Photonic crystals (PhCs) are structured materials with periodic dielectric patterns that enable precise control over light propagation and localization^[1,2]. By functioning as mirrors, waveguides, and resonators, PhCs can effectively reflect and trap light, leading to a wide range of applications, including optical fibers, sensors, and lasers^[3,4]. Defects within these structures play a significant role in their functionality. For instance, introducing point defects into the bulk of a PhC can create localized modes that confine light at specific frequencies^[5,6]. Furthermore, by incorporating a linear defect, it is possible to establish waveguide modes that propagate along the defect itself^[7,8]. Especially, when defects are arranged periodically or in coupled configurations, their interactions give rise to collective modes that exhibit distinctive light behaviors, such as high-quality factors (Q-factors) and slow group velocities^[9]. These phenomena open new avenues for innovative applications, including ultrafast PhC lasers^[10], optical delay components^[11], and nonlinear enhancement^[12].

In recent years, topological photonics has emerged as an exciting advancement in PhC design, drawing inspiration from topological insulators (TIs) in condensed matter physics^[13,14]. This field introduces photonic modes that are resilient to disorder and resistant to backscattering, thanks to the topological properties inherent in the band structure of PhCs. These properties often manifest as protected edge and corner modes^[15–17]. In particular, topological defects—immutable deviations from an ideal crystalline structure—have become a rapidly evolving area of research^[18]. Recent developments include 1D and 0D bound modes in dislocations of TIs^[19,20], fractional charges associated with disclinations in crystalline TIs^[21], and Dirac vortex modes in graphene-like lattices^[22,23]. These advances are paving the way for applications, such as novel nanolasers^[24,25], modifications of the hosting material’s topological properties^[26,27], and interactions with phenomena like non-Hermitian effects^[28,29] and topological pumping^[30].

Resonances in planar micro-nanostructures represent another vibrant area of research in photonics, offering powerful mechanisms for enhancing light–matter interactions on chip^[31]. For example, dipole modes, which are characterized via oscillating dipole moments induced by electromagnetic waves, naturally prefer to radiate and underpin fundamental phenomena in plasmonic resonances and Mie resonances in metasurfaces^[32,33]. In contrast, monopole and quadrupole modes inherently prefer not to radiate, effectively acting as dark modes^[34,35]. Bound states in the continuum (BICs) are confined modes that exist within the continuum of radiative modes yet remain perfectly localized without radiative loss^[36,37]. Perturbations can lead to leaky BICs, known as quasi-BICs, which act as resonances with high Q-factors and long lifetimes, making them valuable for applications in lasers and nonlinear optics^[38]. Recent studies have further unveiled a connection between BICs and topological Weyl semimetals^[39]. In parallel, photonic flatbands have also gained attention due to their potential to support highly localized modes in momentum space, facilitating strong light–matter interactions^[40,41]. Especially, flatbands that exhibit flat features only in a specific portion of the Brillouin zone (BZ) are referred to as quasi-flatbands, which have potential applications in some special scenarios^[42,43]. The combination of resonances with flatbands in topological PhCs presents unique opportunities for realizing photonic modes that are not only strongly confined and robust but also exhibit enhanced stability.

In this Letter, we present an investigation of quasi-flatband resonances and BICs supported by coupled photonic topological defects in a PhC slab. Specifically, we identify anisotropic quasi-flatband resonances and isotropic quasi-flatband symmetry-protected BICs within this topological system, where the topological defects are characterized by nontrivial 2D Zak phases. Our analysis reveals that the resonances originate from dipole corner modes localized in second-order TIs (SOTIs) with shrinking nontrivial regions, while the BICs emerge from quadrupole and monopole corner modes within these regions. The band formation of these modes is theoretically described using a tight-binding (TB) model, and their distinct radiative properties arise from the sub-wavelength scale of the supercells, which significantly suppress diffraction channels. We further explore the physical characteristics of these modes through multipole decomposition, as well as their spectral response under far-field excitations, demonstrating that the quasi-flatband resonances are highly accessible and robust. As a proof of concept, we highlight several potential applications of the quasi-flatband resonances, including angle-robust optical forces, enhanced nonlinear generation of 2D materials, and strong light–matter coupling. These findings offer valuable insights into the interplay between topological photonics, optical resonances, and flatband physics, paving the way for innovative photonic device designs and advanced applications.

