High-quality (Q) factor resonances within metamaterial systems have emerged as a focus of current research due to their wide range of applications in lasing^[1,2], enhancement of nonlinear optics interactions^[3–5], and sensing^[6–11]. Metasurfaces, composed of two-dimensional metamaterials with subwavelength structures, have demonstrated remarkable capabilities in facilitating resonant optical responses. They have found diverse applications, including the generation of perfect vortex beams^[12], wide-angle imaging^[13,14], polarization multiplexing^[15,16], and others^[17–19]. In metallic metasurfaces, the Q-factor of excited surface plasmon resonance is restricted by the considerable intrinsic resistive losses of the metal^[20–23]. Conversely, all-dielectric metasurfaces exhibit the significant potential to confine and enhance the near field inside the nanoparticles, and they have the advantages of low material loss and high laser damage threshold. This characteristic enables the achievement of Mie resonances with high-Q-factor and low absorption loss^[24–27]. It is a resonant scattering phenomenon based on the Mie scattering theory, originating from the superposition of radiation induced by monopoles, dipoles, quadrupoles, octopoles, and higher multipoles^[28]. When electromagnetic waves are incident on dielectric particles, not only electric dipole (P) resonance modes but also magnetic dipole (M) and toroidal dipole (T) resonance modes can be generated under the interaction of electromagnetic fields, thus significantly reducing the radiation loss.

Recent advancements have demonstrated the attainment of ultrahigh-Q resonances in all-dielectric metasurfaces, leveraging optical bound states in the continuum (BICs). BICs denote non-radiating modes residing within the radiation continuum, characterized by an infinite Q-factor and zero linewidth. Among the diverse BICs, symmetry-protected BICs (SP-BICs) emerge as particularly prominent, arising from the unexpected nullification of coupling constants attributed to inherent symmetrical properties. Presently, the methods to induce SP-BICs can be broadly categorized into two main classes: the first involves symmetric metasurfaces, where symmetry breaking can be achieved through various means such as altering the angle of incidence^[29–32], adjusting lattice constants^[33–36], or modifying the dielectric constants of interstructural materials^[37,38]. The second category encompasses asymmetric metasurfaces achieved by altering structural dimensions or shapes to introduce axial or central asymmetry^[3,39–43]. However, the majority of existing studies primarily concentrate on single- or dual-wavelength quasi-BIC resonances at a single polarization. This limitation significantly constrains their applicability to polarization-selective multi-wavelength lasers and multi-channel sensors. Concurrently, quasi-BIC resonances primarily employ a single symmetry-broken method for excitation, thereby restricting the flexibility of structural design and the freedom of resonance tuning for high-Q BIC devices. Therefore, exploring and utilizing the methods of symmetry breaking in arbitrary directions to achieve multiple-wavelength high-Q quasi-BICs in dual polarizations hold significant promise and considerable research interest.

This work focuses on achieving multiple high-Q quasi-SP-BIC resonances by implementing various symmetry-broken approaches in arbitrary directions. For the S-shaped metasurface, three distinct symmetry-broken perturbations are introduced, leading to the presence of triple quasi-BICs dominated by the magnetic quadrupole and electric toroidal dipole in the transverse magnetic (TM) polarization modes with wavelengths of 1071.18, 1098.8, and 1199.6 nm, respectively. Meanwhile, transverse electric (TE) polarization modes support double quasi-BICs dominated by the electric quadrupole and magnetic dipole at wavelengths 1375.9 and 1628.5 nm, respectively. Notably, the excited high-Q quasi-BIC resonances exhibit outstanding sensing capabilities with a maximum sensitivity of 900 nm/RIU. A comparative analysis with previously published studies is provided in Table 1 to highlight its superior sensing performance. The proposed S-shaped structure extends the route to realize multi-wavelength quasi-BICs, making it valuable for designing and applying photonic devices such as biosensors and nonlinear interactions.

