Quasi-bound states in the continuum for electromagnetic induced transparency and strong excitonic coupling

: Advancing on previous reports, we utilize quasi-bound states in the continuum (q-BICs) supported by a metasurface of TiO 2 meta-atoms with broken inversion symmetry on an SiO 2 substrate, for two possible applications. Firstly, we demonstrate that by tuning the metasurface’s asymmetric parameter, a spectral overlap between a broad q-BIC and a narrow magnetic dipole resonance is achieved, yielding an electromagnetic induced transparency analogue with a 50 µ s group delay. Secondly, we have found that, due to the strong coupling between the q-BIC and WS 2 exciton at room temperature and normal incidence, by integrating a single layer of WS 2 to the metasurface, a 37.9 meV Rabi splitting in the absorptance spectrum with 50% absorption efficiency is obtained. These findings promise feasible two-port devices for visible range slow-light characteristics or nanoscale excitonic coupling


Introduction
Metasurfaces -two-dimensional (2D) metamaterials composed of subwavelength resonators (meta-atoms) -can be employed to tailor the amplitude, phase, and polarization of an incident electromagnetic wave [1].Thanks to these remarkable characteristics, metasurfaces have found applications in numerous fields, including polarization conversion, holography, optical vortex generation, lensing, and beam splitting.[2].Here, we focus on the classical analogue of electromagnetically induced transparency (EIT) and strong excitonic coupling as two target applications of dielectric metasurfaces with broken in-plane inversion symmetry.
Dielectric metasurfaces have recently been employed for flat optics wavefront manipulation due to their low resonance-induced heating compared to their plasmonic counterparts, as well as their unique capabilities for controlling the propagation and localization of light, and their compatibility with CMOS fabrication processes [3][4][5][6].Consequently, dielectric metasurface analogues of EIT have garnered significant attention in the field of nanophotonics in recent years, primarily due to their ability to produce high-quality factor (Q-factor) resonances.Such Fano resonances are useful for applications such as low-loss slow-light devices and highly sensitive optical sensors [7,8].EIT is a concept originally observed in atomic physics, arising from quantum interference that results in a narrowband transparency window for light propagating through an initially opaque medium [9].This concept was later extended to classical optical systems using lossy (plasmonic) metasurfaces [10][11][12][13], low-loss (dielectric) metasurfaces [7,8] and coupled waveguide-resonators [14].This extension allowed for experimental implementation with incoherent light and operation at room temperature.The observation of EIT in metasurfaces relies on a Fano-type interference between a broad 'bright'-mode resonance, accessible from free space, and a narrow 'dark' mode resonance, which is less accessible or inaccessible from free space.If these two resonances are brought into close proximity in both the spatial and frequency domains, they can interfere, resulting in an extremely narrow reflection or transmittance window [7,14].However, in this study, in advancing upon the previous one [15], we employ the concept of q-BIC to provide a novel route to achieve the EIT-like feature.
Remarkably, dielectric metasurfaces composed of meta-atoms with broken in-plane inversion symmetry can support high-quality factor (Q) resonances under normal incidence of light.These resonances originate from the distortion of symmetry-protected bound states in the continuum (BIC) [16][17][18][19].In this case, the quasi-BICs (q-BICs) are supported for which both the Q factor and resonance width become finite [20][21][22][23][24][25][26][27][28].A universal relation exists between the radiative Q factor and the asymmetry parameter of the q-BICs, and the transmittance spectrum of the metasurface can be expressed in the form of the classical Fano formula [21].While they can be also supported by the metallic metasurfaces [19,28], high-Q (q)-BICs of the dielectric metasurfaces can boost the electric field enhancement inside the structure with low-losses.They can also be employed in applications such as lasing [18,28], enhanced nonlinear harmonic generation [27], label-free biosensing [23][24][25][26], and strong excitonic coupling in transitional metal dichalcogenides (TMDCs).Furthermore, it has been recently proven that BICs may play an important biological role by boosting light-matter interactions in a biogenic nanostructure: tapetum reflector of a shrimp eye [29].