Figure 1(a) illustrates the geometric configuration of the topological PhC supercell array. The array consists of periodically arranged supercells, each constructed from PhC hole slab structures with a permittivity of ε=11.7. Each supercell comprises a topologically nontrivial unit cell surrounded by half of a trivial unit cell, as indicated by the dashed box in Fig. 1(a). This arrangement ensures that each nontrivial unit cell is separated from others by a row of trivial unit cells, resulting in a supercell periodicity of P=2a. Both the trivial and nontrivial unit cells share a lattice constant a and consist of four compact or expanded identical air holes with w=0.34a, as shown in the insets of Fig. 1(b). The topological properties of these unit cells are governed by the 2D Su–Schrieffer–Heeger (SSH) model, whose Hamiltonian is given by^[16]H=(0ta+tbeikxta+tbe−iky0ta+tbe−ikx00ta+tbe−ikyta+tbeiky00ta+tbeikx0ta+tbeikyta+tbe−ikx0),(1)where ta and tb denote the intra- and inter-cell hopping amplitudes, respectively, which can be tuned by displacing the four air holes along the unit cell diagonals. Within the framework of the 2D SSH model, each supercell can be interpreted as an SOTI featuring shrinking nontrivial regions^[16]. Simultaneously, the PhC slab can also be viewed as a configuration of coupled point defects distributed within a large trivial region, where each point defect is defined by nontrivial topological properties.

Figure 1(b) illustrates the first two photonic bands of both the trivial and nontrivial unit cells within the first BZ, calculated using the finite element method (FEM). The analysis is confined to transverse electric (TE) modes below the light line, as the hole-based structure promotes TE band gaps. Notably, both unit cells exhibit equivalent band structures, featuring a band gap between the first and second bands. The distinction in their topological properties is characterized by their 2D Zak phases, θ=(θx,θy), which are determined based on field parity at the high-symmetry Γ and X(Y) points, expressed as^[44]θα=π(∑iqαimod2),(−1)qαi=η(Xα)η(Γ),α=x,y,(2)where Xα represents either the X or Y point for α=x or y, η denotes the field parity at high-symmetry points relative to the plane at α=a/2, and i indexes the bands below the gap that are restricted to the first band in this scenario. Given the unit cell’s C4 symmetry, it follows that θx=θy. For the trivial unit cell, the first band exhibits even parity at both Γ and X points, as denoted by the green “+” symbols in Fig. 1(b). This leads to a 2D Zak phase of (0,0), identifying it as topologically trivial. In contrast, the nontrivial unit cell demonstrates a 2D Zak phase of (π,π), attributed to the odd parity observed at the X point [blue “−” symbols in Fig. 1(b)]. This parity reversal results from band inversion between the first two bands, a hallmark of the topological phase transition process^[45].

We next focus on the optical properties of the supercell shown in Fig. 1(a). The right side of Fig. 1(c) presents the eigenmode analysis of the supercell at the Γ point, revealing four modes within the band gap of the unit cell. The field distributions of the Hz component at the midplane of the slab are depicted on the right side of Fig. 1(d). These distributions clearly demonstrate that the optical fields of these modes are highly localized within the regions containing nontrivial defects. Notably, these modes exhibit features consistent with topological multipole corner modes^[46]. However, unlike the modes in Ref. [46], where the optical fields are distributed across the four corners as shown on the left side of Fig. 1(d), the fields in this work are localized within the nontrivial defect regions, forming distinct topological defect modes. The evolution of these modes at the Γ point, as the nontrivial regions contract from m=3 to m=1, is shown in Figs. 1(c) and 1(d), providing clear evidence that these topological defect modes originate from topological corner modes. In other words, these defect modes represent specialized cases of corner modes. Accordingly, we designate these modes as quadrupole, monopole, and dipole I and dipole II defect modes, in descending order of eigenfrequency at m=1, as marked by the colorful symbols in Figs. 1(c) and 1(d). More band structure and Q-factor analysis is provided in the Supplement Material.

