Metasurfaces, comprising subwavelength structures with unprecedented electromagnetic manipulation ability^[1–3], have found applications in diverse fields, such as perfect vortex beam generation^[4], wide-angle imaging^[5,6], polarization multiplexing^[7,8], and more^[9]. These efforts push engineering optics into a new era: digital optics^[10]. High-quality (Q) factor resonances produced in metasurfaces play a pivotal role in advancing high-performance nanophotonic devices, owing to their exceptional abilities to trap and modulate light. The Q-factor is one of the key parameters for characterizing resonance strength. A higher Q-factor signifies an extended photon lifetime, greatly enhancing the duration of light–matter interactions. Notably, metasurfaces based on bound states in the continuum (BICs) have recently exhibited remarkable capabilities in generating ultrahigh-Q resonances. Thanks to their potent electromagnetic near-field enhancements and ultrahigh-Q properties, metasurfaces can effectively enhance and manipulate light–matter interactions within the radiative continuum^[11–14]. Consequently, they enable a diverse range of applications, including ultrasensitive biosensing detection^[15–17], strong coupling^[18–20], nonlinear interaction^[21–24], high-Q nanolasers^[25–27], electro-optical modulators^[28], and narrowband wavefront control^[5,6,29].

A genuine BIC can only manifest in ideal lossless infinite structures or under extreme parameter values, representing a resonant state with a vanishing resonance width and an infinite radiative Q-factor^[11,12]. Depending on their generation mechanism, BICs can be classified into two types: (1) Accidental BICs emerge from the interference effect between multiple radiation channels or modes, which can be further subdivided into single resonance parametric^[30], Friedrich–Wintgen (F–W)^[31], and Fabry–Perot BICs^[32]; (2) Symmetry-protected BICs (SP-BICs), emerging from the limitation of external radiation by symmetry, stand as the most prevalent and straightforward type among the current BIC family. F–W BICs are typically formed through destructive interference between two resonances^[33,34]. This mechanism enables the coupling between two non-orthogonal quasi-BIC resonances to enter a regime of strong modal coupling, characterized by crossover avoidance properties, whereas for the induced excitation of high-Q quasi-SP-BICs, they can destroy the symmetry of the excitation field via oblique incidence^[35,36], adjust the relative displacement between the structures^[37–39], employ distinct refractive indices for each structure^[40,41], or introduce the asymmetric way of the structure, such as altering the size, shape, or rotation angle of the structure^[42–46]. However, the high symmetry dependency of quasi-SP-BICs leads to their Q-factors being extremely sensitive to slight variations in the asymmetric parameter, significantly constraining the practical applicability of SP-BICs. Moreover, in practical experiments, additional radiation losses arise from fabrication defects or disorders, compounding with inherent material losses and limitations imposed by constrained sample sizes. These factors significantly diminish the Q-factors of quasi-SP-BICs.

To break the symmetry dependency of SP-BICs, three distinct approaches have been proposed. The first approach involves merging multiple BICs in the momentum space, effectively reducing the radiation loss by combining multiple topological charges in the Brillouin zone and further improving the Q-factors of BICs. For example, Jin et al.^[47] proposed merging multiple BICs, i.e., SP-BIC and accidental BIC in momentum space, to achieve a relationship of Q∝k−6. Their experiments demonstrated topology-enabled quality factors as high as 4.9×105, which exhibits robustness against in-plane fabrication defects. Kang et al.^[48] further enhanced the quality factor by combining multiple BICs with a higher-order BIC. This adjustment increases the scaling property to Q∝k−8, resulting in a larger quality factor for smaller values of k and thereby improving robustness against in-plane defects. The second one is to fold the induced BICs in the Brillouin zone, where the guided modes originally located below the light cone are folded inside the light cone into BICs by introducing periodic perturbations, thus realizing BICs with sustainable ultrahigh-Q in the momentum space. Experimentally, it has been shown that this approach significantly enhances the Q-factor of the resulting BICs in a broader momentum space, with a maximum measured Q-factor of 860^[49]. The third one is to employ a strongly robust high-Q metasurface based on lattice hybridization and two SP-BICs interconversion mechanisms. This approach was employed by researchers to experimentally measure quasi-BIC modes, achieving a maximum Q-factor of 4130 and a minimum Q-factor of 1024, thereby demonstrating its robustness to asymmetric parameters^[50]. Although the initial two approaches have indeed made substantial strides in bolstering the robustness of the Q-factor, their focus is confined to analyses within momentum space, disregarding the influence of corresponding asymmetric parameters on the Q-factor in real space. The third method does improve the robustness of the Q-factor in real space to some degree, offering an efficient way to achieve robust high-Q quasi-BIC metasurfaces. However, the current predominant focus of studies aimed at augmenting the robustness of SP-BICs for high Q resides in the analysis conducted within momentum space, receiving sporadic consideration in real space. Additionally, achieving ultra-high Q metasurfaces with strong robustness across a wider range of asymmetric parameters remains a significant challenge.

