1. Introduction

Electromagnetically induced transparency (EIT) is caused by quantum destructive interference between two different excitation paths in a three-level atomic system. When the energy levels of the interaction resonance coincide, the classical analogue of EIT can evolve from the Fano resonance [1]. The EIT effect with a transparent window in the transmission spectra exhibits the remarkable feature of strong dispersion, which enhances the nonlinear effect [2–5] and leads to the slow light phenomenon [6–9]. However, the scathing experimental requirements of stable lasers and a cryogenic temperature environment hinder its practical applications [10]. To overcome the barriers, the analogue of the EIT effect is implemented in the classical systems such as waveguide systems [11,12], photonic crystals [13], and metamaterials [14–16]. Particularly, due to the easy design of resonances and subwavelength effective mode volumes in metamaterials, much attention has turned to EIT in various oscillator systems [17,18]. With the characteristics of the EIT-like effect realized in metamaterials, it has a great ability to drastically slow down the group delay of light, promising to be used in slow light devices [19]. However, the Q-factor of the EIT-like effect is relatively restricted, thus the group delay in the transparent window is limited.

Bound state in the continuum (BIC) represents perfectly localized states with no leakage even though they coexist with a continuous spectra of radiating waves, which effectively manipulates the light-matter interaction and generates an ultrahigh Q-factor resonance [20,21]. Originally, this concept was explored in quantum mechanics [22], and in recent years, has received much attention in photonics [23–25]. As a dark mode, the optical BIC is invisible in the optical spectra with zero linewidth and infinite Q-factor. When it collapses to the quasi-BIC in the form of Fano resonance or EIT resonance with perturbations, it can be observed experimentally and widely investigated in the linear optical regime [26,27], the nonlinear enhancement [28–30], coherent light generation [31,32], and integrated circuits [33,34]. Using BICs instead of the traditional low-quality optical modes will greatly improve the performance of the optical devices. For example, in the compound grating waveguide structure, the Goos-Hänchen shift [35,36] and harmonic generation can be greatly enhanced [37]. BIC can be divided into two major categories according to the formation of coupling between the resonant modes and all radiation channels in the surrounding space. Symmetry-protected BIC [38,39] is defined if the coupling between light in the free space and eigenmode of the resonator vanishes by symmetry breaking while the resonance-trapped BIC [40–42] like Friedrich-Wingten BIC [24,43–46]) is formed due to the coupling coefficient disappears by tuning the system parameters [47–49]. Particularly, the resonance-trapped BIC can be realized at an arbitrary point of reciprocal space rather than just restricted to the edge of the Brillouin region, realizing a propagating BIC in the continuous range of the free space [41]. So far, their unique properties have attracted much attention and are explored in various material systems, such as nanoresonators (change radius) [27], resonator array [50], and metallic metasurface [47,48]. For the single subwavelength nanoresonator, the BIC can be accessed both numerically and experimentally at normal incidence by using vector beams [27,45,51]. However, most resonance-trapped quasi-BICs in the metasurface are implemented by the oblique incident light in the metasurface, which complicates the measurement of the resonance-trapped quasi-BICs. In addition, metal-based metasurfaces decrease the Q-factor of the quasi-BICs for their inevitable intrinsic high damping losses. Hence, the metasurface used to generate resonance-trapped quasi-BIC with a high Q-factor by normal incident light and improving the group delay of the EIT effect by resonance-trapped quasi-BIC still needs to be further investigated [52].

In this paper, we design an all-dielectric metasurface to achieve resonance-trapped BICs assisted by guiding modes at normal incidence. The double resonance-trapped quasi-BICs in the form of Fano resonance or EIT resonance are realized through selectively exciting different guiding modes in the proposed metasurface for its useful frequency tunability. In addition, the EIT-based quasi-BIC can actively switch to Fano resonance-based quasi-BIC by modulating the polarization of the incident light. Particularly, coupled harmonic oscillator model is employed to analyze the generation mechanism of the quasi-BICs in the form of EIT resonance. Attributing to the high Q-factor of the quasi-BIC, the dispersion in the transparency window of the EIT extremely changes and the group delay is up to 2113 ps, which may offer possible applications for slow light in the near-infrared region.

