1. Introduction

Bound states in the continuum (BICs) in optics are a localized state that coexists with a radiation continuum but completely decouples from the continuum. Therefore, an ideal BIC has a radiative lifetime that diverges to infinity and cannot be inaccessible from external excitations that belong to a same continuum. Nevertheless, in practice, owing to finite size samples, material absorption, and other external perturbations, the state can couple with the external radiation and collapse to a leaky resonance with a finite lifetime––a regime referred to as quasi-BIC [1,2]. Now, quasi-BICs have been utilized to gain very high-Q resonances that possess strong electromagnetic-field enhancement. Such resonances are demonstrated to hold vast potential for a myriad of applications in both linear and nonlinear optics, including lasing [3], sensing [4,5], filters [6], enhanced harmonic generation [7,8], and many others. Thus, it is of paramount importance to access quasi-BICs and harness their functionalities in practical applications. Recently, resonant nanostructures have opened a new paradigm for realization of quasi-BICs in nanophotonics, such as dielectric photonic crystals [9], optical waveguides [10], hybrid photonic-plasmonic systems [11], all-dielectric resonant meta-atoms and metasurfaces [12–15]. Among them, high-refractive-index dielectric metasurfaces with subwavelength confinement properties have been a fascinating platform, which allow us to create BIC states by engineering the structural symmetry of the elementary scattering units supporting Mie resonances.

Of late, transition metal dichalcogenides (TMDCs) have sparked a great deal of attention and interest of research in terms of creating novel dielectric metamaterials with high refractive index. TMDCs are van der Waals materials, and in particular, they exhibit a direct bandgap in the atomic monolayer limit, making them interesting for various optoelectronic applications, such as ultrafast modulation [16], light emission [17], photodetection and light harvesting [18], as well as for fundamental investigations of strong light–matter interactions [19–21]. Bulk TMDC materials possess naturally very high in-plane refractive indices in the visible and near-infrared spectral range, higher than what is found in widely used dielectric materials with high refractive index, as silicon and III–V semiconductors. Compared to these traditional dielectric materials like silicon, TMDCs are capable of simultaneously boosting the second- and third-harmonic generation in the context of nonlinear light emission [22]. Besides, since layered TMDCs have strong van der Waals stiction forces, they can be patterned on low-refractive-index transparent substrates. This is not readily realized for III–V materials. To date, bulk TMDCs have been utilized for designing nanostructures including photonic crystals [23,24], metasurfaces [22,25], and other structures depending on the strong photon–exciton coupling [26]. Nevertheless, it has remained elusive how to excite and control radiationless BIC states in TMDC-based metasurfaces.

In this work, we demonstrate the generation of BICs in TMDC metasurfaces consisting of narrow-gap WS₂ nanobar arrays. The high refractive index in excess of 4 for the bulk WS₂ allows for optical resonances in nanoresonators with small dimensions, enabling densely packed metasurfaces. The WS₂ metasurfaces are demonstrated to support multiple symmetry-protected BICs at the Γ point which can be represented by magnetic dipole (MD) and electric toroidal dipole (TD) modes. Through introduction of defects and perturbations that break the structural in-plane symmetry, such MD- and TD-BIC modes can be transformed into high-Q quasi-BICs with dominating magnetic and toroidal dipole excitations respectively, at the normal incidence of light. Because the quasi-BIC is always accompanied by a giant electromagnetic-field enhancement in the interior of structure, we show that our metasurfaces by themselves provide a natural platform for light–exciton interaction. Specifically, it is observed that MD and TD quasi-BICs strongly coupled to the WS₂ exciton lead to Rabi splittings of up to 134 meV and 162 meV, respectively. We reveal that the difference between Rabi energies for MD and TD BICs can be attributed to their different mode volumes. We demonstrate that a small mode volume allows us for observation of the large spectral splitting, while a large one is more favorable to improving the sensing performance of BIC modes.