In this work, we propose a novel methodology to overcome the challenges associated with the high dependency on symmetry in achieving strong robust quasi-BICs with ultrahigh-Q. As a validation, we utilize a metasurface composed of four square silicon blocks. The excitation of four high-Q quasi-BIC resonances can be achieved by breaking both translational and structural symmetries of the metasurface via spacing and length perturbations, respectively. Moreover, the interconversion between the two SP-BICs is facilitated through length perturbation, successfully breaking the high dependency of SP-BIC on the asymmetric parameter. When the relative asymmetric parameter (ΔL/L) reaches 97.2%, the order of magnitude of the Q-factor of the obtained quasi-BIC resonance remains constant. As displayed in Table 1, our methodology enhances the robustness of the Q-factor for the quasi-BIC by a minimum of 2 orders of magnitude in comparison to the same type of methods, despite our relative asymmetric parameter being approximately 10 times the corresponding work. This work represents a substantial improvement in the high-Q robustness of quasi-BICs, offering a new opportunity and effective method for designing high-performance lasers and nonlinear devices.

The SP-BIC has the potential to attain an infinite Q-factor by eliminating radiation loss in theoretical terms, but it is extremely sensitive to asymmetry. As illustrated in Fig. 1(a), a typical SP-BIC manifests an infinitely high Q at the Γ point of high symmetry in momentum space or when the asymmetric parameter is zero in real space. However, upon the disruption of symmetry, i.e., away from the Γ point or as the asymmetric parameter increases, the Q-factor of the SP-BIC experiences a rapid and substantial decrease. To address this issue, this work proposes a new strategy based on an interconversion mechanism between two SP-BICs to break the high dependence of SP-BICs on symmetry, as shown in Fig. 1(b). The specific structural design is illustrated in Fig. 1(c). The proposed all-dielectric metasurface comprises arrays of amorphous silicon with four square blocks, deposited on a quartz substrate with a refractive index of 1.46. The geometrical dimensions of the unit cell of the metasurface are as follows: Λx=Λy=1000nm, L=360nm, h=250nm, and s=140nm. As depicted in Fig. 1(d), the asymmetric parameters Δs and ΔL represent the variation in distance for the movement of a single square block and the variation in the side length of the right two square blocks along the y axis direction, respectively. Specifically, Δs=0nm indicates that each square block in the unit cell is uniformly distributed with a period of Λx/2 (Λy/2) along both x and y axis directions, forming a symmetric metasurface. When Δs≠0, it indicates that the spacing of the square blocks within the unit cell differs from the spacing between unit cells, breaking the translational symmetry of the metasurface and resulting in leaky quasi-BIC resonances. ΔL=0nm indicates that all four square blocks are equal in size, rendering the unit cell symmetric in the x–y plane. In ΔL≠0, symmetry-breaking is introduced, in which the left two square blocks are not equal in size to the right two square blocks, and the overall structure is asymmetric concerning the y axis, resulting in leaky quasi-BIC resonances. Additionally, manipulating the length perturbation ΔL enables interconversion between the two SP-BICs, thereby breaking the high dependence of the SP-BIC on symmetry, resulting in the strong robust quasi-BIC with a high-Q factor. Numerical simulations of the metasurface are conducted employing the finite element method, utilizing both the COMSOL Multiphysics software and commercial CST software. Within these simulations, the quartz substrate is assumed to be semi-infinite while applying periodic boundary conditions in both x and y directions, along with perfectly matched layers (PMLs) employed in wave propagating direction z.

Initially, introducing spacing perturbation, where the spacing between square blocks varies (Δs≠0nm) while keeping other structural parameters constant, the transmission spectra [depicted in Fig. 2(a)] reveal the emergence of two new leakage quasi-BIC resonance modes within the 1450–1600 nm wavelength range, denoted as MR and TR, respectively. Corresponding Q-factors and resonance wavelengths for MR and TR are presented in Figs. 2(b) and 2(c). As Δs change from ±30 to 0 nm, the resonance wavelengths of MR and TR experience a slight blue shift, while their respective Q factors increase rapidly. Here, the resonance profiles of MR and TR can be described by the Fano formula, with the Q-factor evaluated accordingly (for details, see Sec. S1 of the Supplementary Information). As Δs=0nm, the Q-factors for MR and TR tend toward infinity, resulting in the disappearance of both resonances. At this moment, the square blocks are uniformly distributed with a period of Λx/2 (Λy/2) along both x and y axis directions, defining a symmetric metasurface. As Δs≠0nm, the Q factor peaks of the MR and TR resonances decrease dramatically, exhibiting typical BIC characteristics.

