Abstract

Both toroidal dipoles, electric dipoles and magnetic dipoles belong to one type of electromagnetic excitation. In this paper, we present an all-dielectric metasurface composed of an array of square nanoholes. It can simultaneously generate four resonance responses excited by TD, EQ and MD in the continuous near-infrared band. By introducing the in-plane symmetry breaking of the unit cell, asymmetric dielectric nanohole arrays are used to achieve two quasi-BIC resonance modes with high Q-factors excited by EQ and MD. The paper theoretically analyzes and demonstrates the relationship between structural asymmetry and the radiative Q-factor of two Fano resonances, that are governed by symmetry-protected BICs. And multipole decomposition and near-field analysis are performed to demonstrate the dominant role of various electromagnetic excitations in the four modes. The spectra response is also calculated for different incident polarization angles and medium refractive indices. The proposed metasurface is more feasible and practical compared to other complex nanostructures, which may open avenues for the development of applications such as biochemical sensing, optical switches and optical modulators, and provide a reference for the design of devices with polarization-independent properties.

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

1. Introduction

Toroidal dipole (TD) is an electromagnetic excitation different from the electric dipole and magnetic dipole, which can greatly promote the interaction between light and matter, then realize important optical devices such as low threshold semiconductor lasers [1], high sensitivity sensors [2,3], optical switches [4] and optical modulators [5]. TDs are divided into an electric toroidal dipole and a magnetic toroidal dipole. The former in the static case can be described as a ring pair of electric dipoles (EDs), then in the dynamic case electric toroidal dipole are either dynamic magnetic dipoles (MDs) or mean-square radii of MD, they have the same radiation pattern [6]. And the latter is generated by the poloidal currents flowing on the torus surface along its meridian, i.e., a set of magnetic dipoles (MDs) join head to tail [79]. Usually, TD resonances are considered as the magnetic toroidal dipole excitation. The TD concept was first introduced by Zel’Dovich to solve the problem of parity violation of weak interactions in atomic and nuclear physics [10]. And then in 2010, N. Zheludev’s research group experimentally demonstrated the dynamic TD resonance excitation in the microwave regime for the first time in metamaterials (MMs) [11]. However, in metallic metasurfaces, the contribution of TD radiation in resonance is relatively weaker than the radiation of electric and magnetic dipoles and is often masked by them, which can also gradually dominate by optimizing the structure. All-dielectric TD metasurfaces offer significant advantages because of its lower loss and CMOS system compatibility [12,13], by confining the optical field back into the device, it allows the excitation of high Q-factor magnetic and toroidal responses [8,14]. And the metasurfaces can also support both electric and magnetic dipolar Mie-resonant [2,15,16]. Moreover, the planar asymmetry introduced in the structure brings about the excitation of high Q resonances. Common structural asymmetries can be intrinsic asymmetries of the unit cell, differing from physical properties (e.g., dielectric constant, refractive index) and diversity of atomic arrangements, etc. Some recent work has used different metasurfaces to excite different multipole responses whose unit cells are arrangements and combinations of a number of dielectric meta-atoms. For example, Chen, et al. have achieved TD response for sensors of four dielectric disks in the terahertz range [17]. Terekhov, et al. have achieved magnetic octupole response of four silicon cubes in the near-infrared range [18]. The design of effective metasurface structures to excite and manipulate multipole response has become a current research interest. However, most of the studied dielectric metasurfaces are designed to support only one or two resonances in a single band, which are all excited by the same electromagnetic source (e.g., toroidal dipole) [1921]. This is difficult to find differences between the Fano resonances excited by different electromagnetic sources and to distinguish their behavior. In this paper, we proposed unit cell structure contains a square Si atom, this metasurface support multiple resonances excited by different electromagnetic sources in the near-infrared range.

An efficient method to achieve ultra-high Q Fano resonances is based on bound states in the continuum (BIC) [22,23], which is a localized state with zero line width supported by the continuum spectrum [24,25]. BIC was first developed in quantum mechanics and gradually extended to optics system [2,26,27]. Optical BIC offers a novel way to control light and matter interactions within the radiation continuum because of its ultra-high Q properties and associated electromagnetic near-field enhancement. There are several different types of BIC, one of which is the symmetry-protected BIC obtained at the Γ point in the optical waveguide by Shipman et al. [28] and Bulgakov et al. [29], which is an ideal BIC and has an infinitely high Q-factor and narrow resonance width. It results from the mismatch between the spatial symmetry of the mode and the spatial symmetry of the external radiation waves, so the mode cannot be coupled with the external radiation and observed in practical applications. By breaking the symmetry methods, such as the absorption of the material, oblique incidence of the light source or the introducing symmetry breaks in the structure, the radiation channel with the outside can be constructed, which allows the symmetry-protected BIC is perturbed and converted to a quasi-BIC mode that can radiate to the external continuum, resulting in both the resonance line width and the Q-factor becoming limited. Quasi-BIC leakage resonance has been successfully applied in optical filters, optical emitting devices and the detection of biological and chemical nanomembrane analyzers [30,31]. Recent papers report the high correlation between BICs and high Q-factor TD metasurfaces [32], i.e., the toroidal dipole bound state (TD-BIC) in the Fano resonance excited by the continuum has both TD and BIC characteristics.

In this paper, we present a Mie-resonant all-dielectric metasurface to investigate the properties of BIC excitation and Fano resonance in the near-infrared spectral region. The unit cell containing four asymmetric square nanoholes are etched from Si plate. Firstly, we theoretically study the polarization-independent MD and TD responses supported by the symmetric structure, when the structural symmetry is broken, generating EQ and MD responses with high Q-factors that support symmetry-protected BIC control, and achieving EQ-BIC state on the metasurface. The BIC theory is proved that the asymmetric parameter and the Q-factor follow the quadratic inverse trend. Secondly, the multipole decomposition and near-field distribution are calculated to illustrate the contribution of various electromagnetic excitations to each Fano resonance. Finally, the effect of the polarization angle and the refractive index on the four resonance modes in symmetric and asymmetric structures are discussed, respectively. The four resonance modes can be tuned by varying different geometric parameters. Thus, this work may open avenues for the development of applications such as biochemical sensing, optical switches and optical modulators, and provide a reference for the design of devices with polarization-independent properties.

2. Structure design

In the NIR region, as shown in Fig. 1(a), a metasurface consisting of nanostructure with four square nanoholes is presented. The unit cell is etched from Si plate with a thickness of 120 nm and nanohole arrays are deposited on a SiO2 substrate. The side lengths of the square nanoholes on both sides of the y-axis are represented by d0 and d1, respectively, where d1 is fixed at 80 nm. The center distance l0 between the nanoholes along both the x-axis and y-axis are all 220 nm. The length of the Si plate l1 = 510 nm and the unit cell with structure parameters period x = y = 640 nm. Both Si and SiO2 material parameters can be referred from Palik refractive index database values [33]. The optical properties of this structure are simulated and analyzed by the finite-difference time-domain (FDTD) method, where in the x- and y-directions using periodic boundary conditions, and in the z-direction using perfectly matched layers (PML) boundary conditions. When d0 $\ne $d1, the structure is asymmetric with respect to the y-axis. To excite significant resonant responses, the polarization direction of the incident light must be along an asymmetric axis [8,34]. Thus, we use a y-polarized plane wave propagating along the z-axis to vertically incident on the all-dielectric metasurface.

 figure: Fig. 1.

Fig. 1. (a) Schematic diagram of the structure of a partial metasurface composed of unit cells. (b) The calculated transmittance spectra when d0 changes from 60 nm to 110 nm. Four resonance responses are marked by mode I, II, III, and IV, respectively. (c)-(d) Fano fitting of mode II and mode IV at d0= 60 nm. The solid curve are the results of Fano formula fit, and the dashed curve are the simulation results.

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Additionally, the feasibility of experimental fabrication is analyzed. Figure 2 shows the all process flow for fabricating this nanostructure. Firstly in Fig. 2(b), by using low-pressure physical vapor deposition (LPCVD) method, the Si films can be deposited on SiO2 substrate. Secondly, Fig. 2(c) shows ZEP520A photoresist is then spin-coated on the Si plane and baked. Then employing electron beam lithography (EBL) and development, nanohole arrays can be obtained after inductively coupled plasma (ICP) etching [Fig. 2(d)(e)]. Finally, removal of photoresist and cleaning with deionized water [Fig. 2(f)].

 figure: Fig. 2.

Fig. 2. Process for fabricating the proposed nanostructured metasurfaces.

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3. Results and discussions

Symmetry-protected BIC state has infinitely high Q-factor and the resonance mode excited by structural symmetry breaking is called the quasi-BIC state [3537]. Because radiation channels are established between non-radiated bound states and the free space continuum, it results in more incident light radiating to free space, which manifests as wider resonance peaks and lower Q values in the transmittance spectra. As shown in Fig. 1(b), four Fano resonance modes are exhibited in the transmission curves at different d0, called mode I, II, III, and IV, respectively. Mode I and III have almost identical shapes and extinction ratios and can be observed all the time with d0 increases from 60 nm to 110 nm. It is worth mentioning that these two resonance modes have been formed without etching the nanoholes on the Si metasurface, and a corresponding frequency deviation and shape change with the increase and displacement of the nanoholes. That is because when the effective refractive index of the silicon metasurface decreases, the interaction between light and matter in the resonance cavity gradually decreases, which results in a slight blue-shift of these resonance positions [38]. Meanwhile, the resonance peaks of modes II and IV become sharper as the difference between d0 and d1 decreases, and when d0 = d1 = 80 nm, modes II and IV disappear, which means that there is no leakage of energy from the bound state to the free space continuum. As marked by the circles in Fig. 1(b), the radiation Q-factor tends to infinity at d0 = 80 nm, which proves the existence of two symmetry-protected BICs. Continuing increasing d0, these two modes reappear and become wider. It is because the in-plane symmetry of the unit cell is perturbed when d0 $\ne $ d1, allowing the two BIC modes to transform to two quasi-BIC modes that are radiable and have high Q value. Through the interference between the discrete bound state supported by nanostructure and the continuous radiation from free space, two distinctly different Fano resonances with a slight wavelength shift are formed. We fit the transmittance spectra of modes II and IV by using the typical Fano formula [36,39]:

$${T_{\textrm{Fano}}}(\omega ) = {\left|{{a_1} + j{a_2} + \frac{b}{{\omega - {\omega_0} + j\gamma }}} \right|^2},$$
where ω0 is the resonant frequency, a1, a2, and b are constant real numbers, and γ is the overall damping rate of the resonance cavity and is proportional to the line width of the resonance spectra. For asymmetric line shape Fano resonances in transmittance spectra, calculating their radiative Q values is usually obtained by fitting the spectra to Eq. (1) described above, and then calculating them using $Q = {\omega _0}/2\gamma $. The radiative Q-factor actually determines the asymmetry of the resonance curve. Figures 1(c) and 1(d) show the results of fitting the transmittance curve of the two modes at d0 = 60 nm, where mode II locates at 1073.94 nm and its Q-factor is 3.15 × 104, mode IV lies at 1280.45 nm, with Q-factor of 6625.