The strong coupling between excitons of a quantum emitter and resonances of an optical cavity, characterized by the Rabi splitting measured from the anticrossing branches of the absorptance spectrum spectrum, enables control of light-matter interactions at the nanoscale.
A key aspect of achieving strong coupling lies in enhancing the coupling strength d.In other words, in the strong coupling regime, the rate of coherent energy exchange between excitons and optical resonances exceeds the dissipation rates.This leads to the formation of new hybrid states, known as exciton-polaritons, which exhibit characteristics of both light and matter [30,31].Exciton-polaritons offer a quantum many-body system on a chip, showcasing a plethora of rich physical phenomena for more advanced photonic applications, such as spin switches [32] and ultralow-threshold lasing [33].
To form exciton-polaritons at room temperature, 1L-WS 2 [43][44][45][47][48][49][50][51][52], which exhibits a larger oscillator strength to linewidth ratio compared to other TMDCs such as WSe 2 [37], as well as bulk WS2 [54,55], have been recently employed in previous studies.Reports indicate that when WS2 is integrated into a suspended dielectric metasurface with broken in-plane inversion symmetry, nearly 65 meV of Rabi splitting with 50% light absorptance (A) can be achieved [48,49], owing to the satisfaction of the strong critical coupling condition [56,57].Note that maximizing Rabi splitting while achieving 50% light absorptance are critical factors for the efficient operation of two-port excitonic devices.However, from a practical standpoint, the presence of a substrate must be taken into account in the design of excitonic devices, even if it results in a decrease in the Rabi splitting.To this aim, recent studies reported up to 29.33 meV Rabi splitting with A = 50% for 1L-WS 2 -integrated dielectric metasurfaces [52].It is noteworthy that by encapsulating a monolayer of WS 2 between a defective grating on a SiO 2 spacer layer and a one-dimensional photonic crystal, it is possible to achieve a 70 meV Rabi splitting under oblique incidences [58].However, our aim here is to maximize the Rabi splitting with 50% absorption efficiency in the presence of a simple SiO 2 substrate under normal incidence.In other words, to maximize the optical pumping of the polaritonic system by having the critical coupling in the strong coupling regime.
In this study, our aim is to investigate q-BICs for two potential applications in EIT and strong excitonic coupling.The q-BICs are supported by a feasible CMOS-compatible dielectric metasurface with broken in-plane inversion symmetry that is composed of a periodic arrangement of TiO 2 meta-atoms on an SiO 2 substrate.Firstly, as a novel approach, we demonstrate the possibility of achieving a classical analogue of EIT by overlapping a broad q-BIC resonance with a narrow magnetic dipole (MD) resonance supported by the metasurface.This leads to a group delay of up to 50 µs that makes the suggested design practical for low-loss slow light application in the visible range.Secondly, building upon previous findings, we demonstrate a Rabi splitting of approximately 38 meV with 50% absorption efficiency achieved by integrating a single layer of 1L-WS 2 into the metasurface.This excitonic coupling arises from the strong interaction between the q-BIC and the 1L-WS2 exciton under room temperature and normal incidence.This finding shows promise for the development of feasible two-port devices operating based on room temperature strong excitonic coupling.

Bare dielectric metasurface
We begin our analysis by examining the transmittance spectra and modal characteristics of the q-BIC and MD resonances supported by the bare dielectric metasurface.As shown in Fig. 1(a), the bare metasurface comprises TiO 2 (n = 2.3) nano-resonators on an SiO 2 (n = 1.45) substrate, which can be fabricated with CMOS-compatible processes [6].The unit cell of the metasurface is composed of two elliptical TiO 2 meta-atoms with thickness h, width w = 121 nm and length l = 3 × w that are separated by gap g = 1.62 × w.The in-plane inversion symmetry in each unit cell is broken with angle θ ≠ 0 o as shown in Fig. 1(a1).Note that the asymmetry parameter for this metasurface is defined as α = sin(θ) where α ≅ θ for 1 o <θ<25 o .The periodicities in the X and Y directions are p x and p y where p x = p y = p = 3.33 × w.Here, finite-difference time domain (FDTD) simulations were conducted using an X-polarized plane wave source at normal incidence [59].Periodic boundary conditions were applied in the X and Y directions on the unit cell, while perfectly matched layer boundary conditions were applied in the Z direction.
