1. Introduction

Bound states in the continuum (BIC) are a captivating physical phenomenon that has garnered widespread attention in the field of nanostructures [1–4]. Initially conceptualized in quantum mechanics [5], BIC refers to a counterintuitive situation where a bound state (a state with discrete energy levels) exists in a continuum band, allowing it to remain localized and not decay despite being in a spectrum of lossy states [6]. The incorporation of BIC in nanostructures offers notable advantages like a novel light confinement mechanism, high-quality factors, intense field localization, and enhanced light-matter interactions [7]. Harnessing these characteristics has led to significant advances in areas such as resonator design [8], low-loss optical transmission [9], and efficient nonlinear generation [10,11], improving functionalities in diverse nanostructures including metasurfaces [12–15]. Known for their ability to achieve precise control over light properties such as phase, amplitude, and polarization, metasurfaces are composed of subwavelength nanostructures with extensive design flexibility [16,17]. To date, a variety of BIC-based applications in metasurfaces have leveraged the well-known symmetry-protected BIC in metasurfaces with C_4v group symmetry for their design simplicity and fabrication robustness [18–25]. By introducing specific defects in C_4v metasurfaces, non-radiative BIC can degenerate into quasi-BIC that couple with continuum modes, exhibiting Fano resonances with high-quality factors immensely beneficial for developing high-performance optical sensors [26–28].

Besides, the quasi-BIC can be employed to achieve efficient optical modulation by exploiting its extremely sharp peak in the transmission (reflection) spectrum. To realize such modulation effects, a variety of functional materials have been integrated into dielectric metasurfaces for their unique tunability, including 2D materials [29–32], electro-optic materials [33,34], and phase-change materials [35–37]. Among diverse phase-change materials, Vanadium dioxide (VO₂) has been proven a promising candidate to achieve efficient modulation for its relatively low transition temperature (about 68°) and reversibility nature. These properties enable VO₂-based modulation devices to operate at lower temperatures and offer enhanced durability [4,38–41]. By incorporating VO₂ materials into BIC design, the efficient modulations of quasi-BIC peaks have been achieved in previous works [37,42]. However, while various BIC-based modulation devices have exhibited exceptional tunability, the majority is composed of unit cells with C_s symmetry. Consequently, they may experience significant performance degradation due to polarization deviations [43,44]. This issue could present challenges in practical applications that utilize the resonance peaks of quasi-BICs, such as sensing [45,46], bio-detection [28,47], and dynamic imaging [31,48]. Therefore, it is crucial to design quasi-BIC modulation devices beyond C_s unit cell symmetry with better robustness against polarization deviations. Despite previous demonstrations of polarization-independent quasi-BICs with C₄ and C_4v unit cell symmetries [19,49–54], the integration of existing tuning methods with these modes still awaits further exploration. To the best of our knowledge, our work presents the first attempt to utilize these modes to achieve efficient modulation with enhanced polarization responses.

In this paper, we propose the polarization-insensitive quasi-BIC modulation in Si-VO₂ hybrid metasurfaces. For comparing the polarization response properties, three types of quasi-BICs are designed by introducing specific defects in different metasurfaces with the unit cell symmetries of C_s, C₄, C_4v. These quasi-BICs exhibit strong resonance in the transmission spectrum at different wavelengths, which can be described by the Fano formula. To gain further insights into their physical origins, the far-field scattering of multipoles is calculated, revealing their distinct components with strongly localized fields observed near the resonant wavelengths. Next, we move to investigate the responses under linearly polarized waves with varying directions for three metasurfaces. The resonance peak in the C_s metasurface exhibits significant degradation, while C₄ and C_4v systems maintain their maximum peaks regardless of the polarization direction. To facilitate transmission modulation, a VO₂ layer is integrated into C_4v metasurface aiming to harness its loss change during phase transition. As the temperature rises from 20° to 80°, considerable transmission modulations are observed with a maximum relative modulation depth approaching 350%. Compared to existing quasi-BIC-enabled tunable optical devices [29,31,37,55], our design shows an excellent modulation performance with more fabrication-friendly structures and exhibits identical polarization responses, safeguarding it against performance degradation due to polarization deviation. These features ensure consistent and reliable operation across a range of polarization states, making it a significant advancement in the field of tunable optical devices.

