1. INTRODUCTION

Resonant dielectric nanostructures have received significant attention in the past decade due to their ability to enhance light–matter interaction [1,2]. By exciting Mie resonant modes, the electric and magnetic fields can be intensified and confined within the interior of dielectric nanostructures. Additionally, Mie resonances offer the potential to manipulate the amplitude, phase, and polarization of light, which have been exploited in the design of metasurfaces for wavefront control [3,4]. Among various Mie resonant modes, electric and magnetic multipoles have been extensively studied in dielectric nanostructures. In recent years, toroidal multipoles, which constitute a distinct family of electromagnetic multipoles, have attracted considerable interest owing to their unique electromagnetic characteristics [5,6]. The toroidal dipole (TD) mode, in particular, is an intriguing electromagnetic excitation characterized by poloidal currents and circular magnetic fields. Although the contribution of the TD mode is typically weak in natural materials, it can be significantly enhanced in elaborately designed dielectric nanostructures [7–10]. Initial investigations of dielectric nanostructures revealed low-quality ${Q}$-factors for electric dipole (ED), magnetic dipole (MD), and TD resonances, which limited the intensity of the light–matter interaction. However, recent studies have shown that, by carefully adjusting the geometric parameters of dielectric nanostructures, these Mie resonant modes can be transformed into bound states in the continuum (BICs) with infinite ${Q}$-factors [9–12]. BIC is a localized state that exists within the radiation continuum but possesses an infinite radiative lifetime [13–15]. An ideal BIC cannot be directly excited by incident light; instead, it can be coupled to incident light by converting it into a quasi-BIC, which exhibits a high-${Q}$ resonance in the spectrum. Benefiting from the ultrahigh ${Q}$-factor and extreme electromagnetic field enhancement, BICs based on dielectric nanostructures have been utilized in various applications such as low-threshold lasers [16,17], ultrasensitive sensors [18,19], and harmonic generation [20,21].

 

Fig. 1. (a) Schematic of the hybrid dielectric metasurface integrating Si nanostructures with ${{\rm VO}_2}$. (b) Top and (c) side views of a unit cell of the composite nanostructures. (d) Real part (${\varepsilon _1}$) and (e) imaginary part (${\varepsilon _2}$) of the dielectric constants of ${{\rm VO}_2}$ in the insulating and metallic phases.

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Although resonant dielectric nanostructures have made significant advancements, the optical properties of most dielectric nanostructures remain fixed and difficult to be tuned dynamically. To address this limitation, active metasurfaces have emerged as a promising solution for achieving dynamic tunability [22,23]. One approach to creating active metasurfaces involves integrating them with optically tunable materials. Graphene [24], semiconductors [25], transparent conducting oxides [26], liquid crystals [27], and phase-change materials [28,29] have been extensively studied as tunable materials. By applying thermal, electrical, or optical excitations, the functions of metasurfaces can be dynamically tuned to meet various requirements. Among these tunable materials, vanadium dioxide (${{\rm VO}_2}$) has attracted particular attention due to its phase-change properties [30,31]. ${{\rm VO}_2}$ undergoes an insulator-metal transition around 340 K, accompanied by a structural transition from a monoclinic to a rutile structure. This phase transition is reversible and can be triggered by thermal, electrical, or optical stimuli [30,31]. Further, the refractive index of ${{\rm VO}_2}$ undergoes significant variation during the phase transition. This tunable refractive index has been leveraged in combination with metasurfaces to create a wide range of tunable optical devices [32–39]. Recent studies have demonstrated that the integration of ${{\rm VO}_2}$ with dielectric nanostructures enables the active control of ED and MD modes [40–42]. However, dynamic control of quasi-BICs is rarely investigated in such composite nanostructures. In addition, the achievement of switchable TD modes has not been reported in these works.