To deepen our understanding of the microscopic properties of the dual quasi-BIC resonances, the contributions of the multipole scattered powers are calculated by the multipole decomposition in Cartesian coordinates (for details, see Sec. S2 of the Supplementary Information), which in turn quantitatively analyzes the role of the dipole excitation in the MR (Δs=10nm) and TR (Δs=10nm) resonances. For MR (Δs=10nm), the contribution of the magnetic dipole (M) accounts for the highest proportion, followed by the electric quadrupole (Qe), which constitutes approximately one-quarter of the M [see Fig. 2(d)]. For TR (Δs=10nm), the resonance is dominated by the magnetic toroidal dipole (MT), whereas the magnetic quadrupole (Qm) constitutes the second-largest contribution, approximately one-tenth of the MT; the other multipoles contribute much less [see Fig. 2(e)]. Furthermore, analysis of the x, y, and z components of the scattered power M in MR (Δs=10nm) and the scattered power MT in TR (Δs=10nm) (for details, see Fig. S2 in Sec. S3 of the Supplementary Information) reveals a pronounced dominance of the y component in the scattered power M and the x component in the scattered power MT. These respective components nearly equal the total scattered power of the M and MT in their contributions. Meanwhile, we analyzed the electromagnetic near-field distributions on different cross-sections corresponding to the dual quasi-BIC resonances at the resonance points. For the MR (Δs=10nm) resonance at 1489.6 nm, it can be determined that the M moment is along the y-direction based on the displacement currents and electromagnetic field patterns [see Fig. 2(d)]. However, for the TR (Δs=10nm) resonance at 1537.5 nm, analysis of corresponding electric field distributions in the x–y plane reveals the presence of clockwise and anticlockwise circular displacement currents in the upper and lower square blocks on the left side as well as the upper and lower square blocks on the right side, respectively. Specifically, this pair of typical displacement current loops tends to lead to the excitation of head-to-tail M moments, i.e., MT moment along the x direction [see Fig. 2(e)]. The above results from electromagnetic field map analysis not only align with the principal components of the scattered power from dominant dipoles associated with the MR (Δs=10nm) and TR (Δs=10nm) resonances but also correspond to the field distributions of TE and TM modes in the intrinsic band structures (for details, see Fig. S5 in Sec. S6 of the Supplementary Information), further affirming the origin of MR and TR from BICs leakage. In summary, the dual resonances, MR and TR, induced by spacing perturbations in silicon square block arrays, are generated by M-dominated leaky quasi-BICs and MT-dominated leaky quasi-BICs, respectively.

Next, with a spacing perturbation of Δs=10nm and other structural parameters held constant, length perturbation is introduced, i.e., by varying the lengths of the y direction of the two square blocks above and below on the right side (ΔL≠0nm). The forthcoming discussion will primarily focus on quasi-BIC resonance modes excited by a normally incident plane wave with Ey-polarization. For detailed information about quasi-BIC resonance modes excited by a normally incident plane wave with Ex-polarization, refer to Sec. S9 in the Supplementary Information. The transmission spectra under Ey-polarized incidence are depicted in Fig. 3(a), revealing the excitation not only of the MR and the TR but also of two new leaky quasi-BIC resonance modes. These resonance modes are labeled as λ1, λ2 for ΔL<0nm and λ2′, λ1′ for ΔL>0nm, respectively. The linewidths of λ1−λ2′ and λ2−λ1′ narrow as the absolute value of ΔL decreases. When ΔL=0nm, both λ1−λ2′ and λ2−λ1′ vanish, indicating no energy leakage from the bound state into the free-space continuum. Circles labeled in Fig. 3(a) demonstrate that the radiation quality factor tends to infinity when ΔL=0nm, implying the existence of two BICs in Ey-polarized incident light. By analyzing the transmission spectra for ΔL ranging from −60 to 60 nm, it is evident that the transmission spectra of λ1−λ2′ and λ2−λ1′ exhibit an avoidable crossover feature at ΔL=0 nm. This anti-crossover phenomenon of BIC modes indicates that their interaction enters a state of strong coupling. Within the anti-crossover region, a new F–W BIC mode emerges due to the destructive interference of the two modes. The hybridization behavior of the coupling process between these two BIC modes is evident from the analysis of the distributions of the magnetic fields (Hz) in the x–y plane corresponding to λ1−λ2′ and λ2−λ1′ at various ΔL values [see Fig. 3(b)]. Furthermore, the interconversion process between the two quasi-BIC resonant modes, λ1−λ2′ and λ2−λ1′, is evident from Fig. 3(b). Consequently, the authentic corresponding resonant mode to the quasi-BIC resonance λ1 (ΔL<0nm) is λ1′ (ΔL>0nm), and similarly, the authentic corresponding resonant mode to the quasi-BIC resonance λ2 (ΔL<0nm) is λ2′ (ΔL>0nm). The results of the above analyses can be further validated using the TE modes in the intrinsic band structures (for details, see Fig. S6 in Sec. S6 of the Supplementary Information).

