1. Introduction

Bound state in the continuum (BIC) is a localized state that coexists with the continuum spectrum. Owing to its strong localization and the infinite quality (Q) factor [1,2], BIC has recently attracted great attention. In fact, BIC is a dark mode that has an invisible scattering spectrum and thus cannot be directly measured. In contrast, a quasi-BIC with finite Q factor, collapsing from the genuine BIC, is manifested by a detectable Fano resonance in the spectrum, which has been explored for numerous applications, including sensors [3,4], lasers [5], harmonic wave generation [6] and weak displacement enhancement [7]. How to conveniently generate a BIC is important for the applications of quasi-BICs. Generally, the symmetry-protected BIC is a simple way to achieve quasi-BICs by slightly breaking the geometric symmetry. Many different structures, such as photonic crystal slabs [8–10], gratings [11,12], plasmonic structures [13] and dielectric disk chains [14], were proposed to realize quasi-BICs, while only a single quasi-BIC within a given frequency range is achieved. Rather than a single quasi-BIC, dual quasi-BICs are more noticeable due to the advantage of multiple resonances in a narrow spectrum, which offers more possibilities for designing multifunctional sensors and modulators. Recently, dual-BICs were realized in all-dielectric compound gratings [15–18] by breaking the geometric symmetry, such as the space between gratings, the width of sub-cells and the incident angle. However, the key issue for generating dual quasi-BICs is the lack of tunability, since these designed structures based on the broken geometric symmetry cannot be changed once fabricated, which limits the scope of applications in practice.

Phase-change materials (PCMs) have been recently studied to explore tunable phenomena due to the variable permittivity, which can be reversibly achieved by external stimuli, such as laser, electric and thermal control [19–21]. Inspired by this unique feature of PCMs, here we employ the material asymmetry to design dual quasi-BICs in all-dielectric compound gratings. Different from the common way of the geometric asymmetry, with which these BICs are challenged by nanofabrication defects or angular deviations of incident light [4], this new way originating from the material itself can broaden the degrees of freedom of traditional optical systems, which can enhance the flexibility of BIC-based devices without structural changes in practice and dynamically modulate optical resonances of extremely high Q factors. Interestingly, we find that there are two different types of structural symmetries in the compound gratings, and their existences are determined by the geometric and material asymmetries, respectively. The breaking of the symmetry can result in the leakage of a BIC mode, i.e., a quasi-BIC. Therefore, the even and odd quasi-BIC modes in the compound gratings can appear by operating the geometric asymmetry and material asymmetry, respectively, which enables the generation of dual quasi-BICs. Moreover, owing to the adjacent-mode-coupling of the resulting dual quasi-BICs [22–24], the tunable electromagnetically induced transparency (EIT) is realized in the proposed structure by engineering the material asymmetry, which may find applications in slow light [25], high-sensitive sensors [26] and optical buffers [27]. Considering the lossless PCM of Sb₂Se₃ at the communication wavelength [19], we systematically engineer the Sb₂Se₃-based compound gratings to demonstrate these tunable effects. By developing the high-order Fourier expansion of the grating layer for the rigorous coupled wave analysis (RCWA) [28], analytical calculations are used to reveal these findings, which are consistent with numerical simulations.