In this study, we propose the creation of switchable multiple quasi-BICs by combining dielectric nanostructures with ${{\rm VO}_2}$. The hybrid dielectric metasurface can support three distinct types of BICs for ${{\rm VO}_2}$ in its insulating phase. By introducing asymmetry in the width of the composite nanostructures, one BIC mode can be transformed into a quasi-BIC that exhibits a characteristic of $z$-direction TD under $x$-polarized incidence. Further, by adjusting the width under $y$-polarized incidence, two additional quasi-BIC modes with different electromagnetic properties can be generated. Additionally, the metasurface can also support a high-${Q}$ $y$-direction TD mode, which is directly excited by the $y$-polarized light. All quasi-BIC and TD modes are switched off when ${{\rm VO}_2}$ transitions to its metallic phase, as a result of increased absorption loss. The switchable multiple quasi-BICs are promising for applications in optical modulators, tunable harmonic generation, and biosensors.

2. DESIGN AND SIMULATION

The hybrid dielectric metasurface that combines silicon (Si) nanostructures with ${{\rm VO}_2}$ on a silica substrate is schematically illustrated in Fig. 1(a). The unit cell of this metasurface contains two ${\rm Si} {-} {{\rm VO}_2}$ composite nanobricks with a gap ($g$), as depicted in Figs. 1(b) and 1(c). The ${{\rm VO}_2}$ nanobricks are positioned on top of the Si nanobricks, with respective thicknesses of ${h_1}$ and ${h_2}$. The lengths of the two ${\rm Si} {-} {{\rm VO}_2}$ composite nanobricks are labeled as ${l_1}$ and ${l_2}$; the widths are denoted as ${w_1}$ and ${w_2}$. The period ($P$) is the same along the $x$ and $y$ directions. Previous studies have investigated hybrid metasurfaces with a single ${\rm Si} {-} {{\rm VO}_2}$ nanostructure in the unit cell, demonstrating tunable ED and MD modes [40–42]. However, these structures were unable to induce TD mode and BIC. In contrast, our design incorporates two ${\rm Si} {-} {{\rm VO}_2}$ nanostructures in the unit cell, leading to the discovery of several intriguing phenomena, including two types of TD modes with different dipole moment directions and three types of BICs with diverse origins of dipole moment, which have not been previously reported.

The hybrid dielectric metasurface were studied by numerical simulations using COMSOL Multiphysics software. The simulations employed periodic boundary conditions in the $x$ and $y$ directions, and a perfectly matched layer was utilized in the $z$ direction. The refractive index of Si and silica were sourced from experimental data [43], while the complex dielectric constants of ${{\rm VO}_2}$ in the insulating and metallic phases were obtained from literature [44]. As shown in Figs. 1(d) and 1(e), the real and imaginary parts of the dielectric constants of ${{\rm VO}_2}$ undergo a large variation through the phase transition. The incident light was linearly polarized and directed normally onto the metasurface. The initial structural parameters of the metasurface were set as follows: $P = {2000}\;{\rm nm}$; ${l_1} = {l_2} = {w_1} = {w_2} = {900}\;{\rm nm}$; ${h_1} = {20}\;{\rm nm}$; ${h_2} = {550}\;{\rm nm}$; and $g = {50}\;{\rm nm}$. In this case, the geometric parameters of the two ${\rm Si} {-} {{\rm VO}_2}$ nanobricks are the same, and the two ${\rm Si} {-} {{\rm VO}_2}$ nanobricks are symmetric with respect to the $y$ axis passing through the center of the unit cell. As we will discuss, by adjusting the geometric parameter of one ${\rm Si} {-} {{\rm VO}_2}$ nanobrick in the unit cell to break the structural symmetry, various types of quasi-BICs with different multipole sources can be excited.