From the above analysis, it is clear that by reducing the asymmetry of the structure, ultra-high Q value can be achieved. Several methods can be used for our device. One of them is to decrease the length d0 of the two nanoholes on the left side of the y-axis, and the second one is to increase the area of the Si metasurface (increasing l0) which can indirectly decrease the asymmetry of the nanohole array. Firstly, the radiative Q values of mode II are calculated by decreasing the nanohole with length d0 from 160 nm to 90 nm at l0 = 510 nm, the Q value reaches about 5.2×104 at d0 = 90 nm in Fig. 3(a). Then increasing the area of the Si metasurface, i.e. increasing the l0 from 510 to 540 nm, and reach the Q value of about 6.5×104 at d0 = 90 nm in Fig. 3(b). Further, decreasing the asymmetry of the nanohole arrays can narrow the radiation channels and reduce the energy of the radiation dissipation, which eventually achieves high Q-factor [34]. We also calculated the Q-factor corresponding to when d0 = 81nm, and its value reaches about 2.0 ×106. However, the point can be omitted. It is very challenging to achieve such small asymmetric differences in nanostructures because of the limitations of current fabrication process technology. Indeed, in a previous work, an optimized silicon dimer metasurface can achieve a Q-factor of up to 1010. However, when deviating slightly from the optimized size by a few nanometers, the Q-factor drops by several orders of magnitude [32]. Meanwhile, with the nanohole length decreases, the resonant wavelength all red shifts at different lengths l0 in Fig. 3, which results from an increase in the effective refractive index of the silicon metasurface. Thus, the symmetry breaking of the nanostructure provides a zero-level radiation channel for the metasurface device, which excites a quasi-BIC resonance mode with ultra-high Q values, the Q-factor and line width can be controlled by asymmetric parameters. The high Q value of the resonant cavity can greatly enhance the interaction between light and matter.

 figure: Fig. 3.

Fig. 3. Variation curves of radiative Q-factor of Mode II and resonance wavelength position with side length d0 of nanohole when the side length l0 of the Si metasurface are 510 nm (a) and 540 nm (b), respectively.

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Additionally, the deep relationship between the radiative Q-factors and the asymmetric parameters is also investigated. When the symmetry is perturbed by structural changes, the radiation continuum interacts with the nonradiative bound state and leak out. The symmetry of the interaction matches the symmetry of the free space polarization, compensating for the space symmetry mismatch between the BIC mode and incident polarization, allowing the BIC mode to radiate outside with y-polarized excitation, and exciting ultra-high Q Fano resonance [25]. The x-component of the electric dipole moments has the same amplitude in opposite directions when the electric field is polarized along the y-axis at the Fano resonance (mode II). For the y-component, $Py ={\pm} ({2\Delta S/S} ){P_0}$ is the uncompensated dipole moments in opposite directions supported for the two halves of the nanohole array with respect to the y-axis [40], where S is the area of the symmetric nanohole array structure, ΔS is the reduced area of the broken part of the structure, as shown in Fig. 4(b), and p0 is the electric dipole moment of the right half of the nanohole array. $\alpha = \Delta S/S$ is described as the asymmetric parameter, the radiative Q-factor $Q = {\omega _0}/2\gamma $ of the quasi-BIC state can be expressed as

$$Q = \frac{A}{{2{k_0}}}{|{{p_0}} |^{ - 2}}{(\alpha )^{ - 2}}| ,$$

Then mode II is also confirmed to be inspired by Symmetry-protected BIC, by calculating the Q-factor of the metasurface at different asymmetric parameter α with increasing d0 from 80 to 160 nm. In Fig. 4(a), one can observe that α and the Q value of resonance mode II satisfies the basic condition of symmetry-protected BIC, i.e., the quadratic inverse trend as $Q \propto {\alpha ^{ - 2}}$ [35,40]. The fitting results of the black solid line successfully prove the theory proposed by Koshelev et al. and Cong et al. [35]. And mode IV is similar.

 figure: Fig. 4.

Fig. 4. (a) Plot of Q-factor as a function of the asymmetric parameter α (log-log scale), where the orange points are obtained by the FDTD method, and the black line is fitted to demonstrate the inverse quadratic dependence of α. (b) The schematic diagram of ΔS and S. (c) Transmittance spectra of the metasurface with different d0. The red dashed curve for d0= d1 = 80 nm, the blue solid curve for d0= 130 nm, d1 = 60 nm. (d) The multipole decomposition method obtains five main contributions of multipole excitations.

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The resonance modes in our structure are also a trapped mode excitation [34]. The trapped mode of Mie-resonant excitation actually corresponds to the lowest transmittance near the resonance wavelength [8,34,41]. In Fig. 4(c), we compare the transmittance spectra of the asymmetric and symmetric structures in the wavelength range from 900 nm to 1300 nm. When the unit cell structure is symmetric (d0 = 80 nm), the transmittance curve has two significant Fano resonances near the wavelength of 943.5 nm and 1156 nm, as demonstrated by Fano formula fitting [Eq. (1)]. When an asymmetry exits in the unit cell structure (d0 = 130 nm), a slight blue-shift of the original two Fano transmission peaks comes out and two quasi-BIC Fano resonance peaks are newly formed near λ1 = 935 nm, λ2 = 1023 nm, λ3 = 1122 nm and λ4= 1198 nm, respectively. Their spectral contrasts ratio all reach about 100%, defined as $[{({{T_{peak}} - {T_{antipeak}}} )} ]/[{({{T_{peak}} + {T_{antipeak}}} )} ]\times 100\%$.

To further investigate the properties of each Fano resonance in Fig. 4(c), the multipole decomposition method in Cartesian coordinates is used to calculate the scattering power for different types of multipole excitations. We first calculate the electric, magnetic and toroidal multipole moments by integrating the carrier density ρ(r) or the displacement current density J(r) inside the structural unit cell, which provides a comprehensive characterization of the near-field distribution of the electromagnetic source and their electromagnetic response in the far-field. Then, the sum of the far-field scattered energy at a certain frequency point is calculated for all moments. These moments are defined by ED: ${\boldsymbol {P}} = \frac{1}{{\textrm{i}\omega }}\smallint {\boldsymbol {j}}{\textrm{d}^3}r$, MD: ${\boldsymbol {M}} = \frac{1}{{2c}}\smallint ({{\boldsymbol {r}} \times {\boldsymbol {j}}} ){\textrm{d}^3}r$, TD: ${\boldsymbol {T}} = \frac{1}{{10c}}\smallint [{({{\boldsymbol {r}} \times {\boldsymbol {j}}} ){\boldsymbol {r}} - 2{{\boldsymbol {r}}^2}{\boldsymbol {j}}} ]{\textrm{d}^3}r$, EQ: ${\boldsymbol {Q}}_{\alpha \beta }^{(e )} = \frac{1}{{2i\omega }}\smallint \left[ {{r_\alpha }{j_\beta } + {r_\beta }{j_\alpha } - \frac{2}{3}({{\boldsymbol {r}} \cdot {\boldsymbol {j}}} ){\delta_{\alpha ,\beta }}} \right]{d^3}r$, and MQ: ${\boldsymbol {Q}}_{\alpha \beta }^{(m )} = \frac{1}{{3c}}\smallint [{{{({{\boldsymbol {r}} \times {\boldsymbol {j}}} )}_\alpha }{r_\beta } + {{({{\boldsymbol {r}} \times {\boldsymbol {j}}} )}_\beta }{r_\alpha }} ]{d^3}r$, where r is the position vector and c is the speed of light, $\omega $ is the angular frequency, and α, β = x, y, z. The scattered power of each multipole moment can be obtained from [9,40]. The multipole decomposition results expressed in logarithmic coordinates are shown in Fig. 4(d). The dominant multipole components near each resonant wavelength are MD, EQ, TD and MD, respectively.

Furthermore, to show intuitively the different electromagnetic excitations corresponding to each transmittance dip, the near-field distributions of asymmetric unit cell in the x-y plane near the four wavelength positions and a schematic of the corresponding electromagnetic sources are displayed in Fig. 5. The field patterns are normalized to the incident electromagnetic field. When λ1 = 935 nm [Fig. 5(a-upper)], the vortex arrows present two magnetic field loops rotating in opposite directions in the x-y plane, while the electric field Ez is along the normal of the toroidal in opposite directions. Figure 5(a-lower) shows the electromagnetic excitation exists in the whole unit cell at λ1, a ring of the counterclockwise rotating electric dipoles appears in the y-z plane, which can be identified as a MD response along the x direction.

 figure: Fig. 5.

Fig. 5. The upper shows normalized field patterns (|H/H0| and |E/E0|) of unit cells in the x-y plane near four wavelengths, including Ez (λ1) and Hz (λ2, λ3 and λ4). the lower shows a schematic of the corresponding electromagnetic sources. Arrows represent the instantaneous directions of the magnetic (yellow arrows) and electric (black arrows) field distribution.

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Then at λ2 = 1023 nm [Fig. 5(b)], the field pattern can be identified as an EQ in the x-y plane. Figure 6(a) shows the normalized field distributions in two y-z planes of the unit cell (y-z1 plane at x=d0/2 and y-z2 plane at x = d1/2), where the electric field Ex direction is reversed along the x-axis. That can be identified as two EDs in the x- and –x-direction, respectively. This Fano resonance excited by the electric quadrupole bound state in a continuum (EQ-BIC) exhibits not only EQ characteristics, but also BIC properties. Then looking at λ3 = 1122 nm, the vortex arrows in Fig. 5(c-upper) represent two the displacement current loops with opposite rotational directions, the magnetic field Hz being the left and right along the x-axis in opposite directions. That results from the broken part of the structure and the other part of the structure has a π-phase modulation difference to the incident light. A toroidal electric field distribution is formed in the x-y plane, which results in oscillations of the excited two MDs in the z-direction at λ3, as shown in Fig. 5(c-lower). Additionally, one can observe that the field energy is tightly confined within the structure. Indeed, two current loops circulating in the opposite directions yield a superposition of a magnetic quadrupole (MQ) and a TD. This coincides with the contribution of the multilevel decomposition near wavelength λ3 in Fig. 4(d). However, we can only observe the dominant electromagnetic contribution near each resonance in the near-field pattern, so it does not reflect the MQ response. For TD response, In Fig. 6(b), the electric field Ey oscillates along the center and around of the nanohole array in opposite direction, resulting in the generation of magnetic loop in the x-z plane. That can be identified as a TD response along the –y-direction in the unit cell.

 figure: Fig. 6.