The transmittance spectrum of the bare metasurface for h = 150 nm and θ = 17 o is represented by the solid-blue curve in Fig. 1(a2).This spectrum can be described by the following Fano formula.The Fano parameters are explicitly expressed through the material and geometrical parameters of the metasurface and dimensionless frequency [21] Here the dimensionless frequency ω = ℏ(ω − ω q−BIC )/γ q−BIC , ω q−BIC = 2πc/λ q−BIC , γ q−BIC is the half-linewidth of the q-BIC, and q is the Fano parameter and c is the speed of light in vacuum.T bg and T 0 describe the background contribution of non-resonant modes to the resonant peak amplitude and the offset, respectively.Note that Eq. ( 1) becomes ill-defined for a true BIC, corresponding to a collapse of the Fano resonance when any features in the transmittance spectra disappear.This indicates that the resonant mode is transformed into a dark mode, which does not manifest itself in scattering spectra [21].The Fano fit, depicted as the dashed-red curve in Fig. 1(a2), applied to the numerically obtained T spectrum yields γ q−BIC = 8.62 meV, corresponding to To analyze the type of q-BIC resonance, we performed a Cartesian coordinate multipole decomposition to understand how the multipole moments contribute to the transmittance spectrum T = 1 − |r| 2 where the reflection coefficient is defined as [60][61][62][63][64][65] Here A is the area of a unit cell of the metasurface, and P x , T x , Q xz , m y and M yz refer to the effective electric dipole (ED), toroidal dipole (TD), electric quadrupole (EQ), magnetic dipole (MD), and magnetic quadrupole (MQ), respectively, under X-polarized incident light with amplitude E 0 .Please refer Supplement 1 for more details on the calculations of the multipolar mode contributions.Note that, using Eq. ( 2), the transmittance spectrum shown in Fig. 1(a2) can be reconstructed from the contributions of the multipole elements.As illustrated in Fig. 1(a3), among others, ED (solid-blue curve) shows maximum contribution at λ q−BIC .In agreement with this ED contribution, panels (b2) and (b3) of Fig. 1 confirm that E is oriented in the X direction while H rotates in the Y-Z plane at λ q−BIC , thus confirming the ED nature of the q-BIC.The top-view of E verifies how the electric field is localized and elongated in the Y direction at the q-BIC resonance.In agreement with the previous studies [21][22][23][24][25], it is observed that the dominant electric (magnetic) polarization is along the long (short) elliptical axis of the meta-atoms.
To gain a deeper insight into the characteristics of the supported q-BIC, we investigate the transmittance spectra and Q of the q-BICs as function of θ for h = 150 nm, as shown in Fig. 2(a1-a3).We observe from Fig. 2(a1) that a dark BIC with zero linewidth (infinite Q) at θ = 0 o transforms into high-Q q-BICs, with radiation loss (2γ q−BIC ) increasing with θ, while the resonance location is blue-shifted.To be more specific, as shown in Fig. 2  For the case of θ = 17 o , the examination of the transmittance spectra and Q vs h reveals some interesting features.From Fig. 2(b1) it is observed that, increasing the thickness of the meta-atoms results in a red-shift of the q-BIC locations, while the radiation losses are increased.Another remarkable characteristic of the metasurface is its ability to support extremely narrow resonances, as seen in Fig. 2(b1) for h>160 nm.To elaborate further, in agreement with Fig. 2(b1), the transmittance spectrum at h = 100 nm (solid-red), 150 nm (solid-blue) and 200 nm (solid-black) are compared in Fig. 2(b2).It is evident that for h = 200 nm, the metasurface supports a broad q-BIC (bright) resonance at 631.8 nm with Q = 97.17,and an extremely sharp (dark-like) resonance at λ MD−17 = 617.05nm with Q MD−17 = 6.17 × 10 3 .After analyzing the |E| and |H| local field distributions, it is concluded that the supported dark-like mode inherits an MD nature, as seen in Figs.S1(b1-b3).To further investigate this feature, for the case of h = 200 nm and θ = 0 o , the transmittance spectra are analyzed in Fig. S2(a).Observing the transition from θ 1 o → θ 0 o , it is noted that while the spectral position of the dark-like resonance experiences a slight red-shift, it remains supported.However, the bright resonance (the q-BIC) disappears.In other words, the q-BIC transformed to BIC; i.  (b1-b3)], it is evident that the MD nature of the dark-like mode is preserved.