 

Fig. 2. Transmission spectra with ${{\rm VO}_2}$ in the insulating and metallic phases under (a) $x$-polarized and (b) $y$-polarized incidence, where $P = {2000}\;{\rm nm}$, ${l_1} = {l_2} = {w_1} = {w_2} = {900}\;{\rm nm}$, ${h_1} = {20}\;{\rm nm}$, ${h_2} = {550}\;{\rm nm}$, and $g = {50}\;{\rm nm}$. The solid curves correspond to the direct calculations of the COMSOL software; the dashed curves are obtained based on the multipole decomposition. (c) Electromagnetic field distribution at the wavelength of 3117 nm, where the incident light is polarized along the $x$ direction, and ${{\rm VO}_2}$ is in the insulating phase. Electromagnetic field distributions at the wavelength of 3230 nm for (d) and 3675 nm for (e), where the incident light is polarized along the $y$ direction and ${{\rm VO}_2}$ is in the insulating phase. The $XY$ and $XZ$ planes pass through the centers of two Si nanobricks, while the $YZ$ plane passes through the center of the left Si nanobrick. Cartesian scattered powers with ${{\rm VO}_2}$ in the insulating phase under (f) $x$-polarized and (g) $y$-polarized incidence.

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3. RESULTS AND DISCUSSION

Figure 2(a) shows the calculated transmission spectra with ${{\rm VO}_2}$ in the insulating and metallic phases under $x$-polarized incidence. When ${{\rm VO}_2}$ is in the insulating phase, a dip with a Fano lineshape emerges at a wavelength of 3117 nm. The origin of this resonance can be comprehended by analyzing the electromagnetic field at the resonant wavelength. As depicted in Fig. 2(c), at the wavelength of 3117 nm, an intense electric field enhancement is observed on the top of the gap, and the magnetic field is also enhanced within the ${\rm Si} {-} {{\rm VO}_2}$ nanobricks. Moreover, magnetic fields with different intensities and opposite orientations are formed in every ${\rm Si} {-} {{\rm VO}_2}$ nanobrick, leading to a nonzero net MD moment in the $y$ direction. The analysis of the Fano resonance can be further confirmed through the utilization of the Cartesian multipole decomposition [9,10,45,46]. The Cartesian multipole moments were calculated through the following formulas [9,47]:

Figure 2(f) shows the calculated scattered powers of various multipole moments in Cartesian coordinates with ${{\rm VO}_2}$ in the insulating phase. The scattering power at 3117 nm is primarily governed by the $y$-component MD moment (${{\rm MD}_y}$), with a smaller contribution from the ED moment. Therefore, the dip is mainly attributed to the excitation of the $y$-direction MD mode. However, when ${{\rm VO}_2}$ transitions into the metallic phase, the $y$-direction MD resonance becomes weaker and disappears due to the increased absorption loss of ${{\rm VO}_2}$, as shown in Fig. 2(a). Notably, the transmission of the metasurface can also be calculated based on the multipole moments [47]:

When the incident light is polarized along the $y$-direction, there are two dips with different linewidths in the transmission spectrum for ${{\rm VO}_2}$ in the insulating phase, as presented in Fig. 2(b). The wide dip occurs at 3230 nm; the narrow dip emerges at 3675 nm. At the wavelength of 3230 nm, the magnetic field is found to align with the $x$ direction, while a circular electric field is formed in the $y {-} z$ plane, as depicted in Fig. 2(d). Consequently, the dip at 3230 nm is attributed to the excitation of the MD mode with a dipole moment along the $x$ direction. However, at the wavelength of 3675 nm, two circular electric fields with reversed directions are formed in the $x {-} y$ plane, while a circular magnetic field is formed in the $x {-} z$ plane, as presented in Fig. 2(e). Therefore, the dip at 3675 nm originates from the excitation of TD mode with a dipole moment along the $y$ direction [5,6]. The multipole origins of the two dips can be further confirmed through the multipole decomposition. As shown in Fig. 2(g), the scattering power at 3230 nm is mainly dominated by the MD moment, while the scattering power at 3675 nm is mainly dominated by the $y$-component TD moment (${{\rm TD}_y}$), which is consistent with the previous analysis according to the electromagnetic field. Thus, the two dips indeed arise from the excitation of the MD and $y$-direction TD modes, respectively. Moreover, the transmission spectra under $y$-polarized incidence can also be reconstructed by using Eq. (16), as presented in the dashed curves of Fig. 2(b). It should be noted that the TD mode has not been reported in previous works with one ${\rm Si} {-} {{\rm VO}_2}$ nanostructure in the unit cell [40–42]. Since the TD resonance has a small linewidth, we also calculated the ${Q}$-factors of the TD resonance. The TD resonance can be fitted using the Fano resonance formula [48]