When the asymmetry parameter ΔL varies from −60 to 60 nm, this results in a monotonic redshift of all four resonances, primarily attributed to the increase in the effective refractive index. Specifically, Figs. 3(c) and 3(d) depict the resonance wavelengths (λ1−λ1′ and λ2−λ2′) versus the asymmetry parameter ΔL. Additionally, Figs. 3(c) and 3(d) display the Q-factors of λ1−λ1′ and λ2−λ2′ as a function of the asymmetry parameter ΔL. At ΔL=0nm, the resonances λ1−λ1′ and λ2−λ2′ vanish, and the Q factors become infinite. For ΔL<0nm and ΔL>0nm, the peaks of the resonance Q-factors significantly reduce, exhibiting typical SP-BIC features. Surprisingly, as ΔL varies from 0 to −350nm, the Q-factors corresponding to λ2−λ2′ initially exhibit a decreasing trend (−10nm<ΔL<0nm); see Fig. 3(d). Subsequently, as ΔL varies between −10 and −350nm, the Q factor increases instead of decreasing and remains above 105. When the relative asymmetry parameter reaches 97.2% (ΔL=−350nm), the order of magnitude of the Q-factor (λ2−λ2′) remains above 106 without any decrease, indicating that the Q-factor of quasi-BIC (λ2−λ2′) exhibits remarkable robustness against asymmetry. The unforeseen increase in the Q-factor is primarily attributed to the transition of the BIC mode state from the initial SP-BIC state to the new SP-BIC state. The robustness of the Q-factor (λ2−λ2′) to the asymmetry parameter is enhanced by at least 2 orders of magnitude compared to previous studies^{[22,44,50–55]}, despite our relative asymmetric parameter being approximately 10 times the corresponding works^[44,51]. Likewise, the quasi-BIC (λ3−λ3′) excited under Ex-polarized incidence at ΔL=110nm (with a relative asymmetry parameter of 30.6%) maintains its Q-factor magnitude above 106, experiencing a decrease of merely 1 order of magnitude, also demonstrating significant robustness to asymmetry [for details, see Fig. S10(c) in Sec. S9 of the Supplementary Information]. Meanwhile, given the realization of high-Q metasurfaces, fabrication defects, material losses, and finite dimensions of the arrays can significantly influence both the intensity and Q-factor of high-Q quasi-BIC resonances, we conducted numerical simulations of quasi-BIC resonance transmission spectra under various geometric conditions and examined the impact of material losses on the Q-factor of these resonances (for details, see Secs. S10 and S11 of the Supplementary Information). Additionally, a brief description of the fabrication process is provided (for details, see Sec. S12 of the Supplementary Information). Finally, we demonstrate that the proposed strategy for achieving robust ultra-high Q quasi-BICs can be applied to other spectral ranges, indicating its general applicability (for details, see Sec. S13 of the Supplementary Information).

To enhance our comprehension of the physics underlying the excited quasi-BIC resonances, we analyze the corresponding electromagnetic near-field distributions of λ1′ (ΔL=1nm), λ1′ (ΔL=40nm), λ2′ (ΔL=1nm), and λ2′ (ΔL=40nm) at the resonant wavelengths, as depicted in Figs. 4(a) and 4(d). The electric field patterns of λ1′ (ΔL=1nm), λ1′ (ΔL=40nm), and λ2′ (ΔL=40nm) all exhibit M modes. Nevertheless, the mechanisms by which they correspond to the M moments generated between the square blocks differ. For λ1′ (ΔL=1nm), it generates a single M moment through the joint coupling of four square blocks within the unit cell. λ1′ (ΔL=40nm) generates two M moments that are excited by the mutual coupling between the left and right two square blocks on the upper and lower sides within the unit cell. Conversely, λ2′ (ΔL=40nm) generates two M moments that are excited by the mutual coupling of the upper and lower two square blocks on the left and right sides within the unit cell. Moreover, for λ2′ (ΔL=1nm), the corresponding electric field distribution manifests as strong displacement currents in the form of a typical Qe within the four square blocks.