Analogue of the EIT and slow-light features of the 200 nm-thick metasurface
In continuation of our discussion from the previous section, here we further elaborate on the characteristics of the q-BIC and MD resonances for the case of h = 200 nm.Further investigations confirm that, as observed from Supplement 1, Fig. S3, increasing (decreasing) the periodicity p results in both the MD and q-BIC resonances being red-(blue-) shifted, as expected.However, as illustrated in Fig. 3(a1), the spectral location of the MD resonance remains almost invariant with respect to the asymmetric parameter, while, in agreement with Fig. 2(a1), the q-BIC is blue-shifted with an increase in θ.These two features provide a unique opportunity for the interference of the MD and q-BIC resonances.As seen from Fig. 3(a1), the MD and q-BIC resonances overlap for θ>22 o [see also Fig. S4(a1)].This interference of the so-called dark-like and bright modes leads to the appearance of a transparency window in the transmittance spectrum.In other words, a dielectric metasurface analogue of the EIT is observed.This resembles the classical EIT that can be observed in the metasurfaces composed of dark and bright resonators, where the Fano-type interference between a broadband bright resonance and a narrowband dark one plays a key role.It is noteworthy that by employing a similar approach (i.e.tuning θ), it was reported that the overlap of two bright modes with q-BIC nature can lead to the extreme Huygens' condition [22].
The transmittance spectrum for the case of θ = 23.5 o is illustrated in Fig. 3(a2), where a clear transparency window is observed in the range of 616.4 nm<λ<618.1 nm, accompanied by a sharp transmittance peak (Q EIT = 3.1 × 10 3 and T EIT = 40%) at 617.4 nm.It is shown in Supplement 1, Fig. S4(a2), that by increasing θ to 25 o , it is possible to reach 60% transmittance in the EIT window at the expense of decreasing Q EIT to 344, which is still considered a high value, as compared to the similar studies [7].
To further reveal the effect of the interference between the dark MD and the bright q-BIC resonances, top views of |E| and |H| at the edges and the peak of the transparency window are presented in Figs.3(b1)-3(d2).The field distributions at the troughs (616.4 nm and 618.1 nm) and the peak (617.4 nm) mostly resemble the one shown in Fig. 1(b1), and Supplement 1, Fig. S1(b1), respectively, indicating that the new hybrid modes inherit the nature of the q-BIC and the MD resonances.Furthermore, Fig. S4(b) illustrates the numerical transmittance spectrum and the fitted one obtained by a two-oscillator model.The fitted curve demonstrates that the interference of the dark-like (linewidth 1.3 meV) and the bright mode (linewidth 28.44 meV), in a weak-coupling regime with the coupling strength of κ = 1.5 meV, is responsible for the appearance of the EIT window (see Fig. S5 in the Supplement 1 for more details on the interference mechanism).As mentioned earlier, meta-structures exhibiting an EIT-like response are well-known for their potential applications using slow-light.The slow light effect is achieved due to the strong dispersion properties of the EIT window and can be measured by group delay.The group delay is defined as −dϕ/dω where ϕ stands for the phase of the transmitted light.ϕ and the corresponding group delay are represented in Fig. 3(a3), showing up to 50 µs group delay is obtained in the EIT window.