2. High-order Fourier expansion and guided mode resonance condition in the all-dielectric compound gratings

Figure 1(a) schematically illustrates the configuration of the all-dielectric compound gratings. The grating layer with a period of p and thickness of ${t_g}$, shown in Fig. 1(b), can be regarded as a combination of two individual gratings with permittivities of ${\varepsilon _{g1}}$ and ${\varepsilon _{g2}}$. Here ${\varepsilon _{g1}}$ realized by a common dielectric is fixed and ${\varepsilon _{g2}}$ made of a PCM (Sb₂S₃, Sb₂Se₃ or GSST [19–21]) is variable. The material asymmetry ($\beta $) is defined as ${\varepsilon _{g2}} = ({1 + \beta } ){\varepsilon _{g1}}$. If $\beta = 0$, no material asymmetry is involved. The geometric asymmetry is realized by altering the distance between the two separate gratings, and the original space ${l_b}$ will become $l_b^{\prime}$ that satisfies $l_b^{\prime} = ({1 + \alpha } ){l_b}$, where $\alpha $ represents the geometric asymmetry. If $\alpha = 0$, the system turns to the initial symmetrical state. There are duty ratios describing the structure, i.e., ${f_a} = {l_a}/p,{f_b} = {l_b}/p\; $ and ${f_c} = 1 - 2{f_a} - {f_b} = {l_c}/p$, in which ${l_a}$ is the width of the dielectric blocks. Considering the transverse electric (TE) wave (electric field only polarized along the $z$-axis normally incident on the structure with $\alpha = 0$ and $\beta = 0$, the odd and even BIC modes, defined by the symmetric axis of the supercell are revealed in Fig. 1(c). Generally, the eigenfrequency is a complex number, i.e., $\mathrm{\omega } = {\omega _0} + i\gamma $, and here the eigenfrequencies of two BIC modes with $\; \gamma = 0$ are ${\omega _{odd}}$ = 339.44 THz and ${\omega _{even}}$ = 342.26 THz.

 

Fig. 1. (a) Schematic illustration of the 2D all-dielectric compound gratings, composed of two gratings and a HfO₂ waveguide (${\varepsilon _{wg}} = {\; }{1.98^2}$) layer on a SiO₂ substrate (${\varepsilon _s} = {1.48^2}$). The permittivities of gratings are ${\varepsilon _{g1}} = {1.63^2}$ and ${\varepsilon _{g2}}$. (b) The cross-section of the compound gratings, the thicknesses of gratings and waveguide layer are ${t_g} = 160\; nm$ and ${t_{wg}} = 270\; nm$. The widths of gratings are both ${l_a} = 75\; nm$. The period is $p = 500\; nm$ and the original space of two gratings is ${l_b} = 175\; nm$. (c) Electric field distributions of the odd and even BIC modes in the unit cell with $\alpha = 0{\; }$ and $\beta = 0$ at normal incidence.

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According to the coordinate axis in Fig. 1(b), the geometric symmetry can be broken by setting $\alpha \ne 0$ and the duty ratios are modified as $f_a^{\prime} = l_a^{\prime}/p = {f_a},f_b^{\prime} = l_b^{\prime}/p\; $ and $f_c^{\prime} = 1 - 2f_a^{\prime} - f_b^{\prime} = l_c^{\prime}/p$. In this case, the high-order Fourier harmonics should be considered and the permittivity of the grating layer can be expressed as [29],

Subsequently, to obtain the real and imaginary parts, Equation (2) can be simplified as,

From above, the asymmetric parameters $\alpha $ and $\beta $ can both modulate ${\varepsilon _n}$. Compared with previous works [28,29], Eq. (3) is a general expression involved in RCWA for analytical calculations.

Generally, the BIC modes of the 2D all-dielectric compound gratings originate from the guided mode resonance (GMR) [15], i.e., the conservation of the tangential wave vector between the guided mode in the waveguide layer and the incident light. The tangential component of the wave vector of the incident light in air is ${k_x} = {k_0}sin\theta $, where ${k_0}\; $ is the wave vector in air and $\theta $ is the incident angle. In the waveguide, the propagation constant of the guided mode is $\; {\beta _{GM}}$, which in general is larger than ${k_0}$. Without the grating layer, the incident light cannot couple with the guided mode. However, there is an additional wave vector ${G_m}$ provided by the grating layer, the GMR condition is satisfied as,