2. Results and discussion

The design of our TMDC-based WS₂ metasurface is illustrated in Fig. 1(a). It consists of a periodic square array of two identical nanobars, placed on a low-refractive-index substrate (SiO₂). We indicate as P the period of structure, H the thickness of WS₂ nanoresonator and G the gap width between nanobars. In all cases, both the lengths of two nanobars and the sum of their widths and gap satisfy L = 0.8P, as shown the top-left-corner inset in Fig. 1(a). We first consider the passive case of the metasurface (that is, the Lorentz oscillator is taken out in the dielectric function of WS₂, as illustrated in Appendix 1) and calculate its band structure along the MΓ and ΓX directions using the three-dimensional finite-element-method eigenfrequency solver by Comsol Multiphysics. Here, Γ, X, and M represent the high-symmetry points of the first Brillouin zone for a square lattice. This is shown in Fig. 1(b), where the structural parameters are P = 300 nm, H = 30 nm, L = 240 nm, and G = 0.1P = 30 nm. We observe that there are four modes with different dispersion in the band structure near the Γ point, wherein modes 1, 2 and 3 correspond to the transverse electric (TE)-like mode and mode 4 to the transverse magnetic (TM)-like mode. Figure 1(c) shows the Q factors of these four modes as a function of Bloch wavevector. It is found that the Q factors are appreciably high for modes 1 and 3 and drop sharply away from the Γ point, manifesting that the two modes correspond to symmetry-protected BICs at the Γ point. For modes 2 and 4, they have a large leakage and remain Q factor lower than 10³ over the entire wavevector span of interest, which allows for the probing via far-field measurements.

 

Fig. 1. BIC-based TMDC metasurfaces. (a) Schematic of the WS₂ metasurface on a silica substrate. The structure is composed of a square lattice with unit cells consisting of two identical WS₂ nanobars, the dimensions of which are the period P, the gap between nanobars G, the length of a nanobar L and its width W = (L – G)/2, as shown in the top-left-corner inset. The bottom-right-corner inset sketches the lattice structure of the 2H-stacked WS₂. (b) Dispersion relation and (c) Q factors of four eigenmodes in both the MΓ and ΓX directions as a function of Bloch wavevector. The inset in (a) represents the first Brillouin zone of the square lattice. The calculations are performed at structural parameters: P = 300 nm, L = 0.8P, G = 0.1P, and WS₂ thickness H = 30 nm.

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2.1 Multipolar origination of the BICs

Next, we apply the multipole analysis to our metasurface to get a comprehensive understanding of these modes, as shown in Fig. 2(a–c). Through using the Cartesian multipole decomposition, we calculate the TD (Fig. 2(a)), electric dipole (ED) (Fig. 2(b)) and MD (Fig. 2(c)) moments for the four eigenmodes at the Γ point. It can be seen from Fig. 2(a) that modes 2 and 4 assume a predominant TD character, of which the TD moment is along the y and x axis (that is, TD moment T_y and TD moment T_x), respectively. Such TD modes are different to the nontrivial ones that have a TD moment along z axis while they are dependent on the lattice constant. Since our metasurface is at the sub-diffractive regime and has the C₂ symmetry, the TD moment cannot contribute to the far-field radiation along z-axis direction. However, the ED mode plays a crucial role in opening the radiation channel able to couples to the outgoing waves, as shown in Fig. 2(b). Among the four modes, only modes 2 and 4 have a large ED moment and are with the ED mode P_y and the ED mode P_x, respectively. This manifests that modes 2 and 4 can be respectively excited by the normally incident plane wave with polarization parallel to the y and x direction. For modes 1 and 3, they are radiationless due to their very small ED moments that cannot couple to the radiating waves, and nevertheless, both of them exhibit dominating MD moments along the z direction (see Fig. 2(c)). Such the out-of-plane MD is “dark” and cannot be directly excited by normal incident light. Thus, modes 1 and 3 can also be referred to as the MD BIC.

 

Fig. 2. Multipole analysis to the BIC-based WS₂ metasurface. (a–c) Multipole decomposition results of the eigenmodes at the Γ point, which are normalized to their maximum within each panel. T, P, and M respectively are the toroidal, electric, and magnetic dipole moments. (d) Top row, the xy cross-sectional electric-field profiles |E| for the Γ-point modes, calculated by through the center of the unit cell, z = 0; middle row, corresponding z component of the magnetic field, H_z; lower row, sketches of the magnetic M and the toroidal T dipole moments. The black arrow in the field distributions stands for the electric-field vector.