Symmetry-protected BICs within the family of BICs have been extensively investigated in metasurfaces [13–15]. By manipulating the geometric parameters of metasurfaces to break their structural symmetry, BICs can be transformed into quasi-BICs, which exhibit high-$Q$ resonances in the transmission or reflection spectra. Here, we modify the width of one ${\rm Si} {-} {{\rm VO}_2}$ nanobrick in the unit cell to break the in-plane inversion symmetry. Figure 3(a) displays the calculated transmission spectra with varying widths ${w_2}$ for ${{\rm VO}_2}$ in the insulating phase under $x$-polarized incidence, while keeping the other geometric parameters constant. When the metasurface is in a symmetric configuration (${w_1} = {w_2} = {900}\;{\rm nm}$), only one resonance is observed in the transmission spectrum. The resonance has been previously analyzed and is as shown in Fig. 2(a). However, as ${w_2}$ deviates from 900 nm, a new Fano resonance emerges in the transmission spectrum. The linewidth of the resonance increases with the difference between ${w_1}$ and ${w_2}$, and the resonance experiences a slight redshift with increasing ${w_2}$. When ${w_2}$ is increased to 1100 nm, the Fano resonance shifts to 3381 nm. As depicted in Figs. 3(c) and 3(d), at the wavelength of 3381 nm, two circular electric fields with opposite directions are formed in the $x {-} z$ plane, while a circular magnetic field is formed in the $x {-} y$ plane. Consequently, the resonance arises from the excitation of a TD mode with a dipole moment along the $z$-direction [5,6]. However, when ${{\rm VO}_2}$ transitions into the metallic phase, the resonance vanishes due to the increased absorption loss of ${{\rm VO}_2}$, as shown in Fig. 3(b). It is important to note that a TD resonance can also be induced under the $y$-polarized incidence, as discussed in Fig. 2(b). However, the directions of the two types of TD moments are different. Consequently, two types of TD resonance can be achieved using the hybrid dielectric metasurfaces, and they can be switched off by the phase transition of ${{\rm VO}_2}$.

 

Fig. 3. Transmission spectra with varying widths ${w_2}$ for ${{\rm VO}_2}$ in the (a) insulating and (b) metallic phases under $x$-polarized incidence, where $P = {2000}\;{\rm nm}$, ${l_1} = {l_2} = {w_1} = {900}\;{\rm nm}$, ${h_1} = {20}\;{\rm nm}$, ${h_2} = {550}\;{\rm nm}$, and $g = {50}\;{\rm nm}$. (c) Electrical and (d) magnetic field distributions at the wavelength of 3381 nm, where ${w_2}$ is set to 1100 nm. The $XY$ and $XZ$ planes pass through the centers of two Si nanobricks. (e) Cartesian scattered powers for ${{\rm VO}_2}$ in the insulating phase under $x$-polarized incidence, with ${w_2}$ chosen as 1100 nm. (f) ${Q}$-factors of the -direction TD resonance with and without insulating ${{\rm VO}_2}$. (g) Transmission spectra with varying ${{\rm VO}_2}$ thickness (${h_1}$) for ${{\rm VO}_2}$ in the insulating phase, with ${w_2}$ chosen as 1100 nm. (h) ${Q}$-factors of the $z$-direction TD resonance with varying ${{\rm VO}_2}$ thickness, with ${w_2}$ chosen as 1100 nm.