3. Tunable dual BICs and EITs enabled by the broken material symmetry in all-dielectric compound gratings

Actually, the two BICs shown in Fig. 1(c) are symmetry-protected by the parameters $\alpha $ and β, respectively. In such a more specific case, either the material symmetry or the geometric symmetry remains unbroken, i.e. $\alpha = 0\; \textrm{and}\; \beta = 0$. As a result, the even and odd BICs exist together, yet at different eigenfrequencies. To determine the dominant parameter of each BIC, we have simulated the corresponding reflectivity spectrum and the Q factor variation of each BIC, respectively. As displayed in Fig. 2(a), for $\beta = 0$, the dependence of the reflectivity spectrum on the geometric asymmetry α is investigated. The even BIC only appears at the point of $\alpha = 0$ (about 342.26 THz), which implies that the even BIC is subject to the geometric symmetry. Given that the geometric symmetry is broken, the even BIC will vanish and become a quasi-BIC. When $\alpha = 0$, we show the reflectivity spectrum with the change of $\beta $ in Fig. 2(b). Considering the available permittivity of a PCM, a feasible range of $\beta $ could be taken from -0.2 to 0.2. Analogously, an odd BIC is observed at $\beta = 0$ (about 339.44 THz). Consequently, it is fair to say the odd BIC is protected by $\beta $. Based on the eigen analysis of COMSOL, the Q factor of the eigenmode is calculated by $Q = {\omega _0}/2\gamma $. The Q factors are infinite at $\alpha = 0\; \textrm{and}\; \beta = 0$, and they gradually decrease with the increase of asymmetric parameters (see Figs. 2(c) and 2(d)). Moreover, the Q factors are linear functions of the inverse square of asymmetric parameters, as shown by the fitting lines of ${Q_{even}} \approx 406{\alpha ^{ - 2}} + 60$ and ${Q_{odd}} \approx 460{\beta ^{ - 2}} + 401$ that well match with the simulated results (see Figs. 2(e) and 2(f)). These results reveal that there are two symmetry-protected BICs, and each BIC is protected by the correlated symmetry.

 

Fig. 2. The numerically simulated reflectivity versus the corresponding frequency (a) when $\beta = 0$, $\alpha \in [{ - 1,1} ]$, (b) when $\alpha = 0$, $\beta \in {\; }[{ - 0.2,0.2} ]$, inset pictures represent the corresponding BIC modes. The Q factor varies with (c) the geometric asymmetry $\alpha $ and (d) the material asymmetry $\beta $. The Q factor as a linear function of (e) ${\alpha ^{ - 2}}$, (f) ${\beta ^{ - 2}}$, the solid lines are the fitting results while the dotted lines are the simulated results.