2. Structure design

The all-dielectric metasurface utilized for exciting the quasi-BICs in the form of EIT or Fano resonance and the specific geometrical parameters are schematically illustrated in Figs. 1(a) and 1(b). The periods of the unit cell in the x and y directions are P_x= 1300 nm and P_y= 1050 nm, respectively. The upper layer of a unit cell is the grating layer with three bars, which is utilized to increase the wave vector of the incident light and excite the guiding modes in the middle layer, while the guiding mode is supported by the micro-structure composed of the Si waveguide layer and the Si₃N₄ substrate. The material of the upper and middle layers is Si with the refractive index of n₁= 3.5 while the substrate layer is Si₃N₄ with the refractive index of n₂= 2 [53]. The thickness of the Si grating, guiding layer, and substrate are H = 150 nm, D = 150 nm, and T = 150 nm, respectively. The length of the Si-bars is L = 500 nm. The width of the middle Si-bar W₂ is variable while the side Si-bars are fixed as W₁= 150 nm in the following simulations. All the calculations of the all-dielectric metasurface are numerically simulated by the finite element method (FEM) with the commercial software COMSOL Multiphysics 5.4. The Floquet-periodic boundary conditions are employed in the x and y directions, and the perfectly matching layers are adopted in the z-direction.

 

Fig. 1. Schematic of the unit cell of the all-dielectric metasurface structure under TE-polarized illumination. (a) The unit cell is arranged with a period P_x= 1300 nm, P_y= 1050 nm. The Si grating height H = 150 nm, the Si guiding layer thickness D = 150 nm, and the Si₃N₄ substrate thickness T = 150 nm. (b) Top view of the unit cell. The length of the Si-bars L = 500 nm. Meanwhile, W₁= 150 nm and W₂ represent the width of the Si-bars on both sides and in the middle respectively.

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3. Results and discussions

To demonstrate the mechanism of the BICs in the metasurface, the in-plane band structure of eigenmodes is calculated by a three-dimensional finite element method eigenfrequency solver as shown in Fig. 2(a). Herein, the width of the triple Si bar is fixed as 150 nm. In the investigating range from 1600 nm to 2100 nm, there are three groups of doubly degenerate mode at the $\mathrm{\Gamma }$ point, which completely decouple with the external environment, leading to the formation of three resonance-trapped BICs [41,54]. Figure 2(b) illustrates the corresponding electromagnetic field distributions at the Γ point of the six bands, which are well confined in the proposed metasurface and do not decay. The quasi-BICs will be established away from the $\mathrm{\Gamma }$ point where the modes are no longer degenerate.

 

Fig. 2. (a) Band structure of the metasurface and (b) the corresponding electromagnetic field distribution of eigenmodes at the $\mathrm{\Gamma }$ point. The structural parameters are: W₁= W₂= 150 nm, T = D = H = 150 nm, L = 500 nm, Px = 1300 nm, Py = 1050 nm.

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In order to better understand the optical response of the proposed metasurface, the transmission spectra are investigated with a TE-polarized plane wave propagating along the z-direction. The calculated transmission spectra with varying W₂ are shown in Fig. 3(a), which exhibits two lineshapes corresponding to EIT resonance around 1685 nm and Fano resonance around 1980 nm. When the width of the triple Si bar is equal to 150 nm, the linewidth of the resonance turns to zero and the Q-factor is infinite, which means the establishment of the BICs where no leaky energy from the resonances to the free space attributing to the destructive inference of the guiding modes and grating modes. When the width of the middle Si bars is different from the side Si bars, one can find the Q-factor and linewidth of the EIT resonance and Fano resonance is finite, which means the BICs collapse into quasi-BICs. The linewidth of resonance peaks of the quasi-BICs widen when the W₂ is away from 150 nm. To further confirm the existence of the BICs, we calculate both the transmission spectra and the eigenmodes spectra with W₂ as depicted in Figs. 3(b) and 3(c). For clarity, we define the two eigenmodes as quasi-BIC1 and quasi-BIC2, which radiate as EIT resonance at short wavelength and Fano resonance at long wavelength, respectively.

