole decomposition results also show that the TD-BIC resulted from the destructive interference between the TD and magnetic quadrupole.

Figures 7(a) and 7(b) respectively plot the electric and magnetic near-field enhancements at the resonance wavelength of the TD-BIC. We can see that the electric and magnetic field enhancements can reach up to 23080, and 26600, respectively. The strong electric near-field is mainly concentrated inside and surrounding adjacent rods in *x* direction of the CRRs, while the magnetic field mainly distributes in air environment inside the CRRs. Moreover, the electric and magnetic near-field enhancement factor with respect to distance *s* are given in Figs. 7(c) and 7(d), respectively. It is shown that the electric and magnetic enhancement factors reach a maximum when *s* = 90 and decrease dramatically when s < 90 and s > 90, which are consistent with that of the Q factor shown in Fig. 6(b), i.e., the increase of Q factor always accompanies with strong near-field enhancement.

Finally, considering the realization of high Q metasurface, fabrication, material imperfections or the finite size of the array will greatly influence on the strength and Q factor of the resonance [25], especially for ultrahigh Q BIC resonance [41,42]. The effect of losses is quantified by adding an imaginary part *k* to the Si refractive index and shown in Fig. 6(b). The BIC resonance will disappear when *k *> 10^{−6}, and the leaky resonances with Q value less than 4×10^{4} will not be affected when *k *< 10^{−5}. In fact, achieving strong resonance with Q value over 1000 experimentally in metasurfaces is still a big challenge. So far, high Q value of resonance experimentally obtained from all-dielectric MMs is 483 [52], and quasi-BIC leaky resonance with Q value around 250 has been obtained and used in ultrasensitive hyperspectral imaging and biodetection [43,44].

## 3. Conclusion

In summary, we have presented and numerically demonstrated TD resonance and TD-BIC in an all-dielectric metasurface consisting of dual-SRR array. The TD resonance can be excited, and mainly attributed to contributions of the TD moments both inside the unit cell and between the neighboring unit cells. And the intra- and inter-TD moments can be easily manipulated by altering the gap size or the distance of SRRs, while the ED remains substantially stable. Furthermore, a symmetric unprotected TD-BIC with ultrahigh-Q was achieved in the CRR metasurface, and meanwhile, strong electric and magnetic near-field enhancement can be observed. In addition, nearly non-radiative anapole mode could be realized at relatively broad spectral range in a single pair SRR nanoparticles. The proposed dual-SRR structure provides a good platform for us to better understand the optical coupling of SRRs, which is useful for the design and application of such kind toroidal metasurface in biological sensors, nonlinear interactions and other photonic devices.

## Funding

National Natural Science Foundation of China (61875179, 61875251); Primary Research and Development Plan of Zhejiang Province (2019C03114).

## Disclosures

The authors declare no conflicts of interest.

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