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The excitation of TD mode can be further confirmed through the utilization of the Cartesian multipole decomposition. Figure 3(e) displays the calculated scattered powers of various multipole moments in Cartesian coordinates, with ${w_2}$ chosen as 1100 nm and ${{\rm VO}_2}$ in the insulating phase. The scattered power of the $z$-component TD moment (${{\rm TD}_z}$) exhibits a peak at a wavelength of 3381 nm, which coincides with the resonant wavelength observed in the transmission spectrum. Further, the dominance of the scattered power of the $z$-component TD moment at the wavelength of 3381 nm suggests that the resonance originates from the excitation of the $z$-direction TD mode, consistent with previous discussions on the basis of electromagnetic field distributions. The linewidth of the TD resonance decreases as the ${w_2}$ approaches 900 nm. The ${Q}$-factors of the TD resonance with varying ${w_2}$ can be obtained by fitting transmission spectra using Eq. (17). Additionally, for comparison, the ${Q}$-factors of the TD resonance were also calculated for the metasurface in the absence of ${{\rm VO}_2}$. As shown in Fig. 3(f), when ${{\rm VO}_2}$ is absent, the ${Q}$-factor of the TD resonance significantly increases and diverges as ${w_2}$ approaches 900 nm, indicating the presence of BIC [13–15]. In the presence of ${{\rm VO}_2}$, the ${Q}$-factor of the TD resonance also increases as ${w_2}$ approaches 900 nm but is lower due to the lossy nature of ${{\rm VO}_2}$ in the infrared. Nevertheless, a ${Q}$-factor of 674 is achieved when ${w_2}$ is set to 850 nm, surpassing previous works on ${\rm Si} {-} {{\rm VO}_2}$ composite nanostructures [40–42]. When ${w_2}$ is set as 800 and 1000 nm, the ${Q}$-factors of the Fano resonance are higher than the case with ${w_2}$ of 700 and 1100 nm, as shown in Fig. 3(f). The high-${Q}$ resonance is susceptible to the impact of the absorption loss of the compositive material, which can obscure the high-${Q}$ resonance. The absorption loss of ${{\rm VO}_2}$ leads to a less pronounced dip for ${w_2} = {800}$ and 1000 nm compared with the case when ${w_2} = {700}$ and 1100 nm. We also calculated the transmission spectra with different ${{\rm VO}_2}$ thickness, where ${w_2}$ is chosen as 1100 nm. As shown in Fig. 3(g), the $z$-direction TD resonance redshifts with the increase of ${{\rm VO}_2}$ thickness. However, the ${Q}$-factor of $z$-direction TD resonance decreases as the ${{\rm VO}_2}$ thickness increases, as presented in Fig. 3(h). Given that insulating ${{\rm VO}_2}$ is a lossy material in the infrared, the increased absorption loss of ${{\rm VO}_2}$ with increasing thickness leads to a reduction in the ${Q}$-factor. Therefore, the hybrid dielectric metasurfaces can support a TD-BIC mode for ${{\rm VO}_2}$ in its insulating phase. By modifying the ${w_2}$ to break the structural symmetry, the TD-BIC mode can be turned into a quasi-BIC mode with a high-${Q}$ TD resonance in the transmission spectrum. The high-${Q}$ quasi-BIC can be switched off when ${{\rm VO}_2}$ undergoes a phase transition into its metallic phase. Consequently, a switchable quasi-BIC can be achieved by controlling the phase transition of ${{\rm VO}_2}$.