 

Fig. 3. The transmission spectra and eigenmode analysis of a metasurface under TE-polarization incident light. (a) Evolution of the transmission spectra with different W₂. Transmission spectra and eigenmodes spectra for (b) BIC1 radiating as EIT resonance and (c) BIC2 radiating as Fano resonance with error bars represent the magnitude of the mode inverse radiation lifetime. All the resonance energy is completely restricted to a bound state in the radiation continuum at W₂= 150 nm, which are defined as the BICs. When the linewidths appear in the spectra (W₂≠150 nm), they collapse into quasi-BICs.

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According to the eigenmode analysis shown in Figs. 3(b) and 3(c), the metasurface supports two resonance-trapped BICs that are unstable against perturbations. These leaky modes can be represented by a complex eigenfrequency $\omega = {\omega _0} - i\gamma$, where ${\omega _0}$ and $\gamma$ are resonant frequency and leakage rates of modes, respectively. A slight perturbation induces energy leakage of an ideal bound state, resulting in a quasi-BIC with the observable spectral feature of finite linewidth. The error bars indicate the resonance linewidth/leakage rates, and both eigenmodes disappear at W₂= 150 nm. It is worth mentioning that BIC1 establishes in a cross-section of two modes. Herein, the mode inverse radiation lifetime is calculated for the hybrid mode in the transmission peak, where the dispersion rapidly changes and is helpful for the slow light devices. We can observe from the transmission spectra that BIC1 and BIC2 with infinite Q-factors at W₂= 150 nm transform into high Q-factors quasi-BICs, which verify the eigenmode analysis of BICs. It should be pointed out that the double-BICs are resonance-trapped BICs since the varying W₂ does not break the in-plane inversion symmetry of the metasurface.

The Q-factors of the quasi-BICs can be extracted and further defined as $Q = {{{\omega _0}} / {2\gamma }}$. Since the resonance linewidths of BIC disappear, the general method to verify the existence of BIC is to track the diverging trajectory of Q-factor from the quasi-BICs. As shown in Figs. 4(a) and 4(b), it reveals that the Q-factors of two quasi-BICs of both the EIT and Fano modes tend to be infinite at W₂= 150 nm and the Q-factors diverge when W₂ changes away from 150 nm, which validates the existence of BICs

 

Fig. 4. Extracted Q-factors of two quasi-BICs for (a) EIT and (b) Fano mode.

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The electric field distributions with different W₂ are calculated to investigate the generated mechanism of the quasi-BICs and BICs in the metasurface. One should note that the incident light is above the wavevector of the guiding mode in the bare guiding layer, which means that the guiding mode cannot be excited and leak into the free space [55]. When the grating layer is mounted above the guiding layer, the incident light diffracts in the grating layer and the wave vector of the incident light is modulated to match the guiding modes, leading to the radiation of the guiding modes. For the quasi-BIC1 in the form of EIT resonance, one can find that one of the guiding modes named transverse-electric (TE_4,1) mode is excited and radiating to free space. The middle Si-bar grating provides a leaky channel to the TE_4,1 mode in the shorter wavelength of the dip of EIT resonance while the longer wavelength of TE_4,1 mode is excited by side Si-bar grating as shown in Figs. 5(a) and 5(b).

 

Fig. 5. Electric field distributions in the x-z plane at the resonant wavelength. (a), (b) for two resonant dips of quasi-BIC1 at W2 = 100 nm, (c) for BIC1 in the dip of the transmission spectra at W2 = 150 nm, and (d) for the resonant dip of quasi-BIC2 at W2 = 100 nm. The white arrows indicate the magnetic field directions.