Fig. 6. Calculated normalized field distributions (|E/E0|) of unit cells at λ2 and λ3. (a) The field patterns show the Ex in yz1 and yz2 planes. (b) The field pattern shows the electric field Ey in the x-z plane. The black arrows demonstrate the instantaneous flow direction of the magnetic field.

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The field distribution of λ4 = 1198 is shown in Fig. 5(d), where the displacement current flows present a distinct vortex rotating along a clockwise direction in the x-y plane and the magnetic field is directed along the center and around of the nanohole array in opposite direction. This can be identified as a MD along the –z-direction in the unit cell in Fig. 5(d-lower). This Fano resonance excited by the toroidal dipole bound state in a continuum (MD-BIC) exhibits not only BIC properties, but also MD characteristics.

The near-field distribution results and the scattered power curves in Fig. 5 are verified with each other. It is worth mentioning that the field analysis of λ1 and λ3 can also be used to explain the MD response (943.5nm) and TD response (1156nm) of symmetric structure, as they have the same resonance mode.

To study the sensing performance of the proposed structure, the transmission responses of symmetric and asymmetric nanostructures are calculated in different polarization incident light. Figure 7(a) shows the transmittance of the metasurface at d0 = d1 = 80 nm, the black dashed curves represent the transmittance curves at varied polarization angles. Here, the inset presents the polarization angle θ, which is defined as the angle between the y-axis and the polarization direction of incident electric field. The resonant modes near wavelengths of 943.5 nm and 1156 nm are marked as modes I and III, respectively. The extinction ratios, the transmission coefficients, positions, and shapes are identical for two resonance modes with the θ increasing from 10 to 90 degrees. It demonstrates that the proposed symmetric nanostructure has polarization-independent properties, which is attributed to its complete symmetry of the unit cell.

 figure: Fig. 7.

Fig. 7. (a) Transmittance spectra of modes I, III in different polarization angles at d0 = d1 = 80 nm, the black dashed curve shows the transmission curve. The transmittance magnitude and position are identical. (b)(c) Transmittance for modes II and IV in different polarization angles at d0 = 130nm, d1 = 80nm. The polarization angle θ directions are shown in the insets of (a) and (c).

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Then the transmittance spectra of nanostructure at θ of 10, 30, 50, 70, and 90 degrees are calculated with fixed d0 = 130 nm. As shown in Fig. 7(b) and (c), the resonance modes near wavelengths of 1023 nm and 1198 nm are highlighted in the shaded regions and marked as modes II and IV, respectively. With the increase of the polarization angle θ, the transmittance of modes II and IV gradually decreases and without position shifts. When the polarization direction of the incident plane wave is the same as the x-direction (θ = 90°), the two quasi-BIC resonance responses completely vanish and the maximum transmission amplitudes arrive at 75%. The modulation depths of mode II and IV, defined as $|{({{T_{on}} - {T_{off}}} )/{T_{on}}} |\times 100\%$, all reach almost 100%, where Ton and Toff represent the maximum transmission amplitude about 75% when θ = 90° and the minimum transmission amplitude about 0% when θ = 0°, respectively. It demonstrated that these resonance modes supported by symmetry-protected BIC have polarization dependence, and resonance peaks with high spectral contrasts can only be excited at specific incident polarization states. These properties can provide powerful responses for designs in various device configurations. For example, the high radiation factor, ultra-high modulation depth and spectral contrast of the Fano resonance are exploited to design high-performance optical switches. Compared to the plasmonic metasurface optical switches [42,43], the proposed all-dielectric metasurface will have better performance. In addition, the recent work has achieved imaging through a Fano-Resonant dielectric metasurface governed by Quasi-bound States in the Continuum [44].

Finally, we analyze the effect of an external medium with various refractive indices n on the transmission spectra. When a liquid with n from 1.31 to 1.35 fills the structure gap and surroundings, transmittance spectra show four sharp Fano resonance modes in Fig. 8(a), where the increase of refractive index has little effect on the Q-factor and spectral contrast of the resonant peaks, and the resonance position appears visibly redshift. The figure of merit (FOM) and the refractive index sensitivity (S) are key parameters in measuring the performance of refractive index sensors. S=∂λ/∂n (nm/RIU), where ∂λ and ∂n are the wavelength shift and the refractive index difference values. Then FOM is defined as Sλ, that is the ratio of sensitivity(S) and the line-width of the resonance (Δλ). The calculated sensitivities of the four resonance modes are about 237 nm/RIU, 295 nm/RIU, 175 nm/RIU and 150 nm/RIU, and the FOMs can reach 59, 738, 46 and 55, respectively. The difference in sensitivity between these resonance modes is mainly attributed to the field distribution. In fact, there is work demonstrating that the multipoles’ far-field scattering inside the dielectric particles can be controlled by changing the environmental refractive index, eventually achieving a tailored optical response [45,46]. The device sensitivity can also be enhanced by reducing the structural asymmetry to further improve the FOM. The FOM are higher than the metal or hybrid metal-dielectric sensors [4749]. It is worth mentioning that in the recent work, Buchnev et al. achieved external dynamic control of the Tamm plasma (TP) wavelength on a non-diffracting optical metasurface in a new way, this demonstrates the potential of the TP for refractive index sensing [50]. The hybrid metal-dielectric systems based on Tamm state has high sensitivity and tunability. It is now widely used for refractive index gas and temperature sensing [51,52]. The inherent multi-Fano response and sharp resonance peaks are suitable for the design of multi-channel refractive index sensing devices.

 figure: Fig. 8.

Fig. 8. (a) Transmittance spectra of the four resonance modes when the metasurface is covered by a liquid with different refractive indices. (b) Wavelength shift of the four resonance modes relative to the refractive index.

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4. Conclusion

In summary, an all-dielectric metasurface with multiple Fano resonances is proposed in this paper. The unit cell is etched four asymmetric square nanoholes from the Si metasurface to excite four resonance modes, the electromagnetic sources of modes I, II, III, and IV are MD, EQ, TD, and MD, respectively. Modes I and III derive from symmetric structures, meanwhile modes II and IV are governed by the symmetry-protected BIC which can be transformed into quasi-BIC modes. Additionally, the BIC theory is proved that with the structural asymmetric parameter decreases, the Q-factor follows the quadratic inverse trend. Then multipole decomposition and the field distributions are calculated to demonstrate the contribution of TD and EQ to each Fano resonance mode. Finally, we discuss the effect of polarization angle on the four resonance modes in different structures, where modes I and III are polarization-independent in the symmetric structure, and modes II and IV generated by the asymmetric structure are polarization-dependent, with spectral contrast and modulation depth of about 100%. The multiple sharp Fano resonances exhibited by the structure can be applied to multi-channel refractive index sensing devices, where the highest S and FOM reached 295 nm/RIU and 738, respectively. By declining the asymmetry parameters, higher Q values and further improvement for the FOM can be obtained. Besides, the individual Fano resonances can be conveniently tuned by changing the different structural parameters, making them more suitable for potential applications. The proposed symmetric nanostructure can enhance the study of well-performing polarization-independent resonators, and also have potential applications in photonic switch devices, low threshold lasers, or other ultra-sensitive biosensors.

Funding

National Natural Science Foundation of China (61631014, 62071059).

Disclosures

The authors declare no conflicts of interest.

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9. N. Papasimakis, V. A. Fedotov, V. Savinov, T. A. Raybould, and N. I. Zheludev, “Electromagnetic toroidal excitations in matter and free space,” Nat. Mater. 15(3), 263–271 (2016). [CrossRef]  

10. I. B. Zel’Dovich, “Electromagnetic Interaction with Parity Violation,” J. Exp. Theor. Phys. 6(6), 1184 (1958).

11. T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal Dipolar Response in a Metamaterial,” Science 330(6010), 1510–1512 (2010). [CrossRef]  

12. K. Bi, Y. Guo, X. Liu, Q. Zhao, J. Xiao, M. Lei, and J. Zhou, “Magnetically tunable Mie resonance-based dielectric metamaterials,” Sci. Rep. 4(1), 7001 (2015). [CrossRef]  

13. R. Baets, A. Z. Subramanian, S. Clemmen, B. Kuyken, P. Bienstman, N. Le Thomas, G. Roelkens, D. Van Thourhout, P. Helin, and S. Severi, “Silicon Photonics: silicon nitride versus silicon-on-insulator,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2016), paper Th3J.1

14. Y. Kivshar, “All-dielectric meta-optics and non-linear nanophotonics,” Natl. Sci. Rev. 5(2), 144–158 (2018). [CrossRef]  

15. Q. Zhao, J. Zhou, F. L. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12(12), 60–69 (2009). [CrossRef]  

16. X. Luo, X. Li, T. Lang, X. Jing, and Z. Hong, “Excitation of high Q toroidal dipole resonance in an all-dielectric metasurface,” Opt. Mater. Express 10(2), 358–368 (2020). [CrossRef]  

17. X. Chen and W. Fan, “Ultrahigh-Q toroidal dipole resonance in all-dielectric metamaterials for terahertz sensing,” Opt. Lett. 44(23), 5876–5879 (2019). [CrossRef]  

18. P. D. Terekhov, A. B. Evlyukhin, D. Redka, V. S. Volkov, A. S. Shalin, and A. Karabchevsky, “Magnetic Octupole Response of Dielectric Quadrumers,” Laser Photonics Rev. 14(4), 1900331 (2020). [CrossRef]  

19. J. F. Algorri, D. C. Zografopoulos, A. Ferraro, B. Garcia-Camara, R. Beccherelli, and J. M. Sanchez-Pena, “Ultrahigh-quality factor resonant dielectric metasurfaces based on hollow nanocuboids,” Opt. Express 27(5), 6320–6330 (2019). [CrossRef]  