Apart from the $x$-polarized incidence, we also examined the impact of ${w_2}$ on the transmission spectra under $y$-polarized incidence. Figure 4(a) displays the calculated transmission spectra with varying ${w_2}$ for ${{\rm VO}_2}$ in the insulating phase under $y$-polarized incidence. When ${w_1} = {w_2} = {900}\;{\rm nm}$, a wide dip and a narrow dip are observed in the transmission spectrum, which result from the excitation of $x$-direction MD and $y$-direction TD modes, respectively, as discussed in Fig. 2(b). However, when ${w_2}$ deviates from 900 nm, two additional dips emerge in the transmission spectra, in addition to the MD and TD resonances. When ${w_2}$ is increased to 1100 nm, two new dips occur at 3436 and 3640 nm, respectively. As presented in Fig. 4(c), at the wavelength of 3436 nm, the orientations of the electric field in the two ${\rm Si} {-} {{\rm VO}_2}$ nanobricks are opposite, while the magnetic fields are mainly concentrated in the interior of the two ${\rm Si} {-} {{\rm VO}_2}$ nanobricks and orient in the $z$-direction. Moreover, the scattering power at 3436 nm is primarily governed by the EQ moment, as presented in Fig. 4(g). Therefore, the dip at 3436 nm is mainly attributed to the excitation of the EQ mode. However, at the wavelength of 3640 nm, the electric field is enhanced in the gap between the two ${\rm Si} {-} {{\rm VO}_2}$ nanobricks and forms a loop in the $XY$ plane, while the magnetic field is highly confined within the interior of one ${\rm Si} {-} {{\rm VO}_2}$ nanobrick and orients in the $z$ direction, as illustrated in Fig. 4(d). Additionally, the scattering power of the $z$-component MD moment (${{\rm MD}_z}$) dominates at the wavelength of 3640 nm, as shown in Fig. 4(h). Hence, the dip at 3640 nm primarily arises from the excitation of the $z$-direction MD mode. Further, the enhancement in the electric and magnetic fields of the $z$-direction MD resonance exceeds that of the $x$-direction MD resonance in Fig. 2(d). However, when ${{\rm VO}_2}$ transitions into the metallic phase, all resonances broaden due to the increased absorption loss of ${{\rm VO}_2}$, as depicted in Fig. 4(b).

 

Fig. 4. Transmission spectra with varying widths ${w_2}$ for ${{\rm VO}_2}$ in the (a) insulating and (b) metallic phases under $y$-polarized incidence, where $P = {2000}\;{\rm nm}$, ${l_1} = {l_2} = {w_1} = {900}\;{\rm nm}$, ${h_1} = {20}\;{\rm nm}$, ${h_2} = {550}\;{\rm nm}$, and $g = {50}\;{\rm nm}$. Electromagnetic field distributions at the wavelength of 3436 nm for (c) and 3640 nm for (d), where ${w_2}$ is set to 1100 nm and ${{\rm VO}_2}$ is in the insulating phase. Electromagnetic field distributions at the wavelength of 3160 nm for (e) and 3180 nm for (f), where ${w_2}$ is set to 700 nm and ${{\rm VO}_2}$ is in the insulating phase. The $XY$ and $XZ$ planes pass through the centers of two Si nanobricks; the $YZ$ plane passes through the center of the left Si nanobrick. Cartesian scattered powers in the wavelength range from 3400 to 3500 nm for (g) and the wavelength range from 3600 to 3700 nm for (h), with ${w_2}$ chosen as 1100 nm and ${{\rm VO}_2}$ in the insulating phase. ${Q}$-factors of (i) EQ resonance and (j) $z$-direction MD resonance with and without insulating ${{\rm VO}_2}$.