As illustrated in Fig. 1(a), the proposed all-dielectric metasurface is composed of an S-shaped silicon nanoparticle array with a refractive index n=3.45. Each unit cell consists of three parallel and two vertically connected silicon bars, characterized by the following geometric parameters: d1=d2=660nm, g1=g2=317.5nm, w=165nm, h=250nm, and Λx=Λy=1000nm.

Additionally, three distinct methods of symmetry breaking on the S-shaped structure are considered to induce the excitations of leaky quasi-BICs, as shown in Figs. 1(b)–1(d). In Case I, the symmetry is disrupted by making the lengths of the left and right vertical silicon bars unequal (i.e., g1≠g2), with the degree of asymmetry represented by Δg. In Case II, the symmetry is perturbed by adjusting the lengths of the top and bottom horizontal silicon bars to be unequal (i.e., d1≠d2), with the degree of asymmetry denoted by Δd. In Case III, the symmetry of the excitation field is broken by altering the incident wave angle from normal incidence to oblique incidence, with the degree of asymmetry expressed as θ. To investigate the optical properties of the metasurface, numerical simulations are performed using the finite element method (FEM) with the commercial Comsol Multiphysics and CST software. The simulations employ periodic boundary conditions in both x and y directions and perfectly matched layers (PMLs) for wave propagation along the z axis. The metasurface is illuminated perpendicularly by either an x-polarized or a y-polarized incident wave.

To gain a deeper insight into the physical origin of BICs in S-shaped metasurfaces, we analyze the intrinsic band structures and Q-factors in both TM and TE modes, as illustrated in Figs. 2(a), 2(b) and 2(d), 2(e), respectively. Figure 2(a) reveals that four TM eigenmodes are primarily analyzed in the wavelength range of 1000–1250 nm, denoted as TM1, TM2, TM3, and TM4, respectively. The dispersion curves of these four TM eigenmodes are relatively flat. As shown in Fig. 2(b), the Q-factor of TM3 remains essentially unchanged. However, it is clear to see that TM1, TM2, and TM4 are sensitive to symmetry-broken perturbations. Their Q-factors are infinite at the Γ point but decrease sharply when deviating from it. This observation essentially indicates that the excited TM eigenmodes (TM1, TM2, and TM4) are typical SP-BIC modes. These modes remain stable as long as the S-shaped metasurface maintains the desired symmetry but become excited when symmetry is broken in the S-shaped metasurface. Similarly, the analysis of Figs. 2(d) and 2(e) reveals that the dispersion curve of the TE1 eigenmode is relatively flat. In contrast, for TE2 and TE3 eigenmodes, the dispersion curves tend to intersect, attributable to a strong coupling effect that induces an evasive crossover phenomenon at the Γ point. Additionally, besides TE2, the Q-factor trends of TE1 and TE3 eigenmodes also exhibit the phenomenon of SP-BIC. Figures 2(c) and 2(f) depict the magnetic field distributions of the TM modes at the Γ point and the electric field distributions of the TE modes at the Γ point, respectively. The magnetic field near-field intensity of all four TM eigenmodes is predominantly concentrated within the silicon bars. In contrast, the electric field near-field intensity of all three TE eigenmodes is chiefly concentrated in the outer edges of the silicon bars or the gaps between the silicon bars.

When illuminated by a normally incident plane wave with the electric field along the x axis, the transmission spectra of S-shaped metasurfaces are simulated in both symmetric and asymmetric cases (Δg=10nm, Δd=10nm, and θ=3°) within the wavelength range of 1050 to 1250 nm, as depicted in Fig. 3(a). In symmetric S-shaped metasurfaces, a distinctive strong resonance TM3 is observed at a wavelength of 1125.2 nm. However, for S-shaped metasurfaces when symmetry breaking like cases I, II, and III is introduced, in addition to the TM3 resonance, three new narrow resonances can be seen in Fig. 3(a) at 1071.18, 1098.8, and 1199.6 nm, which are labeled as TM1, TM2, and TM4, respectively. The corresponding dependence of transmission spectra on changes in Δg, Δd, and θ are calculated (for details, see Sec. S1 of the Supplementary Information).