20. J. F. Algorri, D. C. Zografopoulos, A. Ferraro, B. Garcia-Camara, R. Vergaz, R. Beccherelli, and J. M. Sanchez-Pena, “Anapole Modes in Hollow Nanocuboid Dielectric Metasurfaces for Refractometric Sensing,” Nanomaterials 9(1), 30 (2019). [CrossRef]  

21. P. A. Jeong, M. D. Goldflam, S. Campione, J. L. Briscoe, P. P. Vabishchevich, J. Nogan, M. B. Sinclair, T. S. Luk, and I. Brener, “High Quality Factor Toroidal Resonances in Dielectric Metasurfaces,” Acs Photonics 7(7), 1699–1707 (2020). [CrossRef]  

22. K. Fan, I. V. Shadrivov, and W. J. Padilla, “Dynamic bound states in the continuum,” Optica 6(2), 169–173 (2019). [CrossRef]  

23. S. Garmon, K. Noba, G. Ordonez, and D. Segal, “Non-Markovian dynamics revealed at a bound state in the continuum,” Phys. Rev. A 99(1), 010102 (2019). [CrossRef]  

24. R. F. Ndangali and S. V. Shabanov, “Electromagnetic bound states in the radiation continuum for periodic double arrays of subwavelength dielectric cylinders,” J. Math. Phys. 51(10), 102901 (2010). [CrossRef]  

25. C. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljacic, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016). [CrossRef]  

26. D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Bound states in the continuum in photonics,” Phys. Rev. Lett. 100(18), 183902 (2008). [CrossRef]  

27. S. Han, L. Q. Cong, Y. K. Srivastava, B. Qiang, M. V. Rybin, A. Kumar, R. Jain, W. X. Lim, V. C. Achanta, S. S. Prabhu, Q. J. Wang, Y. S. Kivshar, and R. Singh, “All-Dielectric Active Terahertz Photonics Driven by Bound States in the Continuum,” Adv. Mater. 31(37), 1901921 (2019). [CrossRef]  

28. S. P. Shipman and S. Venakides, “Resonant transmission near nonrobust periodic slab modes,” Phys. Rev. E 71(2), 026611 (2005). [CrossRef]  

29. E. N. Bulgakov and A. F. Sadreev, “Bound states in the continuum in photonic waveguides inspired by defects,” Phys. Rev. B 78(7), 075105 (2008). [CrossRef]  

30. A. Tittl, A. Leitis, M. Liu, F. Yesilkoy, D.-Y. Choi, D. N. Neshev, Y. S. Kivshar, and H. Altug, “Imaging-based molecular barcoding with pixelated dielectric metasurfaces,” Science 360(6393), 1105–1109 (2018). [CrossRef]  

31. F. Yesilkoy, E. R. Arvelo, Y. Jahani, M. K. Liu, A. Tittl, V. Cevher, Y. Kivshar, and H. Altug, “Ultrasensitive hyperspectral imaging and biodetection enabled by dielectric metasurfaces,” Nat. Photonics 13(6), 390–396 (2019). [CrossRef]  

32. Y. He, G. T. Guo, T. H. Feng, Y. Xu, and A. E. Miroshnichenko, “Toroidal dipole bound states in the continuum,” Phys. Rev. B 98(16), 161112 (2018). [CrossRef]  

33. A. D. F. Edwards, “Silicon (Si),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, 1985).

34. V. A. Fedotov, M. Rose, S. L. Prosvirnin, N. Papasimakis, and N. I. Zheludev, “Sharp trapped-mode resonances in planar metamaterials with a broken structural symmetry,” Phys. Rev. Lett. 99(14), 147401 (2007). [CrossRef]  

35. K. Koshelev, S. Lepeshov, M. K. Liu, A. Bogdanov, and Y. Kivshar, “Asymmetric Metasurfaces with High-Q Resonances Governed by Bound States in the Continuum,” Phys. Rev. Lett. 121(19), 193903 (2018). [CrossRef]  

36. Z. Sadrieva, K. Frizyuk, M. Petrov, Y. Kivshar, and A. Bogdanov, “Multipolar origin of bound states in the continuum,” Phys. Rev. B 100(11), 115303 (2019). [CrossRef]  

37. K. Koshelev, A. Bogdanov, and Y. Kivshar, “Meta-optics and bound states in the continuum,” Sci. Bull. 64(12), 836–842 (2019). [CrossRef]  

38. C. B. Zhou, G. Q. Liu, G. X. Ban, S. Y. Li, Q. Z. Huang, J. S. Xia, Y. Wang, and M. S. Zhan, “Tunable Fano resonator using multilayer graphene in the near-infrared region,” Appl. Phys. Lett. 112(10), 101904 (2018). [CrossRef]  

39. L. Xu, K. Z. Kamali, L. J. Huang, M. Rahmani, A. Smirnov, R. Camacho-Morales, Y. X. Ma, G. Q. Zhang, M. Woolley, D. Neshev, and A. E. Miroshnichenko, “Dynamic Nonlinear Image Tuning through Magnetic Dipole Quasi-BIC Ultrathin Resonators,” Adv. Sci. 6(15), 1802119 (2019). [CrossRef]  

40. S. Li, C. Zhou, T. Liu, and S. Xiao, “Symmetry-protected bound states in the continuum supported by all-dielectric metasurfaces,” Phys. Rev. A 100(6), 063803 (2019). [CrossRef]  

41. V. R. Tuz, V. V. Khardikov, A. S. Kupriianov, K. L. Domina, S. Xu, H. Wang, and H. B. Sun, “High-quality trapped modes in all-dielectric metamaterials,” Opt. Express 26(3), 2905–2916 (2018). [CrossRef]  

42. L. Zhang, Z. G. Dong, Y. M. Wang, Y. J. Liu, S. Zhang, J. K. W. Yang, and C. W. Qiu, “Dynamically configurable hybridization of plasmon modes in nanoring dimer arrays,” Nanoscale 7(28), 12018–12022 (2015). [CrossRef]  

43. J. N. He, J. Q. Wang, P. Ding, C. Z. Fan, L. Arnaut, and E. J. Liang, “Optical Switching Based on Polarization Tunable Plasmon-Induced Transparency in Disk/Rod Hybrid Metasurfaces,” Plasmonics 10(5), 1115–1121 (2015). [CrossRef]  

44. C. B. Zhou, X. Y. Qu, S. Y. Xiao, and M. H. Fan, “Imaging Through a Fano-Resonant Dielectric Metasurface Governed by Quasi-bound States in the Continuum,” Phys. Rev. Appl. 14(4), 044009 (2020). [CrossRef]  

45. P. D. Terekhov, H. K. Shamkhi, E. A. Gurvitz, K. V. Baryshnikova, A. B. Evlyukhin, A. S. Shalin, and A. Karabchevsky, “Broadband forward scattering from dielectric cubic nanoantenna in lossless media,” Opt. Express 27(8), 10924–10935 (2019). [CrossRef]  

46. P. D. Terekhov, A. B. Evlyukhin, A. S. Shalin, and A. Karabchevsky, “Polarization-dependent asymmetric light scattering by silicon nanopyramids and their multipoles resonances,” J. Appl. Phys. 125(17), 173108 (2019). [CrossRef]  

47. Y. H. Zhan, D. Y. Lei, X. F. Li, and S. A. Maier, “Plasmonic Fano resonances in nanohole quadrumers for ultra-sensitive refractive index sensing,” Nanoscale 6(9), 4705–4715 (2014). [CrossRef]  

48. Z. H. Yong, D. Y. Lei, C. H. Lam, and Y. Wang, “Ultrahigh refractive index sensing performance of plasmonic quadrupole resonances in gold nanoparticles,” Nanoscale Res. Lett. 9(1), 187 (2014). [CrossRef]  

49. X. Zhang, X. S. Zhu, and Y. W. Shi, “An Optical Fiber Refractive Index Sensor Based on the Hybrid Mode of Tamm and Surface Plasmon Polaritons,” Sensors 18(7), 2129 (2018). [CrossRef]  

50. O. Buchnev, A. Belosludtsev, V. Reshetnyak, D. R. Evans, and V. A. Fedotov, “Observing and controlling a Tamm plasmon at the interface with a metasurface,” Nanophotonics 9(4), 897–903 (2020). [CrossRef]  

51. Z. A. Zaky, A. M. Ahmed, A. S. Shalaby, and A. H. Aly, “Refractive index gas sensor based on the Tamm state in a one-dimensional photonic crystal: Theoretical optimisation,” Sci. Rep. 10(1), 9736 (2020). [CrossRef]  

52. A. M. Ahmed and A. Mehaney, “Novel design of wide temperature ranges sensor based on Tamm state in a pyroelectric photonic crystal with high sensitivity,” Phys. E 125, 114387 (2021). [CrossRef]  