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The ${Q}$-factors of the EQ and $z$-direction MD resonances in the insulating phase of ${{\rm VO}_2}$ were retrieved for different ${w_2}$, as depicted in Figs. 4(i) and 4(j), respectively. Additionally, the ${Q}$-factors of the two resonances without ${{\rm VO}_2}$ were calculated for comparison. It was observed that the ${Q}$-factors of the EQ and $z$-direction MD resonances increase significantly as ${w_2}$ approaches 900 nm. Notably, when ${{\rm VO}_2}$ is absent, the ${Q}$-factors of the EQ and the $z$-direction MD resonances tend to infinity for ${w_1} = {w_2} = {900}\;{\rm nm}$. However, in the presence of ${{\rm VO}_2}$, the ${Q}$-factors of the two resonances decrease due to the absorption loss of ${{\rm VO}_2}$. Consequently, the hybrid dielectric metasurfaces can support two BIC modes under $y$-polarized incidence. Unlike the BIC mode with a TD characteristic for $x$-polarized incidence, the two BIC modes for $y$-polarized incidence mainly arise from the EQ and $z$-direction MD modes. Nevertheless, all BICs can be transformed into quasi-BICs by simply changing the ${w_2}$. As ${w_2}$ decreases from 1300 to 500 nm, all resonances experience a blueshift, as illustrated in Fig. 4(a). The blueshift of the EQ resonance is greater than that of the $x$-direction MD resonance, resulting in a reduction in the wavelength separation between the two resonances. Notably, when ${w_2}$ is decreased to 700 nm, an analogy of electromagnetically induced transparency (EIT) is observed at 3160 nm for ${{\rm VO}_2}$ in the insulating phase [49,50], owing to the interaction between the EQ and $x$-direction MD modes. We also calculated the electromagnetic fields at the wavelength of 3160 nm, which reveal the characteristic of the EQ mode, as shown in Fig. 4(e). However, the dip wavelength (3180 nm) around the EIT peak, displaying the feature of the $x$-direction MD mode, is as shown in Fig. 4(f). Therefore, the EIT indeed results from the interaction between the EQ and $x$-direction MD modes. The ${Q}$-factor of the EIT resonance is 359. However, when ${{\rm VO}_2}$ transitions into the metallic phase, the EQ and $x$-direction MD resonances broaden due to the increased absorption loss of ${{\rm VO}_2}$, resulting in the disappearance of the EIT resonance, as depicted in Fig. 4(b). Consequently, a switchable EIT resonance can be achieved using the hybrid dielectric metasurface. However, when ${w_2}$ is decreased to 600 nm, the EQ and $x$-direction MD modes are well separated, resulting in the disappearance of EIT, as shown in Fig. 4(a). Moreover, the EIT phenomenon cannot be formed, as ${w_2}$ is further decreased to 500 nm.

 

Fig. 5. Transmission spectra with varying length ${l_2}$ and widths ${w_2}$ for ${{\rm VO}_2}$ in the (a) insulating and (b) metallic phases under $x$-polarized incidence, and transmission spectra with varying length ${l_2}$ and widths ${w_2}$ for ${{\rm VO}_2}$ in the (c) insulating and (d) metallic phases under $y$-polarized incidence, where $P = {2000}\;{\rm nm}$, ${l_1} = {w_1} = {900}\;{\rm nm}$, ${h_1} = {20}\;{\rm nm}$, ${h_2} = {550}\;{\rm nm}$, and $g = {50}\;{\rm nm}$.

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In addition to modifying the width ${w_2}$ to break the structural symmetry, we also discuss the impact of simultaneously changing the length ${l_2}$ and width ${w_2}$. Figure 5(a) illustrates the calculated transmission spectra with varying lengths and widths for ${{\rm VO}_2}$ in the insulating phase under $x$-polarized incidence while maintaining the other geometric parameters constant. Similar to the case discussed in Fig. 3(a), where the width is simply changed, a quasi-BIC mode can also be excited by simultaneously adjusting the length and width. The quasi-BIC undergoes a redshift with the increase of length and width. When ${{\rm VO}_2}$ transitions into the metallic phase, the quasi-BIC broadens and vanishes due to the increased absorption loss of ${{\rm VO}_2}$, as presented in Fig. 5(b). In addition, as displayed in Fig. 5(c), two quasi-BICs can also be generated by simultaneously changing the length and width when the incident light is polarized along the $y$-direction, which is similar to the case discussed in Fig. 4(a), where the width is simply changed. The two quasi-BICs blueshift as the length and width decrease. Moreover, the two quasi-BICs are switched off when ${{\rm VO}_2}$ transitions into the metallic phase, as shown in Fig. 5(d). Therefore, three quasi-BICs can also be excited under $x$- and $y$-polarized incidence by simultaneously altering the length and width.