Strong excitonic coupling in a hybrid system composed of 1L-WS 2 integrated to the 150 nm-thick metasurface
The optical characteristics of the q-BIC supported by the bare dielectric metasurface for the case of h = 150 nm were already investigated in Figs. 1 and 2. As another application of our suggested designs, in this section, we explore the possibility of obtaining strong excitonic coupling in a hybrid system that is composed of the 1L-WS 2 that is integrated on top of the 150 nm-thick dielectric metasurface.The novelty of this investigation lies in achieving a ℏΩ = 37.4 meV Rabi splitting in the absorptance spectrum for a practical facile hybrid system on a SiO 2 substrate.To our knowledge, this value is the highest among those recently reported [50][51][52].A schematic of the hybrid system is illustrated in Fig. 4(a).In our calculations, the relative permittivity of the 1L-WS 2 is parametrized by the multi-Lorentzian dispersion relation, and the real and imaginary parts are illustrated in Supplement 1, Fig. S5(a).In agreement with Ref. 32 , Fig. S5(b) shows that almost 11% absorptance is achieved for the 1L-WS 2 on an SiO 2 substrate at λ ex , confirming the accuracy of our calculated optical properties.It is noteworthy that the half-linewidth of the A-exciton in the 1L-WS 2 is obtained as γ ex = 11 meV from Fig. S6.Note that, in our FDTD simulation, we modeled the 1L-WS 2 as a 2D sheet with surface conductivity where ϵ 0 is the vacuum permittivity and t WS 2 = 0.618 nm [35].As a standard approach, we used a 3D conformal meshing region with a z-spacing (dz) of 0.5 nm and an x-y spacing (dx = dy) of 10 nm to mesh the 2D layer.The absorptance spectra of the hybrid 1L-WS2-integrated dielectric metasurface, for the case of h = 150 nm, as a function of the asymmetric parameter α ≅ θ are represented in Fig. 4(b1).The spectral locations of the isolated A-exciton and q-BICs are also depicted as dashed-and solid-lines in this figure, highlighting the dispersion-less feature of the A-exciton vs θ, while the q-BICs blue-shift (red-shift) with an increase (decrease) in θ.The maxima of this absorptance spectra illustrates a clear anti-crossing of the lower and upper branches of the created hybrid exciton-q-BIC (exciton-polariton) modes.To further confirm this feature, we employ the coupled-mode theory (CMT) in the Hamiltonian representation to qualitatively fit the eigenenergies of the new hybrid states to the numerically obtained exciton-polariton dispersion [43,56,57] Here, E q−BIC and E ex are the energies for the uncoupled q-BIC mode and the A-exciton resonance, respectively.E 1,2 = ℏω 1,2 represents the energies of the new hybrid states identified by the lower and upper branches in the absorptance spectrum, respectively, where ω 1,2 = 2πc/λ 1,2 .3), the eigenvalues are obtained as where ∆ = E ex − E q−BIC is the detuning energy.At the anti-crossing point, i.e., ∆ = 0, the polaritonic dispersions 1 and 2 exhibit a Rabi splitting, ℏΩ, which is indicated by the black arrow in Fig. 4(b1).In agreement with Fig. 4(b1), absorptance spectra for θ = 1 o (solid-red), θ = 17 o (solid-blue) and θ = 25 o (solid-black) are shown in Fig. 4(b2).As clearly observed, for θ = 17 o , two absorptance peaks occur at λ 1 = 613 nm and λ 2 = 624.7 nm resulting in ℏΩ = 37.9 meV.It is noteworthy that, the solid-red curve in Fig. 4(b2) shows that the ultra-narrow q-BIC supported by the bare dielectric metasurface at λ = 638.5 nm, on the solid line, does not noticeably interact with WS 2 .In contrast, the broader resonance of the q-BIC of the bare metasurface at λ = 604.5 nm [see Fig. 2(a2), solid-black curve], considerably enhances the interaction of light with WS 2 , leading to enhanced absorptance at that wavelength and λ ex , as seen from the solid-black curve in Fig. 4(b2).