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In order to gain a deeper insight into the configuration of these modes, we plot in Fig. 2(d) the electric-field distributions for these four modes. It is obvious that the excitation of the MD mode is accompanied by a circulating electric field (see the mode 1), while the mode 3 exhibits two circulating electric fields with in-phase oscillation in each nanobar, thus leading to two parallel MD moments along the z direction. The TD moment is associated with both the circulating magnetic field and the electric poloidal current distribution. This can be seen from the field profiles of modes 2 and 4. For the direction of their TD moments, it can be visualized from the vivid schematics in the bottom row of Fig. 2(d), which validates the multipole decomposition results in Fig. 2(a).

For the sake of getting a better sense of how light propagates in the metasurface, we calculate the far-field spectra of the structure illuminated by an incident light with different polarizations. In this work, the polarization of light along the y direction is defined as TE and along the x direction as TM. As seen Figs. 3(a) and 3(b), they display the calculated TE- (Fig. 3(a)) and TM-polarized (Fig. 3(b)) transmission spectra with respect to incident angle θ along the ΓX direction. The TE band diagram shows three resonances which are corresponding to eigenmodes 1, 2 and 3 in Fig. 1(b). According to the above-mentioned eigenfrequency analysis, MD modes 1 and 3 are the symmetry-protected BIC at the Γ point and thus exhibit vanishing linewidths at the normal incidence. With increase of the incident angle, they have growing losses and are transformed into the leakage resonances called as quasi-BICs. Under TM-polarized light (Fig. 3(b)), only one resonance appears in the transmission map, corresponding to the TD mode 4. Both TD modes 2 and 4 show a relatively broad resonance at a wide angular range due to their low Q factors. These results are well in accordance with those of the eigenmode analysis.

 

Fig. 3. Far-field spectral features for the WS₂ metasurface. (a, b) Angle-resolved transmission spectra along the ΓX direction for the WS₂ nanostructure illuminated by (a) TE- and (b) TM-polarized plane waves. (c) Top row, detailed transmission of the four modes, with open circles being the transmittance calculated from finite element method and solid lines the fitting results from eq 1; bottom row, normalized radiated power of the modes to which the Cartesian multipoles contribution, corresponding with the panels in top row.

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To further observe their spectral features, we give the transmission for these four resonance modes, as shown in the top-row panels of Fig. 3(c) (open circles), where MD resonances (modes 1 and 3) and TD resonances (modes 2 and 4) are plotted at θ = 10 deg and 0 deg, respectively. Both two resonances manifest themselves as a pronounced asymmetric Fano line shape in the far-field spectral response. As we all know, Fano or Fano-like resonance is formed originating from the interaction between a “bright” state and a “dark” state. For optical resonances studied here, the MD and TD are “dark” while the ED is “bright” and forms a leaky channel that couples to the free-space radiation. This coupling determines the spectral bandwidth. To this end, we first derive the Q factor of each mode from the corresponding transmission spectrum by fitting the transmission T as a function of frequency, to a Fano line shape given by

2.2 Manipulating the BICs under normal excitations

Figure 4(a) shows the transmission of the metasurface excited by a normally y-polarized plane wave, as a function of the gap width between two symmetric nanobars, where other geometrical parameters remain unchanged and the parameter associated with the gap width is η = G/P (see the inset in Fig. 4(a)). It can be seen that at a value of η = 0.2, the spectral linewidth of the TD resonance entirely vanishes. This is attributed to the fact that the structure can be viewed as an equidistant grating when η = 0.2, and in this situation, the resonance cannot be excited due to the momentum mismatch between the incident photon and the waveguide mode. Further, we calculate in Fig. 4(b) the Q factor of the eigenmodes as a function of η, showing that there is an eigenmode with a diverging Q factor at η = 0.2, which manifests the formation of a BIC. The electromagnetic-field distribution in the inset of Fig. 4(b) further reveals the nature of TD for the BIC. It should be noted that the BIC is also symmetry-protected at the Γ point.

 

Fig. 4. Control over the BICs in WS₂ metasurfaces at normal incidence. (a, b) Transmission spectra and Q factors with respect to the parameter η for the symmetric metasurface. (c, d) Transmission spectra and Q factors with respect to the offset Δ of a gap for the asymmetric metasurface, calculated when the width of gap is fixed at G = ηP = 0.2P. Insets in (b, d) show the electric-field profiles for the ideal BICs highlighted by red-dotted circles. All calculations are carried out in the case where the nanostructures are normally illuminated by a plane wave polarized parallel to the long side of nanobars (y axis).