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The above analyses enable an individual modulation to obtain a specific quasi-BIC, thus offering an effective way to realize dual quasi-BICs by destroying both symmetries. When $\beta = 0.1$, as depicted in Fig. 3(a), the reflectivity spectrum has been calculated with the variation of $\alpha $. Compared with Fig. 2(a), an additional spectral line emerges due to $\beta \ne 0$. The odd BIC will turn into an odd quasi-BIC. However, at $\alpha = 0$, the even BIC remains unchanged. Because the even BIC mode is protected by $\alpha $, yet not determined by $\beta $. Except for this point, the even BIC will be transformed into a quasi-BIC for $\; \alpha \ne 0$. Then, dual quasi-BICs exist in the system. From the reflectivity spectrum, the odd quasi-BIC mode has a narrower bandwidth thus leading to a higher Q factor. With the increase of $|\alpha |$, both the eigenfrequency and Q factor of the even mode generally decrease. While for the odd quasi-BIC, the eigenfrequency tends to grow, the change of the Q factor is irregular (see part II in Supplement 1). Next, we will discuss another case to achieve tunable dual quasi-BICs. When $\alpha = 0.2$, the even quasi-BIC is achieved and an additional upper spectral line is found in Fig. 3(b) compared with Fig. 2(b). Analogously, there is an odd BIC mode at $\beta = 0$ and the odd quasi-BIC can be achieved by setting $\beta \ne 0$. In this case, the eigenfrequency of the odd quasi-BIC decreases with the increase of $\beta $ and the Q factor gradually goes down with the increase of $|\beta |$. While the eigenfrequency of the even quasi-BIC is relatively smooth and has a roughly unchanged Q factor. Therefore, the even and odd quasi-BIC modes can be modulated by the geometric asymmetry and material asymmetry respectively. It is because there are two different types of structural symmetries (the geometric and material symmetry) in the compound structure. The odd and even BICs are symmetry-protected by the symmetric axis I and the symmetric axis II, respectively (see Fig. S6 of Supplement 1). When one of them is broken, the corresponding symmetric axis disappears and the other still exists, leading to the generation of the corresponding single quasi-BIC mode. For example, when the material asymmetry is broken, we find that the axis I disappears, but the axis II exists as well. As a result, the odd BIC mode turns into a quasi-BIC that can be tuned by the material asymmetry, and the even BIC mode remains, thus disappearing in the scattering spectrum (see Fig. 2(b)). When the symmetries of geometry and material are broken, the symmetric axes I and II both disappear, leading to the generation of dual quasi-BICs, which are both tuned by the material asymmetry (see Fig. 3(b)). The individual modulation of two BIC modes enables the generation of dual quasi-BICs in a flexible way, which may allow the development of novel optical devices, such as sensors [30,31], lasers [32] and absorbers [33,34]. More details for verifications and explanations for BICs/quasi-BICs caused by two different asymmetries are discussed in part III of Supplement 1.

 

Fig. 3. The numerically simulated reflectivity spectra for (a) $\beta = 0.1$, $\alpha \in [{ - 1,1} ]$ and (b) $\alpha = 0.2$, $\beta \in [{ - 0.2,0.2} ]$. The analytically calculated real parts of the eigenfrequencies for (c) $\beta = 0.1$, $\alpha \in [{ - 1,1} ]$ and (d) $\alpha = 0.2$, $\beta \in $ $[{ - 0.2,0.2} ]$.

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In fact, dual quasi-BICs can occur in other conditions by breaking these two symmetries concurrently. Due to the reconfigurable feature of PCMs, the material asymmetry $\beta $ can be dynamically engineered to achieve tunable dual quasi-BICs. A three-dimensional scatter diagram is depicted in Fig. S7 of Supplement 1. Besides the simulations, we have analytically derived ${\omega _0}$ based on the RCWA and GMR theories. By obtaining the equivalent permittivity ${\varepsilon _{eff}}$ of the grating layer from the Fourier coefficient ${\varepsilon _n}$ in Equation (3), the corresponding ${\omega _0}$ of odd and even BICs/quasi-BICs are calculated by solving the dispersion relationship of the TE₀ guided mode, in which the ±1^st order GMR condition is considered (see part I in Supplement 1). The analytical results (see Figs. 3(c) and 3(d)) agree with the simulated ones in Figs. 3(a) and 3(b). We can find that the odd mode is related to the +1^st order GMR condition and the even mode is ensured by the -1^st order GMR condition. Although the resonant frequencies of the even BIC in Fig. 3(a) and the odd BIC in Fig. 3(b) can be calculated in theory, only its real part is obtained, which implies that the quasi-BIC/BIC modes cannot be distinguished by the analytical calculations.