The results from Figs. 3(c)–3(e) demonstrate that, in a symmetric structure (Δg=0nm, Δd=0nm, and θ=0°), the Q-factors of TM1, TM2, and TM4 resonances are infinite. Conversely, in an asymmetric structure (Δg≠0nm, Δd≠0nm, and θ≠0°), the Q-factors corresponding to these resonances decrease dramatically. Moreover, greater asymmetry values of g, d, and θ lead to smaller Q-factors, indicative of typical SP-BIC characteristics. Notably, the TM1 and TM2 resonances in Case III exhibit high sensitivity to the asymmetric value θ, and they will disappear when θ exceeds 4° and 5°, respectively. The SP-BICs denote a localized state with a zero linewidth embedded in the continuum. The resulting SP-BICs transform into leaky resonances with exceptionally high Q-factors when the symmetry of the S-shaped metasurface is disrupted. In general, the Q-factor is defined as the ratio of the center frequency of the resonance to the bandwidth (FWHM, full width at half-maximum). Here, the Q-factors of the resonances are determined by fitting the transmission spectrum using the classical Fano formula^[49,50], expressed as I∝(Fγ+ω−ω0)2(ω−ω0)2+γ2,(1)where I represents the transmission, F denotes the Fano parameter, and γ and ω0 signify the FWHM and resonant wavelength position of the resonance, respectively. Consequently, Q=ω0/γ. Figure 3(b) displays the fitting results for TM1, TM2, and TM4 in Case I. The corresponding Q-factors for TM1, TM2, and TM4 are 63,761, 11,943, and 6966, respectively.

To further understand the physical mechanisms underlying the generation of the three leaky BIC resonances, TM1, TM2, and TM4, as well as the TM3 resonance, the decomposed scattered powers of the multipole moments are calculated using the Cartesian coordinate system based on the density of induced current inside the metamolecules (for details, see Sec. S2 of the Supplementary Information). Figures 4(a)–4(d) illustrate the six strongest scattered powers around the resonant wavelengths of TM1, TM2, and TM4 in Case I (Δg=10nm) and TM3 in the symmetric case, respectively, as well as the sum of these six scattered powers, including the electric dipole P, magnetic dipole M, magnetic toroidal dipole MT, electric quadruple Qe, magnetic quadruple Qm, electric toroidal dipole ET, and the total scattered powers.

For the leaky BIC resonance TM1, Qm stands as the primary contributor, closely followed by ET. For the leaky BIC resonance TM2, ET is the dominant multipole scattered power, trailed closely by Qm and M. Analysis of the x, y, and z components of the ET scattered power [Fig. S3(a), Supplementary Information] reveals that the z component exhibits the highest strength. However, the scattered power of Qm is very close to that of ET, and thus the leaky resonance TM2 results from the destructive interference between ET and Qm. For the TM3 resonance, M has the highest scattered power, and its y component has the highest intensity [Fig. S3(b), Supplementary Information]. Subsequently, P, MT, and ET closely follow each other in intensity, while the remaining multipoles are suppressed. For leaky BIC resonance TM4, Qm holds the highest proportion among the multipoles. In addition, we conduct a detailed analysis of the electromagnetic field distributions at the resonant wavelengths corresponding to the four modes TM1, TM2, TM3, and TM4, as illustrated in Fig. 4(e). The magnetic field distributions at the resonance wavelengths of these modes align with the previously calculated magnetic field patterns of the TM eigenmodes. It further confirms that the TM1, TM2, and TM4 resonances are generated based on the introduction of symmetry breaking by SP-BIC. Meanwhile, the results of the multipole decomposition computations for TM1, TM2, TM3, and TM4 are in harmony with their respective displacement current distributions.

The transmission spectra of the S-shaped metasurface within the wavelength range of 1300 to 1700 nm are computed for both symmetric and asymmetric cases (Δg=10nm, Δd=10nm, and θ=5°) under the y-polarized incident wave, as depicted in Fig. 5(a).