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When the W₂= W₁= 150 nm, the electric field localizes in the grating layer, and TE_4,1 mode cannot be excited at the 1689 nm as shown in Fig. 5(c), which leads to the establishment of BIC. The phenomenon can be explained as follow: the period of grating in the x-direction decreases rapidly from the triple Si bars to a single Si bar as the effect of them are the same in the unit cell when W₂= W₁= 150 nm, leading the wavevector of grating increases sharply. The matching wavevector frequency point between the TE_4,1 mode and the refraction light will blue shift out of the investigating frequency region, which means that the TE_4,1 mode cannot be excited from 1600 nm to 2100 nm when W₂= W₁= 150 nm. Thus, only the grating mode and single transmission dip remain as shown in Fig. 5(c) and Fig. 3(a). In addition, the electric field distribution of quasi-BIC2 is depicted in Fig. 5(d) where the TE_2,1 mode is excited and radiating to the free space through all the triple Si-bars. Herein, the central wavelength of quasi-BIC2 is larger than quasi-BIC1. For the TE_2,1 mode, the wavevector is larger than the TE_4,1 mode at the same frequency in the guiding layer, Thus, the TE_4,1 mode is excited by grating at a short wavelength, while the TE_2,1 at a long wavelength with the same grating [56].

Using the coupled harmonic oscillator model, the strong coupling regime in the all-dielectric metasurface can be described to explore the formation of quasi-BIC1. The respective contributions of the grating mode and guiding mode can be calculated as [57]:

To further explain the EIT resonance represented as quasi-BIC1, the coupled Lorentz oscillator model is utilized to analyze the interaction between the grating modes and guiding modes. The grating mode in our structure is defined by oscillator 1. The TE_4,1 mode is defined by oscillator 2, which is a dark mode and cannot be directly excited by the incident light. The coupled two-oscillator system to analyze the interaction between two modes is as follows [15]:

As the energy dissipation is proportional to the imaginary part of susceptibility ${\chi _i}$, the transmission T can be described as:

In Figs. 6(a)–6(e), we define the difference between the simulation and the fitting as the error bars. Focus on the transparency window, the error bars are small. The theoretical fitting of transmission spectra marked by the red curve reveals good agreements with simulation results of different W₂ values represented by black points. In addition, the fitting parameters of different W₂ are provided in Fig. 6(f). It is apparent that ${\gamma _2}$ show very slight variations with W₂, while the detuning frequency $\delta $, the damping rate ${\gamma _1}$, and the coupling coefficient $\kappa $ of the grating mode display significant variations. In this case, ${\gamma _2}$ gets several times smaller than ${\gamma _1}$, which means that the excitation of the guiding mode can reach a higher Q-factor. As the W₂ moves away from 150 nm, $\delta $ and $\kappa $ increase and ${\gamma _1}$ decreases due to the stronger coupling of TE_4,1 mode and grating mode, which refers that the switchable establishment between BIC1 and quasi-BIC1 and the active modulation of EIT resonance can be attributed to the change of the damping rate of the grating mode. When W₂= 150 nm, ${\gamma _1}$ becomes large enough to completely suppress the TE_4,1 mode excitation and hampers the destructive interference between the TE_4,1 mode and grating modes, which eventually leads to the disappearance of the EIT resonance peak. Herein, the only grating mode can remain as the ${\gamma _2}$ turns to zero when W₂= 150 nm, which validates the above analysis in Fig. 5(c).

 

Fig. 6. (a)-(e) The numerical transmission spectra (black dots) and corresponding analytically fitted (red curves) transmission spectra with the error bars of the proposed metasurface with different W₂. (f) The fitting parameters $\delta $, $\kappa $, ${\gamma _1}$ and ${\gamma _2}$ of coupled two-oscillator model with the variation of W₂.