2. Results

2.1 Quasi-BIC design in different structures

By introducing defects to disrupt the symmetry of metasurfaces, quasi-BICs in different metasurfaces are engineered. We examine the dielectric metasurfaces composed of silicon cylinder arrays on a SiO₂ substrate with three symmetry unit cells: C_s, C₄, and C_4v, as depicted in Fig. 1.(a1-a3). Through the use of electron-beam lithography (EBL) techniques, several experimentally-oriented studies have reported high-precision fabrications of designed nanostructures with similar patterns to realize quasi-BICs [56–58]. By adopting a parallel fabrication approach, the etched nanodisk structures in Fig. 1.(a1-a3) are also achievable. The C_s symmetry metasurface as depicted in Fig. 1.(a1), consists of two ellipses, each having a major axis L_y of 360 nm, a minor axis L_x of 180 nm, and a height of 200 nm, spaced 500 nm apart at a specific angle. The lattice geometry is P_x =1.26 µm, P_y =900 nm. Using the Finite-Difference Time-Domain (FDTD) method with periodic boundary conditions in the x- and y-directions and perfectly matched layers in the z-direction, the transmission spectrum under x-polarized incident light is calculated for varying θ values. For θ ≠ 0°, distinct resonance peaks around 2 µm emerge, broadening and shifting to shorter wavelengths as θ increases. At θ = 0°, the resonance peak vanishes, indicating a symmetry-protected BIC in Fig. 1.(b1). To explore BIC in more complex structures, we construct metasurfaces with C₄ and C_4v symmetries using four identical cylinders with specific defects arranged in a square grid pattern, as illustrated in Fig. 1.(a2-a3). These cylinders have a radius r = 320 nm and a height of 200 nm, with a length of d = 160 nm and varying width w defect (marked in Fig. 2.(b2-b3)). The lattice geometry is a_x= a_y= a = 860 nm. In Fig. 1.(a2), each cylinder is oriented such that the defect appears to incrementally rotate in a clockwise direction by 90° from one to the next. Such configuration suggests a sequential quarter-turn rotational symmetry to ensure the C₄ symmetry of the unit cell. While maintaining the rotational symmetry, the defect in Fig. 1.(a3) is oriented diagonally at a 45° angle relative to the vertical axis to provide a C_4v symmetry, rather than vertically as in the former case. As shown in Fig. 1.(b2-b3), adjusting the defect widths w in C₄ and C_4v metasurfaces also results in sharp resonance peaks caused by quasi-BICs. A comparison of the transmission spectrum between C₄ and C_4v metasurfaces that share identical parameters but different symmetries reveals close similarity, indicating the resonance peaks are mainly dominated by geometries rather than unit cell symmetries.

 

Fig. 1. Quasi-BIC design in different metasurfaces. (a1-a3) Schematics of metasurfaces with different symmetries. Relevant geometrical parameters are marked with Italics. In (a1), P_x =1.26 µm, P_y =900 nm, L_x= 180 nm, L_y= 360 nm. In (a2-a3), a_x= a_y= a = 860 nm, r = 320 nm, d = 160 nm. (b1-b3) Transmission spectra at varying defect sizes. The absence of a resonance peak when no defects are introduced indicates the existence of BIC when no defect is introduced. (c1-c3) Curve fitting of the scatters with the Fano formula. (d1-d3) Quality factors versus asymmetry parameters. The data are mainly extracted from (b1-b3), with the quality factors decreasing as the asymmetry parameters increase, also indicating the tuning of quasi-BICs.

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Fig. 2. BIC properties in different metasurfaces. (a1-a3) Electric field distributions of quasi-BICs at resonance wavelength. Significant field localization effects are observed. (b1-b3) Multipole decompositions near the resonant wavelength. (c1-c3) Transmission spectra at increasing scaling factors. With the proportionally increasing geometries, the resonance peak exhibits a considerable shift.