More intriguingly, the two branches of the eigenmodes in Figs. 3(a) and 3(c) can intersect with each other and then separate. Near the overlap region, the EITs appear due to the mutual coupling of a high Q mode (odd quasi-BICs) and a low-Q mode (even quasi-BICs) [16,23,24]. Remarkably, such EITs can merely take place at normal incidence. Because the frequency difference between the dual quasi-BICs will increase dramatically as the growth of the incident angle so that the mode interaction fades away, leading to the disappearance of EIT (see part V in Supplement 1). Analogous to the dual quasi-BICs, such EITs share the tunable feature. Considering the feasible permittivity of PCMs, $\beta $ is taken with a step of 0.1 in the range of $\beta \in $ $[{ - 0.2,0.2} ]$, the corresponding transmission spectra have been simulated with $\alpha $ ${\in} $ [0.6,0.9] (see Figs. 4(a)–4(d)). Apparently, the EIT phenomenon exists in each case, and the Q factor correlates well with the two asymmetries ($\alpha $ and $\beta $). However, here we consider a fixed $\alpha $ to achieve tunable EITs by altering $\beta $ as the geometric asymmetry cannot be changed for a fabricated structure. Since the Q factor at $\alpha = 0.76$ is maximum in the EIT range, we show the evolution of transmission for different $\beta $ in Fig. 4(e), where the EIT is modulated by$\; \beta $ and disappears for $\beta = 0$. The corresponding Q factor declines dramatically with the growth of $|\beta |$. Here, the Q factor is calculated by ${f_p}/\Delta f$, where ${f_p}$ is the frequency at the middle transmission peak and $\Delta f$ is the corresponding full width at half maximum. With the increase of $\beta $, the ${f_p}$ slightly shifts about from 340.88 to 341.42 THz (see the black dash line in Fig. 4(e)). The major reason for such a shift is that the eigenfrequencies of the odd quasi-BIC and even quasi-BIC both decreases slightly with the raise of $\beta $, as demonstrated by analytical calculations and numerical simulations. The revealed EIT is explained from the coupled Lorentz oscillator model [18,35] (see details in part V of Supplement 1). In addition, we further demonstrate the EIT effect from its transmission spectrum, phase change and electric field distribution (z component) in the case of α = 0.76 and β = 0.2 in Fig. S10.

 

Fig. 4. Transmission spectra with (a) $\beta ={-} 0.2$, (b) $\beta ={-} 0.1$, (c) $\beta = 0.1$ and (d) $\beta = 0.2$. (e) The evolution of the numerical (solid curves) and analytical (circles) transmission spectra with different $\beta $ for $\alpha = 0.76$. The black dash line shows the slight shift of ${f_p}$.