References

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  13. R. Baets, A. Z. Subramanian, S. Clemmen, B. Kuyken, P. Bienstman, N. Le Thomas, G. Roelkens, D. Van Thourhout, P. Helin, and S. Severi, “Silicon Photonics: silicon nitride versus silicon-on-insulator,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2016), paper Th3J.1
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  18. P. D. Terekhov, A. B. Evlyukhin, D. Redka, V. S. Volkov, A. S. Shalin, and A. Karabchevsky, “Magnetic Octupole Response of Dielectric Quadrumers,” Laser Photonics Rev. 14(4), 1900331 (2020).
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    [Crossref]
  20. J. F. Algorri, D. C. Zografopoulos, A. Ferraro, B. Garcia-Camara, R. Vergaz, R. Beccherelli, and J. M. Sanchez-Pena, “Anapole Modes in Hollow Nanocuboid Dielectric Metasurfaces for Refractometric Sensing,” Nanomaterials 9(1), 30 (2019).
    [Crossref]
  21. P. A. Jeong, M. D. Goldflam, S. Campione, J. L. Briscoe, P. P. Vabishchevich, J. Nogan, M. B. Sinclair, T. S. Luk, and I. Brener, “High Quality Factor Toroidal Resonances in Dielectric Metasurfaces,” Acs Photonics 7(7), 1699–1707 (2020).
    [Crossref]
  22. K. Fan, I. V. Shadrivov, and W. J. Padilla, “Dynamic bound states in the continuum,” Optica 6(2), 169–173 (2019).
    [Crossref]
  23. S. Garmon, K. Noba, G. Ordonez, and D. Segal, “Non-Markovian dynamics revealed at a bound state in the continuum,” Phys. Rev. A 99(1), 010102 (2019).
    [Crossref]
  24. R. F. Ndangali and S. V. Shabanov, “Electromagnetic bound states in the radiation continuum for periodic double arrays of subwavelength dielectric cylinders,” J. Math. Phys. 51(10), 102901 (2010).
    [Crossref]
  25. C. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljacic, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016).
    [Crossref]
  26. D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Bound states in the continuum in photonics,” Phys. Rev. Lett. 100(18), 183902 (2008).
    [Crossref]
  27. S. Han, L. Q. Cong, Y. K. Srivastava, B. Qiang, M. V. Rybin, A. Kumar, R. Jain, W. X. Lim, V. C. Achanta, S. S. Prabhu, Q. J. Wang, Y. S. Kivshar, and R. Singh, “All-Dielectric Active Terahertz Photonics Driven by Bound States in the Continuum,” Adv. Mater. 31(37), 1901921 (2019).
    [Crossref]
  28. S. P. Shipman and S. Venakides, “Resonant transmission near nonrobust periodic slab modes,” Phys. Rev. E 71(2), 026611 (2005).
    [Crossref]
  29. E. N. Bulgakov and A. F. Sadreev, “Bound states in the continuum in photonic waveguides inspired by defects,” Phys. Rev. B 78(7), 075105 (2008).
    [Crossref]
  30. A. Tittl, A. Leitis, M. Liu, F. Yesilkoy, D.-Y. Choi, D. N. Neshev, Y. S. Kivshar, and H. Altug, “Imaging-based molecular barcoding with pixelated dielectric metasurfaces,” Science 360(6393), 1105–1109 (2018).
    [Crossref]
  31. F. Yesilkoy, E. R. Arvelo, Y. Jahani, M. K. Liu, A. Tittl, V. Cevher, Y. Kivshar, and H. Altug, “Ultrasensitive hyperspectral imaging and biodetection enabled by dielectric metasurfaces,” Nat. Photonics 13(6), 390–396 (2019).
    [Crossref]
  32. Y. He, G. T. Guo, T. H. Feng, Y. Xu, and A. E. Miroshnichenko, “Toroidal dipole bound states in the continuum,” Phys. Rev. B 98(16), 161112 (2018).
    [Crossref]
  33. A. D. F. Edwards, “Silicon (Si),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, 1985).
  34. V. A. Fedotov, M. Rose, S. L. Prosvirnin, N. Papasimakis, and N. I. Zheludev, “Sharp trapped-mode resonances in planar metamaterials with a broken structural symmetry,” Phys. Rev. Lett. 99(14), 147401 (2007).
    [Crossref]
  35. K. Koshelev, S. Lepeshov, M. K. Liu, A. Bogdanov, and Y. Kivshar, “Asymmetric Metasurfaces with High-Q Resonances Governed by Bound States in the Continuum,” Phys. Rev. Lett. 121(19), 193903 (2018).
    [Crossref]
  36. Z. Sadrieva, K. Frizyuk, M. Petrov, Y. Kivshar, and A. Bogdanov, “Multipolar origin of bound states in the continuum,” Phys. Rev. B 100(11), 115303 (2019).
    [Crossref]
  37. K. Koshelev, A. Bogdanov, and Y. Kivshar, “Meta-optics and bound states in the continuum,” Sci. Bull. 64(12), 836–842 (2019).
    [Crossref]
  38. C. B. Zhou, G. Q. Liu, G. X. Ban, S. Y. Li, Q. Z. Huang, J. S. Xia, Y. Wang, and M. S. Zhan, “Tunable Fano resonator using multilayer graphene in the near-infrared region,” Appl. Phys. Lett. 112(10), 101904 (2018).
    [Crossref]
  39. L. Xu, K. Z. Kamali, L. J. Huang, M. Rahmani, A. Smirnov, R. Camacho-Morales, Y. X. Ma, G. Q. Zhang, M. Woolley, D. Neshev, and A. E. Miroshnichenko, “Dynamic Nonlinear Image Tuning through Magnetic Dipole Quasi-BIC Ultrathin Resonators,” Adv. Sci. 6(15), 1802119 (2019).
    [Crossref]
  40. S. Li, C. Zhou, T. Liu, and S. Xiao, “Symmetry-protected bound states in the continuum supported by all-dielectric metasurfaces,” Phys. Rev. A 100(6), 063803 (2019).
    [Crossref]
  41. V. R. Tuz, V. V. Khardikov, A. S. Kupriianov, K. L. Domina, S. Xu, H. Wang, and H. B. Sun, “High-quality trapped modes in all-dielectric metamaterials,” Opt. Express 26(3), 2905–2916 (2018).
    [Crossref]
  42. L. Zhang, Z. G. Dong, Y. M. Wang, Y. J. Liu, S. Zhang, J. K. W. Yang, and C. W. Qiu, “Dynamically configurable hybridization of plasmon modes in nanoring dimer arrays,” Nanoscale 7(28), 12018–12022 (2015).
    [Crossref]
  43. J. N. He, J. Q. Wang, P. Ding, C. Z. Fan, L. Arnaut, and E. J. Liang, “Optical Switching Based on Polarization Tunable Plasmon-Induced Transparency in Disk/Rod Hybrid Metasurfaces,” Plasmonics 10(5), 1115–1121 (2015).
    [Crossref]
  44. C. B. Zhou, X. Y. Qu, S. Y. Xiao, and M. H. Fan, “Imaging Through a Fano-Resonant Dielectric Metasurface Governed by Quasi-bound States in the Continuum,” Phys. Rev. Appl. 14(4), 044009 (2020).
    [Crossref]
  45. P. D. Terekhov, H. K. Shamkhi, E. A. Gurvitz, K. V. Baryshnikova, A. B. Evlyukhin, A. S. Shalin, and A. Karabchevsky, “Broadband forward scattering from dielectric cubic nanoantenna in lossless media,” Opt. Express 27(8), 10924–10935 (2019).
    [Crossref]
  46. P. D. Terekhov, A. B. Evlyukhin, A. S. Shalin, and A. Karabchevsky, “Polarization-dependent asymmetric light scattering by silicon nanopyramids and their multipoles resonances,” J. Appl. Phys. 125(17), 173108 (2019).
    [Crossref]
  47. Y. H. Zhan, D. Y. Lei, X. F. Li, and S. A. Maier, “Plasmonic Fano resonances in nanohole quadrumers for ultra-sensitive refractive index sensing,” Nanoscale 6(9), 4705–4715 (2014).
    [Crossref]
  48. Z. H. Yong, D. Y. Lei, C. H. Lam, and Y. Wang, “Ultrahigh refractive index sensing performance of plasmonic quadrupole resonances in gold nanoparticles,” Nanoscale Res. Lett. 9(1), 187 (2014).
    [Crossref]
  49. X. Zhang, X. S. Zhu, and Y. W. Shi, “An Optical Fiber Refractive Index Sensor Based on the Hybrid Mode of Tamm and Surface Plasmon Polaritons,” Sensors 18(7), 2129 (2018).
    [Crossref]
  50. O. Buchnev, A. Belosludtsev, V. Reshetnyak, D. R. Evans, and V. A. Fedotov, “Observing and controlling a Tamm plasmon at the interface with a metasurface,” Nanophotonics 9(4), 897–903 (2020).
    [Crossref]
  51. Z. A. Zaky, A. M. Ahmed, A. S. Shalaby, and A. H. Aly, “Refractive index gas sensor based on the Tamm state in a one-dimensional photonic crystal: Theoretical optimisation,” Sci. Rep. 10(1), 9736 (2020).
    [Crossref]
  52. A. M. Ahmed and A. Mehaney, “Novel design of wide temperature ranges sensor based on Tamm state in a pyroelectric photonic crystal with high sensitivity,” Phys. E 125, 114387 (2021).
    [Crossref]

2021 (1)

A. M. Ahmed and A. Mehaney, “Novel design of wide temperature ranges sensor based on Tamm state in a pyroelectric photonic crystal with high sensitivity,” Phys. E 125, 114387 (2021).
[Crossref]

2020 (7)

O. Buchnev, A. Belosludtsev, V. Reshetnyak, D. R. Evans, and V. A. Fedotov, “Observing and controlling a Tamm plasmon at the interface with a metasurface,” Nanophotonics 9(4), 897–903 (2020).
[Crossref]

Z. A. Zaky, A. M. Ahmed, A. S. Shalaby, and A. H. Aly, “Refractive index gas sensor based on the Tamm state in a one-dimensional photonic crystal: Theoretical optimisation,” Sci. Rep. 10(1), 9736 (2020).
[Crossref]

C. B. Zhou, X. Y. Qu, S. Y. Xiao, and M. H. Fan, “Imaging Through a Fano-Resonant Dielectric Metasurface Governed by Quasi-bound States in the Continuum,” Phys. Rev. Appl. 14(4), 044009 (2020).
[Crossref]

X. Chen, W. Fan, and H. Yan, “Toroidal dipole bound states in the continuum metasurfaces for terahertz nanofilm sensing,” Opt. Express 28(11), 17102–17112 (2020).
[Crossref]

X. Luo, X. Li, T. Lang, X. Jing, and Z. Hong, “Excitation of high Q toroidal dipole resonance in an all-dielectric metasurface,” Opt. Mater. Express 10(2), 358–368 (2020).
[Crossref]

P. D. Terekhov, A. B. Evlyukhin, D. Redka, V. S. Volkov, A. S. Shalin, and A. Karabchevsky, “Magnetic Octupole Response of Dielectric Quadrumers,” Laser Photonics Rev. 14(4), 1900331 (2020).
[Crossref]

P. A. Jeong, M. D. Goldflam, S. Campione, J. L. Briscoe, P. P. Vabishchevich, J. Nogan, M. B. Sinclair, T. S. Luk, and I. Brener, “High Quality Factor Toroidal Resonances in Dielectric Metasurfaces,” Acs Photonics 7(7), 1699–1707 (2020).
[Crossref]

2019 (13)

K. Fan, I. V. Shadrivov, and W. J. Padilla, “Dynamic bound states in the continuum,” Optica 6(2), 169–173 (2019).
[Crossref]

S. Garmon, K. Noba, G. Ordonez, and D. Segal, “Non-Markovian dynamics revealed at a bound state in the continuum,” Phys. Rev. A 99(1), 010102 (2019).
[Crossref]

S. Han, L. Q. Cong, Y. K. Srivastava, B. Qiang, M. V. Rybin, A. Kumar, R. Jain, W. X. Lim, V. C. Achanta, S. S. Prabhu, Q. J. Wang, Y. S. Kivshar, and R. Singh, “All-Dielectric Active Terahertz Photonics Driven by Bound States in the Continuum,” Adv. Mater. 31(37), 1901921 (2019).
[Crossref]