Using d = √︂ (ℏΩ) 2 + (γ ex − γ q−BIC ) 2 /2 and the above-mentioned values of ℏΩ, γ ex and γ q−BIC , the coupling strength is calculated as 18.98 meV, that is noteably higher than the previous reports [50][51][52].The obtained results from Eqs. ( 4) and ( 5) are illustrated by the black circles in Fig. 4(b1), demonstrating an acceptable fit to the peaks of the numerical absorptance spectra.Further investigations also prove that the following strong coupling criteria are satisfied for our design: d>|γ ex − γ q−BIC |/2 and d> √︂ γ 2 ex + γ 2 q−BIC /2 [43].The strong critical coupling condition, where all the incoming energy is converted into polaritons, can be further investigated by comparing the numerical absorptance spectrum to the one obtained using the following relation derived from the CMT [56,57] Here By considering the aforementioned values of γ ex , γ q−BIC , and d for the case of h = 150 nm and θ = 17 • , Eq. ( 5) is plotted as the dashed-red curve in Fig. 4(c), which fits acceptably with the solid-blue curve.This agreement proves the strong critical coupling condition is satisfied for the hybrid two-port system, where the two coupled photonic and excitonic resonances have similar lifetimes.Finally, Fig. 4(d1) clearly illustrates the anti-crossing characteristic of the absorptance spectrum of the hybrid system for θ = 17 o as a function of the meta-atoms' thickness.In this case, the Rabi splitting at h = 150 nm is clearly observed, as indicated by the black arrow.The black circles, calculated from CMT, again also exhibit a good fit to the absorptance maxima.In agreement with Fig. 2(b1), the solid black line indicates the increasing trend of q-BIC wavelength versus h.Furthermore, it is understood from the solid-red curve (the case of h = 100 nm) in Fig. 4(d2) that the enhanced absorptance at λ = 597 nm originates from the interaction of the supported q-BIC [see Fig. 2(b2), solid-red curve] with the ohmic losses of WS 2 .The same mechanism is responsible for the enhanced absorptance observed at λ = 636 nm for the h = 200 nm case.However, it is noteworthy that the support of the MD for this case leads to the enhanced absorptance at λ ex as well.

Conclusion
To conclude, we have employed q-BICs, supported by a dielectric metasurface, for the realization of a classical analogue of EIT and also for achieving strong excitonic coupling.The practical CMOS-compatible dielectric metasurface is composed of a periodic arrangement of TiO2 meta-atoms, with broken in-plane inversion symmetry, on an SiO 2 .We have demonstrated that it is possible to achieve an EIT-like response by tuning the asymmetric parameter of the bare metasurface.Adjusting θ tunes the spectral location of the q-BIC, ultimately leading to its interference with the MD for θ > 22 • .It was observed that, as a consequence of the EIT-like feature, a group delay of up to 50 µs can be achieved in the transparency window.In addition to observing the EIT phenomenon, the integration of 1L-WS 2 into the dielectric metasurface resulted in a 37.9 meV Rabi splitting, evident from the anti-crossing behaviors in the absorptance spectrum.To the best of our knowledge, this value is the largest among those reported for facile integration of a monolayer to a metasurface supported by a substrate.The results presented in this study have practical implications for developing two-port devices operating in the visible spectrum, suitable for applications requiring either low-loss slow-light characteristics or nanoscale room temperature strong light-matter interactions.

Fig. 1 .