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Relevant studies suggest that the resonant peaks in the three cases arise from the coupling between discrete quasi-BICs and modes in the continuous spectrum, which can be described using the typical Fano formula [59,60]:

Here, T₀ is the background transmission, A₀ is the coupling strength between the continuous and discrete modes, λ₀ is the center wavelength of the resonance, Γ is the resonance linewidth, and q represents the Fano parameter, describing the asymmetry of the spectral line. Employing the Matlab curve fitting toolbox, the calculated transmission spectrum is fitted with the Fano formula as shown in Fig. 1.(c1-c3). The obtained regular R² values are 0.9983, 0.9999, and 0.9998, indicating a strong correlation to Fano resonances. Besides, the quality factor Q defined by $\frac{{2\lambda }}{\Gamma }$ yields 4.1 × 10³, 2.85 × 10³, and 2.38 × 10³ for the three resonance peaks respectively. Other values extracted from the lineshapes are summarized in Fig. 1.(d1-d3). As the asymmetry parameters increase, the quality factors drop rapidly, providing another evidence for the existence of quasi-BICs.

To gain deeper insights into the physical origins of BIC, the multipole decompositions near quasi-BIC resonance peaks for the three types of BIC are calculated using Comsol Multiphysics to analyze the dominant multipole components. Additional details about the multipole decomposition method can be found in Ref. [61–65]. Firstly, the electric field distributions at resonance wavelengths are calculated as shown in Fig. 2.(a1-a3). The field intensities exhibit localized enhancement at specific locations, especially the defects in Fig. 2.(a2-a3). Building on this, the computed multipole component contributions for the far-field scattering power are shown in Fig. 2.(b1-b3), revealing significant differences for distinct quasi-BICs. Scattering powers of different orders of multipoles exhibit sharp peaks near the resonance centers, while other wavelengths only have negligible contributions. The quasi-BIC with C_s symmetry predominantly consists of magnetic dipole (MD) and electric quadrupole (EQ). Conversely, the rest are both primarily comprising toroidal dipole (TD) and magnetic quadrupole (MQ) elements.

Moreover, leveraging the robustness of the symmetry-protected BIC [47,66], we highlight that the center resonance wavelengths can be manipulated by uniformly scaling the geometric structures as shown in Fig. 2.(c1-c3).

2.2 Polarization response property of quasi-BICs

The obtained quasi-BIC Fano resonance peaks have considerable high-quality factors with sharp transmission peaks, paving the way to achieve efficient modulations. However, owing to the distinct unit cell symmetries, designed quasi-BICs would have divergent responses to linearly polarized light in different directions.

To explore their polarization responses, the transmission spectra are calculated for the three structures under linearly polarized incident light with four different polarization directions (φ = 0°, 30°, 60°, 90°), as shown with solid lines in Fig. 3.(a1-a3). Considering the anisotropy of C_s symmetric structure, the corresponding background transmission spectrum (θ=0°, no defects are introduced) is also depicted with a dot line for better comparing the peak values of quasi-BICs. From Fig. 3.(a1), the resonance peak progressively weakens as the φ angle increases, and becomes totally undetectable at φ= 90° (corresponding to y-direction). Thus, the decreased resonance peak in the C_s symmetry metasurface will lead to performance degradation in applications depending on the resonance peaks, such as sensing [45,46], bio-detection [28,47], and dynamic imaging [31,48]. In contrast, metasurfaces with C₄ and C_4v symmetries exhibit consistent responses to linearly polarized light from any direction as shown in Fig. 3.(a2-a3). This uniformity enables these structures to achieve maximum quasi-BIC resonance peaks without polarization alignment, thus minimizing the flux losses and contributing to a more compact, robust system.

 

Fig. 3. The polarization response properties of different symmetry systems. (a1-a3) Polarization responses of different metasurfaces. The resonance peak of C_s systems is strongly affected by the polarization directions, while the spectra of C₄ and C_4v systems remain the same regardless of polarization directions. (b1-b3) Electric field distributions at resonance wavelength for polarization direction φ=90°.

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To investigate the physical origins of these varying polarization responses, the electric field distributions at the resonance wavelength for φ=90° (y-polarized incident light) are calculated, as illustrated in Fig. 3.(b1-b3). Analysis of these visualizations shows that y-polarized incident light excites the other two defects in the C₄ structure. Besides, apart from the two vertical nodal lines, the electric field distribution in C_4v structure is almost identical to the case in the x-polarized incident light. Indeed, the electric field distributions shown in Fig. 2.(a2-a3) and Fig. 3.(b2-b3) differ only by a 90° planar rotation. The relative orientation of the electric field does not impact the transmission spectrum. Consequently, this similarity in electric field distributions ensures the identical spectrum of C₄ and C_4v structures in φ=0°, 90°. In contrast, the absence of a direct correlation between the electric field distributions in Fig. 2.(a1) and Fig. 3.(b1) results in distinct responses of the C_s structure.