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We further design the feasible compound gratings consisting of the practical PCM to reveal the existence of tunable dual quasi-BICs and EITs. Owing to the low loss (the extinction coefficient $k = 0$) at the communication wavelength (1550 nm) [19], Sb₂Se₃ is a good candidate to design the all-dielectric compound gratings for the above tunable effects. The permittivities of Sb₂Se₃ in amorphous (A) and crystalline (C) states are $\varepsilon _{g2}^A\; $= (3.285 + 0i)² and $\varepsilon _{g2}^C\; $= (4.050 + 0i)², respectively. By operating the reversible process between amorphous and crystalline states, the refractive index of Sb₂Se₃ is modulated by the crystallization rate (CR) (see Fig. 5(a)). The CR could be changed by applying a pulse voltage of proper magnitude and pulse duration on the nano-heater [36]. Considering the variable range of the refractive index of Sb₂Se₃ (grating II), the medium of grating I is selected as silicon (${\varepsilon _{g1}} = {3.48^2}$). As a result, the material asymmetry $\beta $ can be tuned from -0.109 to 0.354, corresponding to $\textrm{CR} \in [{0{\%},100{\%}} ]$. By carefully engineering the geometry and permittivities of three layers, the Sb₂Se₃-based compound gratings could be realized. The waveguide layer is chosen as ${\varepsilon _{wg}} = {3.4^2}$ (GaAs) and the substrate is ${\varepsilon _s} = {2.6^2}$ (SiC), which are both lossless near the 1550 nm [37,38]. The thicknesses of the grating and waveguide layers are ${t_g} = 220\; nm$ and ${t_{wg}} = 335\; nm$, respectively. The widths of two gratings are both ${l_a} = 75\; nm$ and the period is $p = 500\; nm$. For example, if the geometric asymmetry is $\alpha = 0.2$, the reflectivity spectrum is shown in Fig. 5(b). We can find that when the geometric asymmetry is fixed, dual quasi-BICs can be modulated by the material asymmetry $\beta $. When $\beta ={-} 0.109$ (amorphous state) and $0.354$ (crystalline state), the reflectivity spectra are shown in Figs. 5(c) and 5(d), respectively. It is obvious that the EITs occur in the range of $|\alpha |\in [{0.7,0.9} ]$ . To get the maximum Q factor, $\alpha $ is chosen to be 0.8 and the evolution of the transmission spectra for the Sb₂Se₃-based compound gratings with $\textrm{CR} = 0{\%},{\; }25{\%},{\; }50{\%},{\; }75{\%}$ and $100{\%}$ (corresponding to $\beta = {\; } - 0.109,{\; } - 0.022,{\; }0.081,{\; }0.204$ and $0.354$) is shown in Fig. 5(e). With these selected CRs in Fig. 5, the Q factor of the EIT declines dramatically from $1.76 \times {10^5}$ (CR = 25%, $\beta ={-} 0.022$) to $1.5 \times {10^3}$ (crystalline state, $\beta = 0.354$) with the increase of $|\beta |$, which shows a tunable range that covers two orders of magnitude. This tunable range could be further increased by choosing other appropriate CRs or $\alpha $. The tunable EIT effect at optical wavelengths may boost applications [25–27] in slow light, high-sensitive sensors and optical buffers. The frequency of the EIT peak also has a slight shift owing to the decreased eigenfrequencies of the two modes when $\beta $ goes down. Additionally, the corresponding evolution of transmission spectra for the Sb₂Se₃-based compound gratings with $\alpha = 0.76$ is shown in part VIII of Supplement 1. Therefore, owing to the coupling of two quasi-BICs, the proposed Sb₂Se₃-assisted scheme can realize the tunable EIT effect.

 

Fig. 5. (a) Refractive index and extinction coefficient of Sb₂Se₃ with the change of CR. The reflection spectrum of the Sb₂Se₃-based all-dielectric compound gratings (b) when $\alpha = 0.2$ and $\beta \in [{ - 0.109,0.354} ]$, (c) when $\beta ={-} 0.109$ (amorphous state) and $\alpha $ ${\in} $ [-1,1], (d) when $\beta = 0.354$ (crystalline state) and $\alpha \in [{ - 1,1} ]$. (e) The evolution of the numerical (COMSOL, solid curves) and analytical (RCWA, circles) transmission spectra for the Sb₂Se₃-based compound gratings with $\beta = {\; } - 0.109,{\; } - 0.022,{\; }0.081,{\; }0.204$ and $0.354$ when $\alpha = 0.8$. The dashed line represents the tiny shift of the peak (from 193.69 THz to 193.38 THz).

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4. Conclusion

Dual quasi-BICs in such all-dielectric compound gratings have been demonstrated by introducing an additional degree of freedom of the material asymmetry, apart from the geometric asymmetry. Interestingly, the even and odd BIC modes in the structure are individually determined by the geometric asymmetry and the material asymmetry, owing to the two types of structural symmetries in the compound structure. When both symmetries are broken, the odd and even quasi-BICs can take place. Owing to the dynamic reconfiguration capability of PCMs, the generated dual quasi-BICs here possess a superior performance of tunability. Particularly, the EIT with tunable Q factor, which results from the mode coupling of dual quasi-BICs, can be realized by controlling the material asymmetry $\beta $. The practical Sb₂Se₃-based compound gratings structure is also demonstrated to achieve tunable quasi-BICs and EITs by changing the crystallization rate. With these advantages, our results may boost the capacity to dynamically modulate the quasi-BIC in a passive nanostructure, which may have fascinating applications in sensing, lasing, switching, and so on.