A clear broad resonance TE2 located at the central wavelength of 1619.3 nm can be easily seen for a symmetric S-shaped metasurface. However, once symmetry-broken perturbations are introduced to the S-shaped metasurface, as in Cases I, II, and III, in all three cases we observe a clear electromagnetically induced transparency (EIT)-like resonance TE3 near 1628.5 nm. The EIT-like resonances in Cases I, II, and III exhibit high transparencies of 96%, 85%, and 98%, respectively. Cases I, II, and III will also generate a new narrow resonance TE1 around 1375.9 nm. However, the TE1 resonance in Case II is notably less pronounced with an exceedingly weak amplitude compared to Cases I and III. The transmission spectra for varying Δg, Δd, and θ are provided in Fig. S2 of the Supplementary Information. As shown in Fig. 5(b), the transmission profiles of TE1 and TE3 resonances in Case I are fitted using the Fano formula to calculate the corresponding Q-factors. Figures 5(c)–5(e) reveal the Q-factors corresponding to different asymmetry values (Δg, Δd, and θ) for the TE1 and TE3 resonances in the three cases, shedding light on the dependence of the Q-factors on these parameters. The results indicate a sharp decrease in the Q-factor as the absolute values of Δg, Δd, and θ increase, approaching infinity when these parameters are zero. From the above, it underscores that the two leaky resonances of TE1 and TE3 are generated based on the SP-BIC.

Similarly, to understand the microscopic nature of the resonant multipoles, we analyzed the multipole decomposition and electromagnetic field distribution for the two leaky BIC resonances TE1 and TE3 in case I and the TE2 resonance in the symmetric case. For the leaky BIC resonance TE1, Qe increases considerably near the resonance wavelength and outperforms the other multipoles in the far-field scattered powers [see Fig. 6(a)]. The corresponding electric field distribution [see Fig. 6(d)] can be seen in the top and bottom silicon bars, as well as in the bars on both sides, where strong displacement currents can be observed in the form of Qe, thus confirming that the leaky BIC TE1 resonance mode is Qe. For the TE2 resonance, MT is dominant until at the wavelength 1609.35 nm, followed by P; while after 1609.35 nm it is P that has the strongest far-field scattering power, with MT in the second place, which implies that both P and MT are contributing to the TE2 resonance but that P plays a major role at the resonant wavelength [see Fig. 6(b)]. Analysis of the x, y, and z components of the scattered powers for MT and P [Figs. S4(a)–S4(b), Supplementary Information] demonstrates that the y component is most pronounced in P, while the x component is most pronounced in MT. Furthermore, it is shown from the distributions of the electric and magnetic fields [see Fig. 6(e)] that the MT moment along the x direction is generated in the S-shaped metasurface. For leaky BIC resonance TE3, M is dominant in the multipoles, followed by Qe and ET, respectively, and the scattered powers of Qe and ET are very close [see Fig. 6(c)]. Analysis of the x, y, and z components of the M reveals that the z component has the highest intensity [Fig. S4(c), Supplementary Information]. The corresponding field distributions in Fig. 6(f) also indicate that the M moment along the z direction is generated in the S-shaped metasurface.

Given the realization of high-Q metasurfaces, fabrication imperfections, material defects, or finite dimensions of the arrays can markedly affect the strength and Q-factor of the resonances, particularly concerning ultra-high-Q quasi-BIC resonances. Thus, we first computed the transmission spectra of the asymmetric metasurface structure under different geometric conditions in Case I (Δg=10nm) to investigate the dependence of the quasi-BIC resonances excited in the TM mode of Sec. 2.3 and the TE mode of Sec. 2.4 on various geometrical parameters. All other geometric parameters remained consistent with those illustrated in Fig. 1(a), with only the variable parameter changing as indicated in each figure. (for details, refer to Sec. S4 of the Supplementary Information). Next, it is expected that material loss plays a crucial role in such high-Q quasi-BIC resonance structures. While silicon exhibits negligible losses within the studied wavelength range, practical scenarios may introduce absorption and scattering losses due to defects in the fabrication process or surface roughness. Thus, we quantified the effect of losses on quasi-BIC resonances by incorporating the loss tangent (tan) into the refractive index of silicon (for details, see Sec. S5 of the Supplementary Information). Finally, considering the implementation of S-shaped metasurfaces and their practical applications, we also chose the case with quartz substrate for our study (for details, see Sec. S6 of the Supplementary Information).