For a more in-depth analysis of the polarization response property, define a 2 × 2 matrix S that relates the transmitted electric fields E_x', E_y' to the incident electric fields E_x, E_y:

The scattering matrix related to different metasurfaces is derived using group representation theory, $S = \left( {\begin{array}{{cc}} {{S_{11}}}&0\\ 0&{{S_{22}}} \end{array}} \right)$, $\left( {\begin{array}{{cc}} {{S_{11}}}&{ - {S_{12}}}\\ {{S_{12}}}&{{S_{11}}} \end{array}} \right)$, $\left( {\begin{array}{{cc}} {{S_{11}}}&0\\ 0&{{S_{11}}} \end{array}} \right)$ for C_s, C₄, C_4v respectively, with each configuration being dominated by the unit cell symmetries. These varying configurations of scattering matrix cause the distinct polarization responses of different metasurfaces. For the C_s symmetry, the non-zero elements in its scattering matrix are confined to the diagonal with different values (S₁₁≠ S₂₂). The varying diagonal elements lead to the polarization response sensitivity in Fig. 3.(a1). Conversely, the scattering matrices for C₄ and C_4v symmetries have equal diagonal elements, rendering them polarization-insensitive properties in Fig. 3.(a2-a3). A notable distinction of scattering matrix for C₄ lies in its opposite off-diagonal elements (S₁₂=-S₂₁), potentially exhibiting circular dichroism beyond polarization insensitivity [43].

2.3 Polarization-insensitive quasi-BIC modulation by VO₂ thin films

With the simplest scattering matrix, the constructed C_4v metasurface is a promising choice to achieve polarization-independent quasi-BIC modulation. By employing the pulsed laser deposition (PLD) technique, thin-film VO₂ with a thickness of several tens of nanometers have been fabricated with excellent performance characteristics [38,67]. These studies provide solid foundation for designing VO₂ hybrid systems. In consideration of the potential impact of VO₂ layer thickness on modulation effects, our study initially focuses on examining these effects by integrating a 10 nm thick VO₂ layer onto each cylinder within a silicon array, as shown in Fig. 4.(a). The modulations of resonance peaks are achieved by leveraging the phase-change property of VO₂, which switches from a low-loss dielectric state to a high-loss metallic state at a specific temperature. This transition significantly affects the loss of VO₂ layers, thus facilitating effective tuning of quasi-BIC resonance peaks. Our simulations adopt the temperature-dependent refractive index and extinction coefficient data of VO₂ thin films from Ref. [68]. Considering the VO₂ layer's thin thickness, a refined mesh at the VO₂-Si interface is implemented to ensure computational precision. The calculated transmission spectra across rising temperatures are shown in Fig. 4.(b). At 20°C, the VO₂ remains in its dielectric state with minimal loss. The lower loss enables quasi-BIC to maintain a strong Fano resonance with a significant transmission peak. As the temperature increases, the extinction coefficient of VO₂ gradually rises, diminishing the resonance peak and causing spectral broadening due to the higher loss. Upon the temperature surpassing the phase transition threshold (about 68°C), VO₂ enters its metallic phase with dramatically increased loss, causing a significant modulation with negligible resonance effect observed in the transmission spectrum. For a more quantitative measurement, the relative modulation depths are calculated using the following formula [37,69] (Fig. 4.c):

From the results, ΔT shows sharp peaks near the resonance wavelength, corresponding to the modulation effect of quasi-BIC induced by temperature changes. While at other wavelengths, only very low changes are observed. The modulation depths of the two temperatures hold 202%, and 342% respectively, delivering a notable progress compared to existing researches [37,42].

 

Fig. 4. Resonance peak modulation induced by VO₂ layers. (a) Schematic of the Si-VO₂ hybrid metasurface. The VO₂ layer is on each Si cylinder. (b) Transmission spectra at different temperatures. (c) Calculated relative modulation depths ΔT for 55°C and 80°C, with a maximum depth of 202% and 342%, respectively.