To further elucidate the role of dipole excitations analyzed above in λ1′ (ΔL=1nm), λ1′ (ΔL=40nm), λ2′ (ΔL=1nm), and λ2′ (ΔL=40nm), we have similarly utilized multipole decomposition calculations to obtain the contributions of the corresponding resonant multipole scattered powers, as illustrated in Figs. 4(e)–4(h). The results indicate that for ΔL=1nm in λ1′, the M exhibits the highest scattered power, emerging as the dominant multipole. It is closely followed by the electric toroidal dipole (ET) and the Qm. Conversely, for ΔL=1 nm in λ2′, the Qe takes the first place, with the M and Qm following closely. For λ1′ and λ2′ at ΔL=40nm, the M takes the first place in both scenarios, followed closely by the Qe and the ET. Analysis of the x, y, and z components of the M scattered power (for details, see Fig. S3 in Sec. S4 of the Supplementary Information) reveals that the z component exhibits the highest strength in λ1′ (ΔL=1nm), λ1′ (ΔL=40nm), and λ2′ (ΔL=40nm). In summary, the silicon square block array induces dual quasi-BIC resonances, λ1−λ1′ and λ2−λ2′, when subjected to length perturbation under Ey-polarized light. These resonances arise from M-dominated quasi-SP-BICs when ΔL deviates from 0. Conversely, as ΔL approaches 0, the strong coupling effect, attributed to the avoidable crossover property, results in the generation of resonant modes dominated by M and Qe quasi-F-W-BICs.

Finally, we conducted an analysis of the electromagnetic field distributions in the x–y plane for the quasi-BIC resonance mode λ2 when varying ΔL from −50 to −350nm, as shown in Fig. 5. The change in the electric field distributions indicates a gradual decrease in electric field strength in the gap between the square blocks on the left and right sides of the unit cell as ΔL varies from −50 to −350nm. Initially, two M moments are generated by the upper and lower two square blocks on both sides, but later this changes to one M moment generated by only the upper and lower two square blocks on the left side. Analysis of the magnetic field distributions reveals that initially, the main distribution of magnetic field strength occurs in the gap between the upper and lower two square blocks on the left and right sides of the unit cell, as well as at the edges of the square blocks. Subsequently, the magnetic field strength becomes only concentrated in the gap between the upper and lower two square blocks on the left side, including both the edges and the interior. When ΔL=−350nm, this corresponds to a length of 10 nm for the right upper and lower two square blocks in the y direction. From the electromagnetic field distributions, it is evident that the quasi-BIC resonance λ2 predominantly arises due to the mutual coupling effect between the upper and lower two square blocks on the left side. At this time, we can disregard the upper and lower square blocks on the right side. Consequently, we calculate the TE mode in the intrinsic band structures with only the upper and lower two square blocks on the left side of the unit cell (for details, see Fig. S8 in Sec. S7 of the Supplementary Information), and we observe that the electromagnetic field distributions of the calculated TE mode align with that at ΔL=−350nm. The variation in its Q-factor indicates a typical BIC mode, precisely explaining the change in the Q-factor associated with the quasi-BIC resonance λ2 that first decreases and then increases when ΔL varies from 0 to −350nm.

In summary, we propose a novel strategy to realize the strong robustness of quasi-BIC to the asymmetric parameter. The results demonstrate that four leakage quasi-BIC resonances can be induced by breaking the translational symmetry and structural symmetry of the metasurface through spacing perturbation and length perturbation, respectively. Additionally, the interconversion mechanism between two SP-BICs based on the length perturbation significantly decreases the sensitivity of the SP-BICs to asymmetry, enhancing the robustness of high-Q quasi-BICs to the asymmetric parameter. Even with a relative asymmetric parameter of 97.2%, the order of magnitude of the Q-factor of the obtained quasi-BIC remains constant. The obtained results significantly improve the tolerance of ultrahigh-Q quasi-BICs to micro-nano fabrication errors. We expect that the work in this paper can provide a new direction and idea for the studies and applications of ultrahigh-Q metasurfaces that realize strong robustness.