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Additionally, we demonstrate that introducing the symmetric defects to the structure can provide an efficient way opening the leaky channel so that the MD quasi-BIC can be excited under normal incidence. As shown in the inset of Fig. 4(c), the Δ is defined as a parameter determining the amount of the structural symmetry-breaking, which is the offset of the air gap away from the center of the unit cell along the x direction (short side of nanobars). Here, we fix the gap width at G = 0.2P to avoid the excitation of TD resonances, and calculate the transmission map as a function of Δ for the asymmetric metasurfaces under y polarized light at incident angle of θ = 0 deg. As shown in Fig. 4(c), the transmission dip vanishes at the zero offset due to the lack of the coupling between the incident light and the BIC. A narrow dip occurs in the transmission spectra as the structural symmetry is broken, and it becomes broader with increase of the offset value. This corresponds to the formation of the MD quasi-BIC and a decrease of the Q factor (Fig. 4(c)), respectively.

2.3 Strong coupling between BICs and exciton

The generation of BICs is accompanied by the giant electromagnetic-field enhancement inside structures, indicating that they can be used for boosting light–matter interactions. To this end, strong coupling between BICs and the exciton is explored in our metasurface. To observe this coupling, we consider the active structure for which the WS₂ exciton is turned back on. Figures 5(a–c) show the optical response for three structural configurations that respectively support the TD quasi-BIC (Fig. 5(a)), the TD resonance (Fig. 5(b)), and the MD quasi-BIC (Fig. 5(c)), where the absorption at normal incidence is plotted as a function of the period P. For the TD-BIC metasurface where the gap width between two nanobars is fixed at G = 0.1P, the absorption map exhibits clear anticrossing between the TD quasi-BIC and exciton modes, splitting into upper and lower energy branches with dispersive character. This marks the strong coupling and the formation of the mixed optical state consisting of BIC and exciton. In particular, the mix state in the lower (upper) energy branch takes on a more BIC (exciton) character for sparse arrays (large period). Moreover, the anticrossing appears at the spectral position at which the uncoupled BIC and exciton modes are tuned to intersect, which results in a spectral splitting of 162 meV (see Figs. 5(a) and 5(d)). In order to confirm this, the strong coupling between a BIC and exciton mode is modeled using the following coupled oscillator model

 

Fig. 5. Strong coupling of nonradiating states with excitons in WS₂ metasurfaces. (a–c) Absorption spectral maps as a function of the lattice period for three WS₂ metasurfaces. Insets show schematic of the structure designs and the polarization configurations. (d–f) Theoretical dispersion curves of coupled upper/lower polariton branches (solid red/black curves) resulting from strong coupling between bare optical resonance/exciton dispersion curves (dotted blue/green lines), corresponding with panels (a–c), respectively.

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In addition to observation of the clear anticrossing, that strong coupling is confirmed requires to satisfy the following inequality:

The coupling between photon and exciton scales as $\mathrm{\Omega } \propto \sqrt {nf/{V_{eff}}} $, where n, f, and V­_eff represent oscillator numbers per volume, exciton oscillator strength, and effective mode volume, respectively. For bulk WS₂, the effective concentration of excitons has been determined at the fabrication stage (that is, n and f are a definite value), and thus, the mode volume is an appropriate figure of merit that can provide a quantitative insight into the difference of the Rabi splitting of these coupled systems. The definition of mode volume V­_eff can be given using the equation

 

Fig. 6. Effective modal volume of the three optical states in WS₂ metasurfaces as a function of period. The solid lines are to guide the eyes. The dashed line indicates the actual volume of nanobars.