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Figure 4.(c) demonstrates the relative modulation depth depends on the loss of VO₂ layers. To further examine the effects of different VO₂ thickness on light modulation, the transmission spectra for VO₂ layers with thicknesses of 3 nm and 50 nm are calculated at three representative temperatures (Fig. 5.a1-a2), and their corresponding modulation depths are determined (Fig. 5.b1-b2). By comparing the results, it is observed that thinner VO₂ layers restrict the loss caused by the VO₂ phase transition, leading to less degradation as the temperature rises. As a result, the transmission spectra can still maintain distinguishable resonance peaks despite VO₂ entering its metal phase. The relative modulation depth of 20°C to 55°C holds only very low values near the resonance wavelength due to an inadequate peak change.

 

Fig. 5. (a1-a2) Transmission spectra at three representative temperatures for 3 nm (a1) and 50 nm (a2) thick VO₂ layers. (b1-b2) Corresponding relative modulation depths.

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Conversely, excessively thick VO₂ layers incur considerable loss even at 20°C, adversely reducing the peak and quality factor of the transmission spectrum. The significant spectrum degradation at 20°C leads to a much smaller ΔT than the previous two cases. Therefore, controlling the thickness of VO₂ layers is crucial for achieving efficient modulation. The 10 nm thick VO₂ is a better choice compared to the other two for providing sufficient loss change to strike a balance between maintaining a strong resonance in the dielectric state and achieving efficient temperature-switching modulation capabilities.

By adding the VO₂ layer to each Si cylinder, significant transmission modulation is observed. However, this approach requires modifying numerous VO₂ thin layers on Si cylinders, potentially increasing fabrication difficulty and influencing the thickness uniformity. Hence, we propose another more fabrication-friendly configuration that yields comparable modulation effects. As depicted in Fig. 6.(a), this approach involves growing the VO₂ layer directly on a SiO₂ substrate. Without additional VO₂ shaping processes, this method streamlines the fabrication procedure and enhances the uniformity of VO₂ thin films. Following a similar manner as above, the transmission spectrum (Fig. 6.b1-b2), and their relative modulation depths (Fig. 6.c1-c2) are calculated with VO₂ thicknesses of 3 nm and 10 nm. Compared to the previous method, using VO₂ as a direct-contact substrate results in a more pronounced impact on the resonance peak at the same VO₂ thicknesses, which can be attributed to the increased loss due to a larger VO₂ area. Therefore, to achieve comparable modulation depth, a thinner VO₂ layer is required. Despite vibrant advances in VO₂ growth technology, producing such thin VO₂ layers on SiO₂ substrates still awaits more opportunities.

 

Fig. 6. (a) Schematic of Si-VO₂ hybrid metasurface. The VO₂ layer is only on the SiO₂ substrate. (b1-b2) Transmission spectrum at three representative temperatures for 3 nm (b1) and 10 nm (b2) thick VO₂ layers. (c1-c2) Corresponding relative modulation depths.

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Overall, each proposed approach has distinct advantages and limitations. The choice of the most appropriate structure should be guided by practical fabrication conditions.

3. Conclusion

In summary, we have achieved the modulation of polarization-insensitive quasi-BICs via the Si-VO₂ hybrid metasurfaces with C_4v symmetry. By exploiting the symmetry-broken induced quasi-BICs, we design the high-Q Fano resonance peaks in metasurface structures with C_s, C₄, and C_4v symmetries, and investigate their various properties including the localized enhanced fields, dominant multipole components as well as the tunning effects induced by proportional geometry scaling. The comparison of their responses to differently polarized incident light reveals that the metasurfaces with C₄ and C_4v symmetries respond uniformly, maintaining maximum resonance peaks regardless of the polarization orientation. Building on this, the modulations in two specially designed Si-VO₂ hybrid metasurfaces with C_4v symmetry are achieved by leveraging the phase transition loss of VO₂. Quantitative calculations show that the relative modulation depth approaches 350%. Beyond the loss modulation induced by VO₂ materials, our method can be easily extended to other modulation schemes like refractive index tuning devices by integrating the polarization-insensitive quasi-BICs with C_4v unit cell symmetry into such structures [33], which can enhance their device performance with better polarization response robustness. We believe our work will offer valuable insights for the development of designing polarization-insensitive quasi-BICs in metasurfaces.