Fig. 1. (a1) Schematic of the unit cell (top) and the dielectric metasurface (bottom).The meta-atoms with thickness h, length l, width w, and the asymmetric parameter θ, are separated by the gap g.For the results shown in this figure h = 150 nm and θ = 17 o .(a2) The numerical transmittance spectra of the metasurface, solid-blue, that is fitted by the dashed-red curve fitted by the Fano formula Eq. (1).Note that for this case, Q 17 o = 117.For reference, the A-exciton wavelength of WS 2 , λ ex = 617.6 nm, is indicated by the vertical dashed-black line.Here, a normally incident X-polarized plane wave excites the metasurface.(a3) The amplitudes of the decomposed multipolar contributions, under the Cartesian coordinates, contributing to the q-BIC mode.The amplitudes include the electric dipole (ED or P), the toroidal dipole (TD or T), the electric quadrupole (EQ or Q), the magnetic dipole (MD or m), and the magnetic quadrupole (MQ or M).(b1) Top-view of |E| at z = h, and (b2,3) side-view of |E| (overlaid with E arrows) and |H| (overlaid with H arrows) at λ ex .The vertical dashed-red line in the schematic (bottom-left) indicates the cross-section of the Y-Z plane in which the side-view mode profiles are plotted.
(a2), for θ 1 o → θ 25 o , the q-BIC at λ 1 o = 617.6 nm with Q 1 o = 3.5 × 10 4 (solid-red curve) is shifted to λ 25 o = 604.4nm with Q 25 o = 50.54(solid-black curve).Note that here, the solid-blue curve, corresponding to the case of θ = 17 o , is provided as a reference [the same applies to Fig. 2(b2)].In Fig. 2(a3), the numerically calculated values of Q num are plotted as a function of θ (red circles) and fitted with the analytical Q ana = Q 1 o θ −2 (solid-blue line), confirming the universal nature of the supported q-BICs.

Fig. 2 .
Fig. 2. (a1) Numerical transmittance spectra of the bare dielectric metasurface vs the asymmetric parameter θ for h = 150 nm.Note that here θ ∼ sin(θ).(a2) The transmittance spectra for θ = 1 o (solid-red), θ = 17 o (solid-blue), and θ = 25 o (solid-black).(a3) Dependence of the Q factor on θ for the q-BIC resonance.The red circles represent the numerically calculated Q that are extracted from (a1), and the solid-blue curve is the fitted one according to Q ana = Q 1 o θ −2 where Q 1 o = 3.5 × 10 4 corresponds to Q for θ = 1 o .(b1) Numerical transmittance spectra of the bare dielectric metasurface vs the thickness of the meta-atoms h at θ = 17 o .(b2) The transmittance spectra at h = 100 nm (solid-red), h = 150 nm (solid-blue), and h = 200 nm (solid-black).Note that at h = 200 nm a magnetic dipole resonance is supported at λ MD−17 o = 617.1 nm with Q MD−17 o = 7.5 × 10 3 .(b3) Dependence of the Q of the q-BIC resonance on h for.Note that the vertical dashed-black lines in panels (a2) and (b2) indicate λ ex as a reference.
e. a dark mode with infinite Q. Comparing the field distributions of the MD for the case of θ = 0 o [shown in Figs.S2(b1-b3)] to those of θ = 1 o [see Figs.S1

Fig. 3 .
Fig. 3. (a1) Transmittance spectra of the bare dielectric metasurface vs the asymmetric parameter θ for h = 200 nm.The q-BIC and MD resonances are labeled accordingly.The vertical dashed-black line indicates θ = 23.5 o .(a2) The transmittance spectrum at θ = 23.5 o .The red (616.4nm) and black (618.1 nm) stars indicate the EIT window in which the transmittance peak with Q EIT = 3.1 × 10 3 is indicated by the blue start (617.4nm).λ ex is also highlighted by the vertical dashed-line.In agreement with (a2), phase (ϕ) and group delay (− dφ dω ) of the transmitted light at θ = 23.5 o are shown in (a3) by solid-red and solid-blue curves, respectively.Top-view profiles of |E| [(b1), (c1), and (d1)] and |H| [(b2), (c2), and (d2)] at the troughs and the peaks of the EIT window are also illustrated accordingly.Here, the mode profiles plotted at z = h.