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To this end, we study how the far-field spectrum for BIC-based metasurfaces is modified upon changing the refractive index of the incident medium. Figure 7(a) shows the variations within the absorption spectral response of the TD-BIC metasurface, demonstrating that as the refractive index increases, the TD quasi-BICs (indicated by an orange shadow) shift to longer wavelengths. This change can also be observed in the MD-BIC metasurface (Fig. 7(d)), whereas the MD quasi-BIC clearly shows larger spectral shifts than the TD quasi-BIC as due to its larger mode volume. A larger mode volume implies the electromagnetic field is confined within the dielectric layer with higher evanescent leakage, thus leading to a higher sensitivity. This leakage can be further increased by reducing the thickness of the metasurface, which can be used for boosting the sensitivity of photonic sensors. As shown in Figs. 7(b) and 7(e), we plot the linear dependence of the resonance wavelength of TD quasi-BIC (Fig. 7(b)) and MD quasi-BIC (Fig. 7(e)) to the refractive index of the incident medium at different thicknesses of metasurfaces, and calculate corresponding refractive index sensitivity (S) that is defined as the ratio of the resonance wavelength to the change in the refractive index of the adjacent medium (that is, S = Δλ/Δn). These results clearly show that reducing the thickness can effectively improve the sensitivity. Also, we show the influence of the gap width G (Fig. 7(c)) and the offset Δ (Fig. 7(f)) on the sensitivity. As can be seen, the sensitivity is robust against these two parameters, which in turn indicates that the effective mode volume dose not vary much (see Appendix 4). However, since the resonance Q factor strongly depends on these two parameters (see Figs. 4(b) and 4(d)), we can by properly choosing parameters obtain an extremely high sensing figure of merit (not shown), a crucial performance-defining parameter obtained by dividing the sensitivity by the full width at half maximum of the resonance. Finally, we would like to remark that there is still a plenty of room for improving the results reported here by, for example, reducing the thickness of WS₂ to the monolayer level or by removing the substrate. This is because at the atomically thick WS₂ membrane, the field leakage will be substantially increased. And suspending the structure in space can effectively increase the interaction volume between light and analyte. In order to illustrate this idea, we take the TD-BIC metasurface as an example, and show in the Appendix 5 its refractive index sensitivity when suspended in the environment. It is evident that the freestanding structure exhibits a higher sensitivity than the one placed on a substrate. More remarkably, a sensitivity of up to 480 nm/RIU can be achieved in the structure with a few-layer WS₂ thickness, which can be comparable to that of plasmonic nanostructures.

 

Fig. 7. BIC-based metasurfaces for refractive-index sensing. (a, d) Normal-incidence absorption spectra of (a) the TD-BIC metasurface (initial parameters: P = 300 nm, L = 0.8P, η = 0.1, Δ = 0 nm, and H = 30 nm) and (d) the MD-BIC metasurface (initial parameters: P = 300 nm, L = 0.8P, η = 0.2, Δ = 10 nm, and H = 30 nm) for different values of the refractive index of the incident medium. (b, c) Linear plot of TD quasi-BIC shifts as a function of refractive index for (b) different values of H and (c) for different values of η. (e, f) Linear plot of MD quasi-BIC shifts as a function of refractive index for (b) different values of H and (c) for different values of Δ. Note that when changing the structural parameters of interest (H, η, and Δ), other initial parameters are fixed.

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3. Conclusion

In summary, we have demonstrated that the ultrathin resonant WS₂ metasurface supports MD BICs with out-of-plane MD moments and TD BICs with in-plane TD moments. Through either tuning of the system parameters or manipulating the symmetry of the excitation field, we can transform the BICs into the high-Q leaky resonances. Using the multipole decomposition approach, we have shown that the ED moment plays an important role in opening a radiation channel and transforming the BICs into the leaky resonances. Since the BICs can produce strong light confinement within the nanoresonators, the strong coupling regime was observed in our metasurface where the WS₂ serves as a host for both excitonic and cavity modes, without the need of an external optical cavity. Finally, we demonstrated the promise of the WS₂ metasurface for sensing applications. Our studies could pave the way for making integrated polaritonic devices, as well as for TMDC-based light harvesting.

Appendix 1: Numerical simulations

The simulations of transmission and absorption maps of structures have been done using the rigorous coupled wave analysis [27]. The band structures, Q-factor, multipole analysis, and electromagnetic-field simulations have been carried out using a commercial software based on finite element code (COMSOL Multiphysics). All results were obtained by simulating a unit cell with Floquet periodic boundary conditions in the two lattice-vector directions and perfectly matched layers at the boundaries in the out-of-plane direction.