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We use cartesian electromagnetic multipole decomposition to analyze the contribution of different multipole components of quasi-BICs in far-field radiation, which can be defined as:

As shown in Figs. 7(a) and 7(b), one can find that the scattering power of the ED moment and the EQ moment are dominant for quasi-BIC1 while the MQ moment and the TD moment for quasi-BIC2 play a dominant role at W₂= 100 nm. For quasi-BIC1, the ED moment was attributed to the guiding mode waves with energy fluxes. The generation of transmission dips around 1670 nm is mainly due to the destructive interference of the ED moment and the EQ moment owing to their similar far-field scattering patterns. Due to the large scattering power difference between the ED moment and the EQ moment, the destructive interference effect weakens creating a peak between the two dips. For quasi-BIC2, the scattering of the MQ moment is significant, while the scattering of the ED moment is suppressed dramatically. Owing to the scattered power difference between the MQ moment and the TD moment, the radiation channel is provided, resulting in a Fano resonance around 1976 nm.

 

Fig. 7. Multipole decomposition analysis of the metasurface of (a) quasi-BIC1 and (b) quasi-BIC2 when W₂= 100 nm, respectively.

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Then the transmission spectra with a TM-polarized plane wave propagating along the z-axis are calculated to investigate the responses of the metasurface to polarization. As shown in Fig. 8(a), the transmission spectra with varying W₂ infer that two Fano resonances excite around 1800 nm and 1970 nm. Compared with Fig. 3(a), it can be noted that the switch between Fano and EIT mode BIC is realized. It is obvious that the switch between EIT and Fano modes can be achieved not only by altering different wavelengths of the same polarization but also by converting polarization. As the geometric parameter W₂ changes from 50 nm to 250 nm, two Fano resonances with vanished linewidth demonstrate the existence of the BIC states with infinite Q-factors at W₂= 150 nm in Figs. 8(b) and 8(c). For clarity, we define the two states as BIC3 and BIC4. According to the eigenmode analysis, the linewidths of the two resonances become disappeared at W₂= 150 nm, and the corresponding Q-factors also diverge as the W₂ gradually changes, validating two resonance-trapped BICs. The Q-factor of the metasurface as a function of the W₂ is presented in Fig. 8(d), it becomes an ideal BIC at W₂= 150 nm.

 

Fig. 8. The transmission spectra and eigenmode analysis of the metasurface under TM-polarization incident light. (a) Evolution of the transmission spectra with different W₂. Transmission spectra and eigenmodes spectra for (b) BIC3 and (c) BIC4 radiating as Fano resonances with error bars represent the magnitude of the mode inverse radiation lifetime. All the resonance energy is completely restricted to a bound state in the radiation continuum at W₂= 150 nm, which are defined as the BICs. (d) Extracted Q-factors of two quasi-BICs for Fano modes respectively.

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To gain insight into the generated mechanism of two resonances, we analyze the contributions of different multipole components in the far-field radiation. The magnetic field distribution in the grating layer is excited around 1790 nm as shown in Fig. 9(a), which leads to the formation of quasi-BIC3. Different from the TE_4,1 mode in TE-polarization, the guiding mode named transverse-magnetic (TM_2,1) mode is excited inside the guiding layer in TM-polarization, resulting in the switching from EIT to Fano resonance at the longer wavelengths [59,60]. According to the Ref. [60], the relationship of the propagation constant of the eigenmode is ${\beta _{\textrm{T}{\textrm{E}_{2,1}}}}$>${\beta _{\textrm{T}{\textrm{M}_{2,1}}1}}$>${\beta _{\textrm{T}{\textrm{E}_{4,1}}}}$ above the light line under the same wavelength in the grating, which means that the central wavelength of the resonance dominated by guided modes is ${\lambda _{\textrm{TE}2,1}}$>${\lambda _{\textrm{TM}2,1}}$>${\lambda _{\textrm{TE}4,1}}$ under the same excitation condition (the same grating). Therefore, the TM_2,1 mode is excited at a longer wavelength in contrast to the TE_4,1 mode. Similar quasi-BIC features are observed in the Fano resonance at the long wavelengths in different polarization directions, which means they originate from the same mechanisms. The electric field distributions clearly display the mode pattern of the quasi-BICs in Fig. 9(b). The quasi-BIC4 is excited around 1968 nm while quasi-BIC2 is excited around 1976 nm. As the order is three, the effective refractive index of TM mode is smaller than the TE mode, so the resonance wavelength of TE_2,1 mode is greater than it of TM_2,1 mode under the excitation of the same propagation constant. The dominant role of scattering power distribution changes from the ED moment to the MD moment, and the MD moment and EQ moment play a dominant role in forming quasi-BIC3 at W₂= 100 nm, as shown in Fig. 9(b). Meanwhile, it can be seen in Fig. 9(d) that the coupling affects the formation of quasi-BIC4 is still dominated by the TD moment and MQ moment at W₂= 100 nm. Owing to the scattered power difference between the two dominant moments, respectively, the radiation channel is provided, resulting in Fano resonances around 1790 nm and 1967.5 nm.