The leaky quasi-SP-BICs resonances excited in TM and TE modes are promising for a wide range of applications in ultra-sensitive sensors due to their ultrahigh-Q-factor and significant near-field enhancement. As shown in Fig. 7, we further assess the performance of the S-shaped metasurface in refractive index sensing detection. Specifically, we investigate the influence of the refractive index of the environment surrounding the S-shaped metasurface on the spectra in TM and TE modes under Case I (Δg=10nm).

As depicted in Fig. 7(a), when the refractive index n of the surroundings varies from 1 to 1.1, the resonance wavelengths of TM1, TM2, and TM4 undergo a noteworthy redshift. Notably, the amplitude modulation depth of the TM1 and TM4 resonances exhibit greater sensitivity to changes in the refractive index. Specifically, the amplitude modulation depth of the TM1 resonance demonstrates irregular variation as the refractive index increases, while the amplitude modulation depth of the TM4 resonance decreases with the rising refractive index. Conversely, the amplitude modulation depth of TM2 resonance remains relatively stable under refractive index variations. The resonant wavelengths of TM1, TM2, and TM4 at different refractive indices are illustrated in Fig. 7(b). The spectral sensitivity is defined as the wavelength shift of the resonance peak with a change in unit refractive index, which can be calculated by linear fitting. The TM1, TM2, and TM4 exhibit sensitivities of approximately 400, 220, and 300 nm/RIU, respectively. Compared to TM modes, the leaky quasi-SP-BIC resonances excited in TE modes exhibit heightened sensitivity in refractive index sensing detection, and their transmittance spectra demonstrate greater stability in the face of refractive index variations. Figures 7(c) and 7(d) reveal that resonance wavelengths of TE1 and TE3 experience significant redshifts at a refractive index of 1 to 1.1 in the surroundings, with sensitivities as high as about 900 and 620 nm/RIU for TE1 and TE3, respectively. Notably, TE1 exhibits heightened sensitivity and superior sensing capabilities compared to similar articles published previously^[34,44–48]. Since the sensitivity of nanophotonic structures is intricately linked to the enhancement of the near-field and its spatial overlap with the analyte, it can be found from Fig. 6(d) that the electromagnetic near-field enhancement of the TE1 resonance primarily concentrates along the edges and gaps of the silicon rods, which facilitates its close interaction with matter in space, leading to its high refractive index sensitivity. Meanwhile, we also analyzed the detection limit (DL), which is defined as the ratio of spectral resolution to sensitivity as one of the most important metrics for refractive index sensors. If the resolution of the spectrum analyzer is 1 pm and the metasurface sensitivity is 900 nm/RIU, the detection limit should be DL=1.1×10−6RIU^[51,52]. Furthermore, the sensitivities of the leaky quasi-SP-BIC resonances excited by TM and TE modes in symmetry-broken Case II and Case III closely resemble that of Case I.

In summary, we achieve multi-wavelength high-Q quasi-BIC resonances at dual polarizations by employing arbitrary symmetry-broken perturbations. Within the S-shaped metasurface, three distinct structural asymmetry manners are introduced and excite the same quasi-BIC modes. This leads to triple quasi-BICs dominated by Qm and ET at wavelengths of 1071.18, 1098.8, and 1199.6 nm in TM modes, and two quasi-BICs dominated by Qe and M at wavelengths of 1375.9 and 1628.5 nm in TE modes. Moreover, the obtained high-Q quasi-BIC resonances demonstrate a sensitivity of up to 900 nm/RIU. Our work provides more degrees of freedom for the excitation and tuning of quasi-BICs. The resulting quasi-BICs hold promising applications in nonlinear optics and biosensing.