Appendix 2: Bulk WS₂ model

The dielectric function of bulk WS₂, used in numerical simulations and analytical calculations, is given by [28]

(5)$$\begin{array}{c} {\epsilon (E )= {\epsilon _0} + \frac{{{f_0}}}{{E_{ex}^2 - {E^2} - i{\gamma _{ex}}E}},} \end{array}$$

where ${\epsilon _0}$ = 20 is the permittivity of the passive structure, ${f_0}$ = 0.8 eV² is the oscillator strength of WS₂ exciton, and E_ex = 2 eV and ${\gamma _{ex}}$ = 50 meV are its energy and linewidth, respectively. For the WS₂ metasurface, when taking the Lorentz term out, it is called the passive structure, otherwise it is called the active structure.

Appendix 3: Cartesian multipole decomposition

Cartesian multipoles are computed by integrating over the charge density $\rho ({\mathbf r} )$ or current density ${\mathbf J}({\mathbf r} )$ distribution within the unit cell of the metasurface. And the dipole moments can be calculated by [29]:

(6)$$\begin{array}{c} {{P_\alpha } = \frac{1}{{i\omega }}\smallint {d^3}r\; {J_\alpha },} \end{array}$$

(7)$$\begin{array}{c} {{M_\alpha } = \frac{1}{{2c}}\smallint {d^3}r\; {{[{{\mathbf r} \times {\mathbf J}} ]}_\alpha },} \end{array}$$

(8)$$\begin{array}{c} {{T_\alpha } = \frac{1}{{10c}}\smallint {d^3}r\; [{({{\mathbf r} \cdot {\mathbf J}} ){r_\alpha } - 2{r^2}{J_\alpha }} ].} \end{array}$$

Here, ω is the angular frequency, c is the speed of light and r specifies the location where the induced current J is evaluated and α, β =x, y, z. The radiation power of each pole can be calculated as: ${\sigma _P} = \frac{{{\mu _0}{\omega ^4}}}{{12\pi c}}{|{\mathbf P} |^2}$, ${\sigma _M} = \frac{{{\mu _0}{\omega ^4}}}{{12\pi c}}{|{\mathbf M} |^2}$, and ${\sigma _T} = \frac{{{\mu _0}{\omega ^4}{k^2}}}{{12\pi c}}|{\mathbf T} |$, with ${\mu _0}$ being the permeability of vacuumnd k the wavevector.

Appendix 4: Effective mode volume for TD and MD BICs

In order to interpret the variation in the sensitivity of our metasurfaces, we give in Fig. 8 the effective mode volume of TD and MD modes as a function of structural parameters corresponding with those of Fig. 7. This confirms the intrinsic link between the mode volume and the sensitivity.

 

Fig. 8. (a) The effective eigenmode volume for TD and MD BICs versus the WS₂ thickness H. (b) The TD-BIC mode volume versus the parameter η associated with the gap width, and the MD-BIC mode volume versus the offset Δ.

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Appendix 5: Sensing performance for freestanding WS₂ metasurfaces

Figure 9 presents detailed calculations with respect to the refractive index sensitivity for freestanding TD-BIC metasurfaces with different WS₂ thickness. The structural parameters for the metasurface in Fig. 9(a) are same to those in Fig. 7(a). Particularly, given that the WS₂ material possesses a very low absorption loss in the atomic monolayer limit, thus leading to a small absorption amplitude, and the TD quasi-BIC resonance shows an evident blueshift as the WS₂ thickness is substantially reduced, the period and thickness of the metasurface in Fig. 9(b) are set as P = 500 nm and H = 4 nm, respectively.

 

Fig. 9. (a, b) Absorption spectra of the freestanding TD-BIC metasurfaces with a WS₂ thickness of (a) H = 30 nm and (b) H = 4 nm. The different curves correspond to different values of the refractive index of the surroundings. (c, d) Resonance wavelength as a function of refractive index for the TD quasi-BIC mode in the metasurfaces with (c) H = 30 nm and (d) H = 4 nm. The black lines are liner fit to the dots.

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Funding

National Natural Science Foundation of China (62105095, U1804261); Natural Science Foundation of Henan Province (202300410238); National Scientific Research Project Cultivation Fund of Henan Normal University (20210381, 2021PL22); Program for Innovative Research Team (in Science and Technology) in University of Henan Province (23IRTSTHN013).

Disclosures

The authors declare no conflicts of interest.

Data availability

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

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