Bound states in the continuum (BICs) are resonant modes that exist within the radiation continuum yet remain perfectly confined, exhibiting near-zero radiation loss and theoretically infinite quality factors (Q-factors). They hold immense application potential in lasers, filters, sensing, and other fields. Among them, symmetry-protected BICs (SP-BICs), whose Q-factors can be flexibly tuned via symmetry breaking, represent a hot research topic in the metasurface field.
Current mainstream research largely focuses on monolayer metasurfaces with in-plane C2 symmetry. The Q-factors of such structures are highly sensitive to minute wavevector perturbations; even slight fabrication defects cause severe Q-factor degradation, resulting in narrow usable parameter windows and high fabrication thresholds, severely limiting practical device applications. Meanwhile, the physical mechanisms of SP-BICs beyond C2 symmetry in higher-order symmetric systems remain insufficiently explored. Multilayer architectures that introduce out-of-plane degrees of freedom often require ultra-high-precision nanofabrication, where processing deviations directly deteriorate device performance. Achieving both high-Q resonance characteristics and fabrication error tolerance remains a critical challenge to be addressed.
To address these pain points, the research team proposed a bilayer terahertz metasurface design strategy based on S4 point-group symmetry. This new type of SP-BIC structure consists of two identical monolayer metasurfaces rotated by 90° along the z-axis and combined with broken inversion symmetry, exhibiting strong robustness. It breaks through the bottleneck of conventional QBICs—poor robustness and extremely narrow fabrication tolerance—and contributes to understanding the physical essence of SP-BIC generation in three-dimensional configurations. Simulations and experiments reveal that when the two upper elliptical cylinders are parallel to each other (θ = 0°), S4 symmetry is satisfied, forming two discrete dark modes—SP-BICs. Introducing angular perturbation to break S4 symmetry transforms both ideal BICs into QBICs, producing characteristic sharp Fano resonance dips in the transmission spectrum.
Through group-theoretical selection rules and scattering matrix/effective Hamiltonian analysis, the team further elucidated the distinct physical roles of S4 and its C2(z) subgroup in BIC formation: S4 simultaneously protects two sets of BIC modes at the overall representation level, while C2(z) independently suppresses radiation within each layer—an effect that does not rely on interlayer interference. Using lateral interlayer displacement as a quantifiable symmetry-breaking parameter, both numerical simulations and experimental measurements confirmed that the architecture maintains stable resonance performance under fabrication misalignment and structural perturbations, significantly relaxing precision requirements for micro-/nanodevice fabrication.
1. S4 symmetry couples in-plane rotation with out-of-plane reflection to realize dual BIC modes
Extending traditional monolayer in-plane symmetry to a bilayer architecture, two sets of SP-BICs with distinct intrinsic properties are simultaneously constructed via S4 point-group operations. At the Γ-point, the free-space continuum belongs to the two-dimensional degenerate E representation of S4, while TM1 and TM2 modes correspond to one-dimensional irreducible representations A(χ(S4) = +1) and B(χ(S4) = -1), respectively. According to the Wigner–Eckart theorem, the mode–radiation channel overlap integrals are strictly zero, yielding dual BIC modes. The evolution of Q-factors with asymmetry parameter α = sinθ strictly follows the inverse-square power law (Q ∝ α-2), consistent with the classical BIC-to-QBIC transition. The two modes exhibit counter-winding polarization distributions near the Γ-point in the Brillouin zone, with topological charges V1 = -1 and V2 = +1, respectively.
2. High Q-factors maintained under wide-range displacement perturbation, substantially enhanced resistance to fabrication defects
Simulating common interlayer alignment errors in nanofabrication, the lateral displacement parameter Dx was introduced for robustness testing. Within a wide tolerance range of |Dx| ≤ 42 μm, the resonance frequency shifts of both QBICs remain below 0.01 THz, with relative fluctuations less than 0.33%; Q-factors are stably maintained above 102, avoiding the order-of-magnitude Q-factor collapse observed in conventional structures under perturbation. Multipole decomposition analysis reveals that QBIC1 is dominated by magnetic dipole resonance and QBIC2 by electric quadrupole resonance; their distinct local electromagnetic responses collectively endow the structure with excellent misalignment immunity.
3. Experimental validation of robustness with excellent agreement between measurement and simulation
Three bilayer metasurface samples with different interlayer misalignments—zero offset, left-shift of -15 μm, and right-shift of 20 μm—were fabricated and characterized using terahertz time-domain spectroscopy (THz-TDS). Even under significant interlayer offsets, the sample spectra clearly exhibit double-resonance transmission dips, consistent with simulation predictions.