To date, our research has achieved three types of BICs with different multipole sources using a hybrid dielectric metasurface. By adjusting the geometric parameters ${w_2}$ to break the structural symmetry, these BICs are transformed into quasi-BICs under $x$- or $y$-polarized incidence. Further, these quasi-BICs can be switched off through the phase transition of ${{\rm VO}_2}$. It is worth noting that our study also reveals two types of TD resonances with different dipole moment orientations, which have not been previously reported in ${\rm Si} {-} {{\rm VO}_2}$ nanostructures [40–42]. Importantly, all BICs occur under the same geometric parameters as mentioned earlier and do not exhibit any resonance in the transmission spectrum. Additionally, we conducted eigenmode analysis to study the modal properties of the hybrid dielectric metasurface [9,16]. The band diagram in Figs. 6(a) and 6(b) reveal the presence of three transverse electric (TE)-like modes and three transverse magnetic (TM)-like modes, denoted as ${{\rm TE}_1}$, ${{\rm TE}_2}$, ${{\rm TE}_3}$, ${{\rm TM}_1}$, ${{\rm TM}_2}$, and ${{\rm TM}_3}$, respectively. The electromagnetic fields of these eigenmodes at the $\Gamma$ point are depicted in Figs. 6(c) and 6(d). It is observed that ${{\rm TE}_1}$, ${{\rm TM}_2}$, and ${{\rm TM}_3}$ modes exhibit similar electromagnetic field distributions to the $y$-direction TD, $x$-direction MD, and $y$-direction MD modes, as discussed in Fig. 2, which can be directly excited by $x$- or $y$-polarized light. Additionally, the electromagnetic fields of ${{\rm TM}_1}$, ${{\rm TE}_2}$, and ${{\rm TE}_3}$ modes resemble those of $z$-direction TD, $z$-direction MD, and EQ modes, as discussed in Figs. 3 and 4. The ${Q}$-factors of ${{\rm TM}_1}$, ${{\rm TE}_2}$, and ${{\rm TE}_3}$ modes at the $\Gamma$ point are approximately ${{10}^3}$, which are limited by the absorption loss of ${{\rm VO}_2}$. When ${{\rm VO}_2}$ is removed, the ${Q}$-factors of these modes can reach ${{10}^7}$. Although the ${{\rm TM}_1}$, ${{\rm TE}_2}$, and ${{\rm TE}_3}$ modes are presented when the structure is symmetric, they cannot be available due to the symmetry mismatch. Instead, by adjusting the width of one ${\rm Si} {-} {{\rm VO}_2}$ nanostructure to destroy the structural symmetry, the ${{\rm TM}_1}$, ${{\rm TE}_2}$, and ${{\rm TE}_3}$ modes are transformed into three quasi-BICs with high-${Q}$ resonances in the transmission spectra.

 

Fig. 6. Band diagrams of (a) TE and (b) TM modes of the hybrid dielectric metasurface with ${{\rm VO}_2}$ in the insulating phase, where $P = {2000}\;{\rm nm}$, ${l_1} = {l_2} = {w_1} = {w_2} = {900}\;{\rm nm}$, ${h_1} = {20}\;{\rm nm}$, ${h_2} = {550}\;{\rm nm}$, and $g = {50}\;{\rm nm}$. Electric and magnetic fields of (c) TE and (d) TM modes at the $\Gamma$ point.

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4. CONCLUSION

In summary, we have demonstrated the generation of switchable multiple quasi-BICs by utilizing the phase transition properties of ${{\rm VO}_2}$. Through manipulation of the geometric parameters, we produced three types of high-${Q}$ quasi-BICs with distinct multipole origins under $x$- and $y$-polarized incidence. Notably, the metasurface also supports two TD modes with dipole moments oriented in different directions. Further, these quasi-BIC and TD modes can be switched off through the phase transition of ${{\rm VO}_2}$. While our findings were validated through numerical calculations, it is worth noting that recent works have demonstrated the experimental fabrication of ${\rm Si} {-} {{\rm VO}_2}$ nanostructures [40,42]. Additionally, the photoinduced phase transition of ${{\rm VO}_2}$ can occur on a femtosecond timescale [51], enabling ultrafast control of these quasi-BICs. The switchable multiple quasi-BICs could be applied in optical modulators, tunable harmonic generation, and biosensors.