F. Yesilkoy, E. R. Arvelo, Y. Jahani, M. K. Liu, A. Tittl, V. Cevher, Y. Kivshar, and H. Altug, “Ultrasensitive hyperspectral imaging and biodetection enabled by dielectric metasurfaces,” Nat. Photonics 13(6), 390–396 (2019).
[Crossref]

Z. Sadrieva, K. Frizyuk, M. Petrov, Y. Kivshar, and A. Bogdanov, “Multipolar origin of bound states in the continuum,” Phys. Rev. B 100(11), 115303 (2019).
[Crossref]

K. Koshelev, A. Bogdanov, and Y. Kivshar, “Meta-optics and bound states in the continuum,” Sci. Bull. 64(12), 836–842 (2019).
[Crossref]

J. F. Algorri, D. C. Zografopoulos, A. Ferraro, B. Garcia-Camara, R. Beccherelli, and J. M. Sanchez-Pena, “Ultrahigh-quality factor resonant dielectric metasurfaces based on hollow nanocuboids,” Opt. Express 27(5), 6320–6330 (2019).
[Crossref]

J. F. Algorri, D. C. Zografopoulos, A. Ferraro, B. Garcia-Camara, R. Vergaz, R. Beccherelli, and J. M. Sanchez-Pena, “Anapole Modes in Hollow Nanocuboid Dielectric Metasurfaces for Refractometric Sensing,” Nanomaterials 9(1), 30 (2019).
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X. Chen and W. Fan, “Ultrahigh-Q toroidal dipole resonance in all-dielectric metamaterials for terahertz sensing,” Opt. Lett. 44(23), 5876–5879 (2019).
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P. D. Terekhov, H. K. Shamkhi, E. A. Gurvitz, K. V. Baryshnikova, A. B. Evlyukhin, A. S. Shalin, and A. Karabchevsky, “Broadband forward scattering from dielectric cubic nanoantenna in lossless media,” Opt. Express 27(8), 10924–10935 (2019).
[Crossref]

P. D. Terekhov, A. B. Evlyukhin, A. S. Shalin, and A. Karabchevsky, “Polarization-dependent asymmetric light scattering by silicon nanopyramids and their multipoles resonances,” J. Appl. Phys. 125(17), 173108 (2019).
[Crossref]

L. Xu, K. Z. Kamali, L. J. Huang, M. Rahmani, A. Smirnov, R. Camacho-Morales, Y. X. Ma, G. Q. Zhang, M. Woolley, D. Neshev, and A. E. Miroshnichenko, “Dynamic Nonlinear Image Tuning through Magnetic Dipole Quasi-BIC Ultrathin Resonators,” Adv. Sci. 6(15), 1802119 (2019).
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S. Li, C. Zhou, T. Liu, and S. Xiao, “Symmetry-protected bound states in the continuum supported by all-dielectric metasurfaces,” Phys. Rev. A 100(6), 063803 (2019).
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2018 (10)

V. R. Tuz, V. V. Khardikov, A. S. Kupriianov, K. L. Domina, S. Xu, H. Wang, and H. B. Sun, “High-quality trapped modes in all-dielectric metamaterials,” Opt. Express 26(3), 2905–2916 (2018).
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X. Zhang, X. S. Zhu, and Y. W. Shi, “An Optical Fiber Refractive Index Sensor Based on the Hybrid Mode of Tamm and Surface Plasmon Polaritons,” Sensors 18(7), 2129 (2018).
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Y. Kivshar, “All-dielectric meta-optics and non-linear nanophotonics,” Natl. Sci. Rev. 5(2), 144–158 (2018).
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M. Gupta, Y. K. Srivastava, and R. Singh, “A Toroidal Metamaterial Switch,” Adv. Mater. 30(4), 1704845 (2018).
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S. Q. Li and K. B. Crozier, “Origin of the anapole condition as revealed by a simple expansion beyond the toroidal multipole,” Phys. Rev. B 97(24), 245423 (2018).
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V. R. Tuz, V. V. Khardikov, and Y. S. Kivshar, “All-Dielectric Resonant Metasurfaces with a Strong Toroidal Response,” ACS Photonics 5(5), 1871–1876 (2018).
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C. B. Zhou, G. Q. Liu, G. X. Ban, S. Y. Li, Q. Z. Huang, J. S. Xia, Y. Wang, and M. S. Zhan, “Tunable Fano resonator using multilayer graphene in the near-infrared region,” Appl. Phys. Lett. 112(10), 101904 (2018).
[Crossref]

A. Tittl, A. Leitis, M. Liu, F. Yesilkoy, D.-Y. Choi, D. N. Neshev, Y. S. Kivshar, and H. Altug, “Imaging-based molecular barcoding with pixelated dielectric metasurfaces,” Science 360(6393), 1105–1109 (2018).
[Crossref]

K. Koshelev, S. Lepeshov, M. K. Liu, A. Bogdanov, and Y. Kivshar, “Asymmetric Metasurfaces with High-Q Resonances Governed by Bound States in the Continuum,” Phys. Rev. Lett. 121(19), 193903 (2018).
[Crossref]

Y. He, G. T. Guo, T. H. Feng, Y. Xu, and A. E. Miroshnichenko, “Toroidal dipole bound states in the continuum,” Phys. Rev. B 98(16), 161112 (2018).
[Crossref]

2017 (1)

A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kanté, “Lasing action from photonic bound states in continuum,” Nature 541(7636), 196–199 (2017).
[Crossref]

2016 (2)

N. Papasimakis, V. A. Fedotov, V. Savinov, T. A. Raybould, and N. I. Zheludev, “Electromagnetic toroidal excitations in matter and free space,” Nat. Mater. 15(3), 263–271 (2016).
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C. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljacic, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016).
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2015 (4)

K. Bi, Y. Guo, X. Liu, Q. Zhao, J. Xiao, M. Lei, and J. Zhou, “Magnetically tunable Mie resonance-based dielectric metamaterials,” Sci. Rep. 4(1), 7001 (2015).
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W. Y. Zhao, H. Jiang, B. Y. Liu, Y. Y. Jiang, C. C. Tang, and J. J. Li, “Fano resonance based optical modulator reaching 85% modulation depth,” Appl. Phys. Lett. 107(17), 171109 (2015).
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L. Zhang, Z. G. Dong, Y. M. Wang, Y. J. Liu, S. Zhang, J. K. W. Yang, and C. W. Qiu, “Dynamically configurable hybridization of plasmon modes in nanoring dimer arrays,” Nanoscale 7(28), 12018–12022 (2015).
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J. N. He, J. Q. Wang, P. Ding, C. Z. Fan, L. Arnaut, and E. J. Liang, “Optical Switching Based on Polarization Tunable Plasmon-Induced Transparency in Disk/Rod Hybrid Metasurfaces,” Plasmonics 10(5), 1115–1121 (2015).
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2014 (3)

Y. H. Zhan, D. Y. Lei, X. F. Li, and S. A. Maier, “Plasmonic Fano resonances in nanohole quadrumers for ultra-sensitive refractive index sensing,” Nanoscale 6(9), 4705–4715 (2014).
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Z. H. Yong, D. Y. Lei, C. H. Lam, and Y. Wang, “Ultrahigh refractive index sensing performance of plasmonic quadrupole resonances in gold nanoparticles,” Nanoscale Res. Lett. 9(1), 187 (2014).
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L. Y. Guo, M. H. Li, X. J. Huang, and H. L. Yang, “Electric toroidal metamaterial for resonant transparency and circular cross-polarization conversion,” Appl. Phys. Lett. 105(3), 033507 (2014).
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2013 (1)

Q. W. Ye, L. Y. Guo, M. H. Li, Y. Liu, B. X. Xiao, and H. L. Yang, “The magnetic toroidal dipole in steric metamaterial for permittivity sensor application,” Phys. Scr. 88(5), 055002 (2013).
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2010 (2)

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal Dipolar Response in a Metamaterial,” Science 330(6010), 1510–1512 (2010).
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R. F. Ndangali and S. V. Shabanov, “Electromagnetic bound states in the radiation continuum for periodic double arrays of subwavelength dielectric cylinders,” J. Math. Phys. 51(10), 102901 (2010).
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2009 (1)

Q. Zhao, J. Zhou, F. L. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12(12), 60–69 (2009).
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2008 (2)

D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Bound states in the continuum in photonics,” Phys. Rev. Lett. 100(18), 183902 (2008).
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E. N. Bulgakov and A. F. Sadreev, “Bound states in the continuum in photonic waveguides inspired by defects,” Phys. Rev. B 78(7), 075105 (2008).
[Crossref]

2007 (1)

V. A. Fedotov, M. Rose, S. L. Prosvirnin, N. Papasimakis, and N. I. Zheludev, “Sharp trapped-mode resonances in planar metamaterials with a broken structural symmetry,” Phys. Rev. Lett. 99(14), 147401 (2007).
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2005 (1)

S. P. Shipman and S. Venakides, “Resonant transmission near nonrobust periodic slab modes,” Phys. Rev. E 71(2), 026611 (2005).
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1958 (1)

I. B. Zel’Dovich, “Electromagnetic Interaction with Parity Violation,” J. Exp. Theor. Phys. 6(6), 1184 (1958).

Achanta, V. C.

S. Han, L. Q. Cong, Y. K. Srivastava, B. Qiang, M. V. Rybin, A. Kumar, R. Jain, W. X. Lim, V. C. Achanta, S. S. Prabhu, Q. J. Wang, Y. S. Kivshar, and R. Singh, “All-Dielectric Active Terahertz Photonics Driven by Bound States in the Continuum,” Adv. Mater. 31(37), 1901921 (2019).
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Ahmed, A. M.

A. M. Ahmed and A. Mehaney, “Novel design of wide temperature ranges sensor based on Tamm state in a pyroelectric photonic crystal with high sensitivity,” Phys. E 125, 114387 (2021).
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Z. A. Zaky, A. M. Ahmed, A. S. Shalaby, and A. H. Aly, “Refractive index gas sensor based on the Tamm state in a one-dimensional photonic crystal: Theoretical optimisation,” Sci. Rep. 10(1), 9736 (2020).
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Algorri, J. F.

J. F. Algorri, D. C. Zografopoulos, A. Ferraro, B. Garcia-Camara, R. Beccherelli, and J. M. Sanchez-Pena, “Ultrahigh-quality factor resonant dielectric metasurfaces based on hollow nanocuboids,” Opt. Express 27(5), 6320–6330 (2019).
[Crossref]

J. F. Algorri, D. C. Zografopoulos, A. Ferraro, B. Garcia-Camara, R. Vergaz, R. Beccherelli, and J. M. Sanchez-Pena, “Anapole Modes in Hollow Nanocuboid Dielectric Metasurfaces for Refractometric Sensing,” Nanomaterials 9(1), 30 (2019).
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Altug, H.