 

Fig. 9. (a) Magnetic and (c) electric field distributions in the x-z plane at the resonant wavelength for both quasi-BIC1 and quasi-BIC2 at W₂= 100 nm. (b), (d) Multipole decomposition analysis of the metasurface, which shows the scattered power contribution of different multipoles at W₂= 100 nm.

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Characterized by the group time delay, the slow light phenomenon accompanied by the spectral response is a significant feature of the EIT resonance response for the metasurfaces. Based on the simulated phase shift $\phi $ spectra, the group delay is described as $G\textrm{ = } - \partial \phi (\omega )/\partial \omega$, where $\phi (\omega )$ and $\omega = 2\pi f$ represent the transmission phase and angular frequency of the light, respectively [61]. Figure 10(a) plots the transmission spectra of the light passing through the metasurface with different structure parameters W₂. It can be obviously shown that the large group time delay occurs near the transparent window. When the width of the middle bar is close to W₂= 150 nm gradually, the tunable group delay is greatly enhanced correspondingly in Fig. 10(b). Figure 10(c) shows the wavelength of transmission peaks and corresponding group delays with W₂ varying from 145 nm to 150 nm. As W₂ is set to 149.5 nm, the maximum group delay can run up to around 2113 ps near the transparency window. In this way, we can control the group delay accompanied by the EIT effect, which can be further investigated for slow light application.

 

Fig. 10. (a) The transmission spectra and (b) group delays of EIT-like effect under the excitation of TE-polarized incident wave, where the group delay data is plotted by logarithm. (c) The wavelength of transmission peaks and corresponding group delays with W₂ varying from 145 nm to 150 nm.

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For the fabrication of the proposed metasurface, we discuss the possibility of utilizing the standard nanofabrication techniques. The material Si and Si₃N₄ are used to epitaxially grow the 300 nm Si layer and 150 nm Si₃N₄ layer on the InP substrate. Then, using electron-beam lithography and reactive ion etching to define the 150 nm grating layer with the help of a mask. The InP substrate can be ultrasonically removed with the help of photolithography and chemical solution. Our structure can be interconnected by a network of supporting bridges without the glass substrate like the Ref. [41].

4. Conclusions

In summary, we present a numerical and theoretical analysis of double resonance-trapped BIC in the form of EIT or Fano resonance in an all-dielectric metasurface. The eigenmodes analysis reveals the existence of resonance-trapped BICs with infinite leakage and the multipole analysis exhibits the dominant pole of the resonance-trapped BICs upon the proper choice of structure parameters. To further insight into the physical mechanism of the two BICs, the coupled harmonic oscillator model and coupled Lorentz oscillator model are employed and the electric distribution is calculated, which reveals the grating can selectively excite different guiding modes in the guiding layer for the realization of the two BICs. In addition, the BIC based on Fano and EIT resonance can simultaneously exist at different wavelengths. Particularly, the group delay in the transparency window of the EIT can effectively reach 2113 ps attributing to the ultrahigh Q-factor resonance caused by the resonance-trapped BICs. Therefore, these findings provide a new feasible platform for slow light applications in the near-infrared region.