F. Yesilkoy, E. R. Arvelo, Y. Jahani, M. K. Liu, A. Tittl, V. Cevher, Y. Kivshar, and H. Altug, “Ultrasensitive hyperspectral imaging and biodetection enabled by dielectric metasurfaces,” Nat. Photonics 13(6), 390–396 (2019).
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A. Tittl, A. Leitis, M. Liu, F. Yesilkoy, D.-Y. Choi, D. N. Neshev, Y. S. Kivshar, and H. Altug, “Imaging-based molecular barcoding with pixelated dielectric metasurfaces,” Science 360(6393), 1105–1109 (2018).
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Aly, A. H.

Z. A. Zaky, A. M. Ahmed, A. S. Shalaby, and A. H. Aly, “Refractive index gas sensor based on the Tamm state in a one-dimensional photonic crystal: Theoretical optimisation,” Sci. Rep. 10(1), 9736 (2020).
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Arnaut, L.

J. N. He, J. Q. Wang, P. Ding, C. Z. Fan, L. Arnaut, and E. J. Liang, “Optical Switching Based on Polarization Tunable Plasmon-Induced Transparency in Disk/Rod Hybrid Metasurfaces,” Plasmonics 10(5), 1115–1121 (2015).
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Arvelo, E. R.

F. Yesilkoy, E. R. Arvelo, Y. Jahani, M. K. Liu, A. Tittl, V. Cevher, Y. Kivshar, and H. Altug, “Ultrasensitive hyperspectral imaging and biodetection enabled by dielectric metasurfaces,” Nat. Photonics 13(6), 390–396 (2019).
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Baets, R.

R. Baets, A. Z. Subramanian, S. Clemmen, B. Kuyken, P. Bienstman, N. Le Thomas, G. Roelkens, D. Van Thourhout, P. Helin, and S. Severi, “Silicon Photonics: silicon nitride versus silicon-on-insulator,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2016), paper Th3J.1

Bahari, B.

A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kanté, “Lasing action from photonic bound states in continuum,” Nature 541(7636), 196–199 (2017).
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Ban, G. X.

C. B. Zhou, G. Q. Liu, G. X. Ban, S. Y. Li, Q. Z. Huang, J. S. Xia, Y. Wang, and M. S. Zhan, “Tunable Fano resonator using multilayer graphene in the near-infrared region,” Appl. Phys. Lett. 112(10), 101904 (2018).
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Baryshnikova, K. V.

Beccherelli, R.

J. F. Algorri, D. C. Zografopoulos, A. Ferraro, B. Garcia-Camara, R. Beccherelli, and J. M. Sanchez-Pena, “Ultrahigh-quality factor resonant dielectric metasurfaces based on hollow nanocuboids,” Opt. Express 27(5), 6320–6330 (2019).
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J. F. Algorri, D. C. Zografopoulos, A. Ferraro, B. Garcia-Camara, R. Vergaz, R. Beccherelli, and J. M. Sanchez-Pena, “Anapole Modes in Hollow Nanocuboid Dielectric Metasurfaces for Refractometric Sensing,” Nanomaterials 9(1), 30 (2019).
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O. Buchnev, A. Belosludtsev, V. Reshetnyak, D. R. Evans, and V. A. Fedotov, “Observing and controlling a Tamm plasmon at the interface with a metasurface,” Nanophotonics 9(4), 897–903 (2020).
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Bi, K.

K. Bi, Y. Guo, X. Liu, Q. Zhao, J. Xiao, M. Lei, and J. Zhou, “Magnetically tunable Mie resonance-based dielectric metamaterials,” Sci. Rep. 4(1), 7001 (2015).
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Bienstman, P.

R. Baets, A. Z. Subramanian, S. Clemmen, B. Kuyken, P. Bienstman, N. Le Thomas, G. Roelkens, D. Van Thourhout, P. Helin, and S. Severi, “Silicon Photonics: silicon nitride versus silicon-on-insulator,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2016), paper Th3J.1

Bogdanov, A.

Z. Sadrieva, K. Frizyuk, M. Petrov, Y. Kivshar, and A. Bogdanov, “Multipolar origin of bound states in the continuum,” Phys. Rev. B 100(11), 115303 (2019).
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K. Koshelev, A. Bogdanov, and Y. Kivshar, “Meta-optics and bound states in the continuum,” Sci. Bull. 64(12), 836–842 (2019).
[Crossref]

K. Koshelev, S. Lepeshov, M. K. Liu, A. Bogdanov, and Y. Kivshar, “Asymmetric Metasurfaces with High-Q Resonances Governed by Bound States in the Continuum,” Phys. Rev. Lett. 121(19), 193903 (2018).
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Borisov, A. G.

D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Bound states in the continuum in photonics,” Phys. Rev. Lett. 100(18), 183902 (2008).
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Brener, I.

P. A. Jeong, M. D. Goldflam, S. Campione, J. L. Briscoe, P. P. Vabishchevich, J. Nogan, M. B. Sinclair, T. S. Luk, and I. Brener, “High Quality Factor Toroidal Resonances in Dielectric Metasurfaces,” Acs Photonics 7(7), 1699–1707 (2020).
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P. A. Jeong, M. D. Goldflam, S. Campione, J. L. Briscoe, P. P. Vabishchevich, J. Nogan, M. B. Sinclair, T. S. Luk, and I. Brener, “High Quality Factor Toroidal Resonances in Dielectric Metasurfaces,” Acs Photonics 7(7), 1699–1707 (2020).
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Buchnev, O.

O. Buchnev, A. Belosludtsev, V. Reshetnyak, D. R. Evans, and V. A. Fedotov, “Observing and controlling a Tamm plasmon at the interface with a metasurface,” Nanophotonics 9(4), 897–903 (2020).
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Bulgakov, E. N.

E. N. Bulgakov and A. F. Sadreev, “Bound states in the continuum in photonic waveguides inspired by defects,” Phys. Rev. B 78(7), 075105 (2008).
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Camacho-Morales, R.

L. Xu, K. Z. Kamali, L. J. Huang, M. Rahmani, A. Smirnov, R. Camacho-Morales, Y. X. Ma, G. Q. Zhang, M. Woolley, D. Neshev, and A. E. Miroshnichenko, “Dynamic Nonlinear Image Tuning through Magnetic Dipole Quasi-BIC Ultrathin Resonators,” Adv. Sci. 6(15), 1802119 (2019).
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Campione, S.

P. A. Jeong, M. D. Goldflam, S. Campione, J. L. Briscoe, P. P. Vabishchevich, J. Nogan, M. B. Sinclair, T. S. Luk, and I. Brener, “High Quality Factor Toroidal Resonances in Dielectric Metasurfaces,” Acs Photonics 7(7), 1699–1707 (2020).
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Cevher, V.

F. Yesilkoy, E. R. Arvelo, Y. Jahani, M. K. Liu, A. Tittl, V. Cevher, Y. Kivshar, and H. Altug, “Ultrasensitive hyperspectral imaging and biodetection enabled by dielectric metasurfaces,” Nat. Photonics 13(6), 390–396 (2019).
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Chen, X.

Choi, D.-Y.

A. Tittl, A. Leitis, M. Liu, F. Yesilkoy, D.-Y. Choi, D. N. Neshev, Y. S. Kivshar, and H. Altug, “Imaging-based molecular barcoding with pixelated dielectric metasurfaces,” Science 360(6393), 1105–1109 (2018).
[Crossref]

Clemmen, S.

R. Baets, A. Z. Subramanian, S. Clemmen, B. Kuyken, P. Bienstman, N. Le Thomas, G. Roelkens, D. Van Thourhout, P. Helin, and S. Severi, “Silicon Photonics: silicon nitride versus silicon-on-insulator,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2016), paper Th3J.1

Cong, L. Q.

S. Han, L. Q. Cong, Y. K. Srivastava, B. Qiang, M. V. Rybin, A. Kumar, R. Jain, W. X. Lim, V. C. Achanta, S. S. Prabhu, Q. J. Wang, Y. S. Kivshar, and R. Singh, “All-Dielectric Active Terahertz Photonics Driven by Bound States in the Continuum,” Adv. Mater. 31(37), 1901921 (2019).
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Crozier, K. B.

S. Q. Li and K. B. Crozier, “Origin of the anapole condition as revealed by a simple expansion beyond the toroidal multipole,” Phys. Rev. B 97(24), 245423 (2018).
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Ding, P.

J. N. He, J. Q. Wang, P. Ding, C. Z. Fan, L. Arnaut, and E. J. Liang, “Optical Switching Based on Polarization Tunable Plasmon-Induced Transparency in Disk/Rod Hybrid Metasurfaces,” Plasmonics 10(5), 1115–1121 (2015).
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Domina, K. L.

Dong, Z. G.

L. Zhang, Z. G. Dong, Y. M. Wang, Y. J. Liu, S. Zhang, J. K. W. Yang, and C. W. Qiu, “Dynamically configurable hybridization of plasmon modes in nanoring dimer arrays,” Nanoscale 7(28), 12018–12022 (2015).
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Evans, D. R.

O. Buchnev, A. Belosludtsev, V. Reshetnyak, D. R. Evans, and V. A. Fedotov, “Observing and controlling a Tamm plasmon at the interface with a metasurface,” Nanophotonics 9(4), 897–903 (2020).
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Evlyukhin, A. B.

P. D. Terekhov, A. B. Evlyukhin, D. Redka, V. S. Volkov, A. S. Shalin, and A. Karabchevsky, “Magnetic Octupole Response of Dielectric Quadrumers,” Laser Photonics Rev. 14(4), 1900331 (2020).
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P. D. Terekhov, H. K. Shamkhi, E. A. Gurvitz, K. V. Baryshnikova, A. B. Evlyukhin, A. S. Shalin, and A. Karabchevsky, “Broadband forward scattering from dielectric cubic nanoantenna in lossless media,” Opt. Express 27(8), 10924–10935 (2019).
[Crossref]

P. D. Terekhov, A. B. Evlyukhin, A. S. Shalin, and A. Karabchevsky, “Polarization-dependent asymmetric light scattering by silicon nanopyramids and their multipoles resonances,” J. Appl. Phys. 125(17), 173108 (2019).
[Crossref]

Fainman, Y.

A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kanté, “Lasing action from photonic bound states in continuum,” Nature 541(7636), 196–199 (2017).
[Crossref]

Fan, C. Z.

J. N. He, J. Q. Wang, P. Ding, C. Z. Fan, L. Arnaut, and E. J. Liang, “Optical Switching Based on Polarization Tunable Plasmon-Induced Transparency in Disk/Rod Hybrid Metasurfaces,” Plasmonics 10(5), 1115–1121 (2015).
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Fan, K.

Fan, M. H.

C. B. Zhou, X. Y. Qu, S. Y. Xiao, and M. H. Fan, “Imaging Through a Fano-Resonant Dielectric Metasurface Governed by Quasi-bound States in the Continuum,” Phys. Rev. Appl. 14(4), 044009 (2020).
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Fedotov, V. A.

O. Buchnev, A. Belosludtsev, V. Reshetnyak, D. R. Evans, and V. A. Fedotov, “Observing and controlling a Tamm plasmon at the interface with a metasurface,” Nanophotonics 9(4), 897–903 (2020).
[Crossref]

N. Papasimakis, V. A. Fedotov, V. Savinov, T. A. Raybould, and N. I. Zheludev, “Electromagnetic toroidal excitations in matter and free space,” Nat. Mater. 15(3), 263–271 (2016).
[Crossref]

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal Dipolar Response in a Metamaterial,” Science 330(6010), 1510–1512 (2010).
[Crossref]

V. A. Fedotov, M. Rose, S. L. Prosvirnin, N. Papasimakis, and N. I. Zheludev, “Sharp trapped-mode resonances in planar metamaterials with a broken structural symmetry,” Phys. Rev. Lett. 99(14), 147401 (2007).
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Feng, T. H.

Y. He, G. T. Guo, T. H. Feng, Y. Xu, and A. E. Miroshnichenko, “Toroidal dipole bound states in the continuum,” Phys. Rev. B 98(16), 161112 (2018).
[Crossref]

Ferraro, A.

J. F. Algorri, D. C. Zografopoulos, A. Ferraro, B. Garcia-Camara, R. Beccherelli, and J. M. Sanchez-Pena, “Ultrahigh-quality factor resonant dielectric metasurfaces based on hollow nanocuboids,” Opt. Express 27(5), 6320–6330 (2019).
[Crossref]

J. F. Algorri, D. C. Zografopoulos, A. Ferraro, B. Garcia-Camara, R. Vergaz, R. Beccherelli, and J. M. Sanchez-Pena, “Anapole Modes in Hollow Nanocuboid Dielectric Metasurfaces for Refractometric Sensing,” Nanomaterials 9(1), 30 (2019).
[Crossref]

Frizyuk, K.

Z. Sadrieva, K. Frizyuk, M. Petrov, Y. Kivshar, and A. Bogdanov, “Multipolar origin of bound states in the continuum,” Phys. Rev. B 100(11), 115303 (2019).
[Crossref]

Garcia-Camara, B.

J. F. Algorri, D. C. Zografopoulos, A. Ferraro, B. Garcia-Camara, R. Vergaz, R. Beccherelli, and J. M. Sanchez-Pena, “Anapole Modes in Hollow Nanocuboid Dielectric Metasurfaces for Refractometric Sensing,” Nanomaterials 9(1), 30 (2019).
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J. F. Algorri, D. C. Zografopoulos, A. Ferraro, B. Garcia-Camara, R. Beccherelli, and J. M. Sanchez-Pena, “Ultrahigh-quality factor resonant dielectric metasurfaces based on hollow nanocuboids,” Opt. Express 27(5), 6320–6330 (2019).
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S. Garmon, K. Noba, G. Ordonez, and D. Segal, “Non-Markovian dynamics revealed at a bound state in the continuum,” Phys. Rev. A 99(1), 010102 (2019).
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P. A. Jeong, M. D. Goldflam, S. Campione, J. L. Briscoe, P. P. Vabishchevich, J. Nogan, M. B. Sinclair, T. S. Luk, and I. Brener, “High Quality Factor Toroidal Resonances in Dielectric Metasurfaces,” Acs Photonics 7(7), 1699–1707 (2020).
[Crossref]

Gu, Q.

A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kanté, “Lasing action from photonic bound states in continuum,” Nature 541(7636), 196–199 (2017).
[Crossref]

Guo, G. T.

Y. He, G. T. Guo, T. H. Feng, Y. Xu, and A. E. Miroshnichenko, “Toroidal dipole bound states in the continuum,” Phys. Rev. B 98(16), 161112 (2018).
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S. Li, C. Zhou, T. Liu, and S. Xiao, “Symmetry-protected bound states in the continuum supported by all-dielectric metasurfaces,” Phys. Rev. A 100(6), 063803 (2019).
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C. B. Zhou, X. Y. Qu, S. Y. Xiao, and M. H. Fan, “Imaging Through a Fano-Resonant Dielectric Metasurface Governed by Quasi-bound States in the Continuum,” Phys. Rev. Appl. 14(4), 044009 (2020).
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X. Zhang, X. S. Zhu, and Y. W. Shi, “An Optical Fiber Refractive Index Sensor Based on the Hybrid Mode of Tamm and Surface Plasmon Polaritons,” Sensors 18(7), 2129 (2018).
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J. F. Algorri, D. C. Zografopoulos, A. Ferraro, B. Garcia-Camara, R. Vergaz, R. Beccherelli, and J. M. Sanchez-Pena, “Anapole Modes in Hollow Nanocuboid Dielectric Metasurfaces for Refractometric Sensing,” Nanomaterials 9(1), 30 (2019).
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J. F. Algorri, D. C. Zografopoulos, A. Ferraro, B. Garcia-Camara, R. Beccherelli, and J. M. Sanchez-Pena, “Ultrahigh-quality factor resonant dielectric metasurfaces based on hollow nanocuboids,” Opt. Express 27(5), 6320–6330 (2019).
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ACS Photonics (2)

V. R. Tuz, V. V. Khardikov, and Y. S. Kivshar, “All-Dielectric Resonant Metasurfaces with a Strong Toroidal Response,” ACS Photonics 5(5), 1871–1876 (2018).
[Crossref]

P. A. Jeong, M. D. Goldflam, S. Campione, J. L. Briscoe, P. P. Vabishchevich, J. Nogan, M. B. Sinclair, T. S. Luk, and I. Brener, “High Quality Factor Toroidal Resonances in Dielectric Metasurfaces,” Acs Photonics 7(7), 1699–1707 (2020).
[Crossref]

Adv. Mater. (2)

S. Han, L. Q. Cong, Y. K. Srivastava, B. Qiang, M. V. Rybin, A. Kumar, R. Jain, W. X. Lim, V. C. Achanta, S. S. Prabhu, Q. J. Wang, Y. S. Kivshar, and R. Singh, “All-Dielectric Active Terahertz Photonics Driven by Bound States in the Continuum,” Adv. Mater. 31(37), 1901921 (2019).
[Crossref]

M. Gupta, Y. K. Srivastava, and R. Singh, “A Toroidal Metamaterial Switch,” Adv. Mater. 30(4), 1704845 (2018).
[Crossref]

Adv. Sci. (1)

L. Xu, K. Z. Kamali, L. J. Huang, M. Rahmani, A. Smirnov, R. Camacho-Morales, Y. X. Ma, G. Q. Zhang, M. Woolley, D. Neshev, and A. E. Miroshnichenko, “Dynamic Nonlinear Image Tuning through Magnetic Dipole Quasi-BIC Ultrathin Resonators,” Adv. Sci. 6(15), 1802119 (2019).
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Figures (8)

Fig. 1.
Fig. 1. (a) Schematic diagram of the structure of a partial metasurface composed of unit cells. (b) The calculated transmittance spectra when d0 changes from 60 nm to 110 nm. Four resonance responses are marked by mode I, II, III, and IV, respectively. (c)-(d) Fano fitting of mode II and mode IV at d0= 60 nm. The solid curve are the results of Fano formula fit, and the dashed curve are the simulation results.
Fig. 2.
Fig. 2. Process for fabricating the proposed nanostructured metasurfaces.
Fig. 3.
Fig. 3. Variation curves of radiative Q-factor of Mode II and resonance wavelength position with side length d0 of nanohole when the side length l0 of the Si metasurface are 510 nm (a) and 540 nm (b), respectively.
Fig. 4.
Fig. 4. (a) Plot of Q-factor as a function of the asymmetric parameter α (log-log scale), where the orange points are obtained by the FDTD method, and the black line is fitted to demonstrate the inverse quadratic dependence of α. (b) The schematic diagram of ΔS and S. (c) Transmittance spectra of the metasurface with different d0. The red dashed curve for d0= d1 = 80 nm, the blue solid curve for d0= 130 nm, d1 = 60 nm. (d) The multipole decomposition method obtains five main contributions of multipole excitations.
Fig. 5.
Fig. 5. The upper shows normalized field patterns (|H/H0| and |E/E0|) of unit cells in the x-y plane near four wavelengths, including Ez (λ1) and Hz (λ2, λ3 and λ4). the lower shows a schematic of the corresponding electromagnetic sources. Arrows represent the instantaneous directions of the magnetic (yellow arrows) and electric (black arrows) field distribution.
Fig. 6.
Fig. 6. Calculated normalized field distributions (|E/E0|) of unit cells at λ2 and λ3. (a) The field patterns show the Ex in yz1 and yz2 planes. (b) The field pattern shows the electric field Ey in the x-z plane. The black arrows demonstrate the instantaneous flow direction of the magnetic field.
Fig. 7.
Fig. 7. (a) Transmittance spectra of modes I, III in different polarization angles at d0 = d1 = 80 nm, the black dashed curve shows the transmission curve. The transmittance magnitude and position are identical. (b)(c) Transmittance for modes II and IV in different polarization angles at d0 = 130nm, d1 = 80nm. The polarization angle θ directions are shown in the insets of (a) and (c).
Fig. 8.
Fig. 8. (a) Transmittance spectra of the four resonance modes when the metasurface is covered by a liquid with different refractive indices. (b) Wavelength shift of the four resonance modes relative to the refractive index.

Equations (2)

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$${T_{\textrm{Fano}}}(\omega ) = {\left|{{a_1} + j{a_2} + \frac{b}{{\omega - {\omega_0} + j\gamma }}} \right|^2},$$
$$Q = \frac{A}{{2{k_0}}}{|{{p_0}} |^{ - 2}}{(\alpha )^{ - 2}}| ,$$

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