We further explore the band structures of these coupled topological defect modes at m=1 in the vicinity of the Γ point. The numerically calculated iso-frequency surfaces of the four bands with their color-mapped Q-factors are presented together in Fig. 2(a), with the band structures along the −X−Γ−X direction shown in Fig. 2(b). Interestingly, the quadrupole and monopole defect modes exhibit isotropic flatbands with localized partial flatness near the Γ point. In contrast, the bands of the two dipole defect modes feature locally confined directional flatbands along the Γ−X or Γ−Y directions, with dispersive behavior elsewhere, and degenerate eigenfrequencies at the Γ point. These distinct band characteristics arise from the unique symmetry properties and mode-coupling mechanisms in the supercell. The monopole defect mode exhibits isotropic localization of optical fields in real space, driven by its highly symmetric field distribution, which lacks directional preference. This isotropy suppresses dispersion in all directions within the BZ, resulting in an isotropic flatband near the Γ point. Similarly, the quadrupole defect modes possess a radiation pattern with four lobes governed by C4 symmetry, which results in equal radiation along the four Γ−M directions. This symmetry ensures uniform coupling to all four neighboring sites in a given lattice direction, facilitating the flatband formation. In contrast, the dipole defect modes exhibit C2 radiation symmetry, characterized by a two-lobe field distribution with strong directional anisotropy in real space. This anisotropy promotes coupling along specific high-symmetry directions in the BZ, breaking the isotropy and giving rise to flat ridges along these directions. Theoretically, since the array includes only one site per supercell, the band dispersion can be expressed in the TB approximation using the Bloch theorem^[47]: ω(k)−ω0=−∑Rt(R)eik·R,(3)where k=(kx,ky) is the Bloch wave vector, R is the lattice translation vector, and ω0 is the on-site angular eigenfrequency. The coupling terms t(R) represent the interactions between sites separated by R. Our analytical results align with the numerical results only when considering up to third-nearest-neighbor coupling, as detailed in Fig. 2(d). This reveals the long-range interaction nature of these coupled defect modes, which are governed not only by local interactions but also by distant lattice sites. These features distinguish them from modes associated with local flatbands, where both near- and long-range interactions are negligible. The nonlocal flatbands in this work demonstrate more compact resonant field localization, leading to a higher local field density that is highly beneficial for enhancing light–matter interactions. The detailed form of Eq. (3) is discussed in the Supplement Material.

Despite being situated beyond the light line, the defect modes studied here exhibit distinct radiative behaviors. As shown by color maps in Fig. 2(a), the quadrupole and monopole defect modes achieve infinitely high Q-factors at the Γ point and follow the scaling rule of Q∝1/k2, a hallmark of BICs^[36]. In contrast, the dipole defect modes exhibit finite Q-factors and display conventional resonance characteristics. These differences arise from the interplay between flatband characteristics and the symmetry of the defect modes. The flatbands associated with the quadrupole and monopole defect modes near the Γ point correspond to an ultraslow group velocity of light, which significantly extends the decay time of these modes and enhances field localization within the supercell. Additionally, the sub-diffraction-limited period of the supercell allows only a single radiative channel at the Γ point for coupling energy into the far field. The quadrupole and monopole defect modes, governed by higher-order rotational symmetries, exhibit optical field distributions that lack a net dipole moment. Consequently, these modes are protected from radiative losses and manifest as symmetry-protected quadrupole BICs (qBICs) and monopole BICs (mBICs).

In contrast, the dipole defect modes characterized by C2 radiation symmetry align with the field patterns of the propagating electromagnetic waves in free space, facilitating efficient coupling to the far-field radiative channel. Furthermore, the limited flatband characteristics of the dipole modes reduce field localization, leading to a conventional resonance behavior with lower Q-factors. These combined factors give rise to the formation of two paired dipole resonances (dRes I and dRes II, respectively). To gain deeper insight into the origin of these defect modes, we calculate their scattering powers at the Γ point for various multipoles in the Cartesian coordinate system using the multipole scattering theory in the Supplement Material, which further reveals the rich multipole landscape of topological defect modes in the PhC slabs. We also extend our investigation into the properties of the two BICs by examining their far-field radiation characteristics in momentum space, finding that the qBIC possesses a topological charge of q=+2, while the mBIC yields a topological charge of q=−2. These results provide a clear demonstration of the distinct topological characteristics of the two BICs.

To explore the potential applications of this PhC slab under external illumination, plane waves are introduced to perform far-field excitation, with full-wave calculations conducted using the finite-difference time-domain (FDTD) method. Here, we focus on the transmission spectra along the −X−Γ−X direction, where the incident angle is characterized by the polar angle θi, as indicated in Fig. 1(a). The magnitude of the wave vector k on the surface plane is given by k=(ω/c)sinθi, where c is the speed of light in a vacuum^[48]. The transmission spectra of the PhC slab for the p- and s-polarized plane wave incidences at angles θi ranging from −30° to 30° are presented in Figs. 3(a) and 3(b), respectively. Under p-polarized wave excitation, a significant resonant dip is observed near the frequency f=0.31a/λ, exhibiting angle-invariant transmission properties across the entire range of incident angles. In addition, a resonant dip with a much narrower linewidth is detected near f=0.36a/λ, which becomes highly accessible only at larger incident angles. These two transmission features align well with the dipole II and quadrupole band structures shown in Fig. 2(c), confirming the nature of these resonances. Two other transmission trajectories, corresponding to PhC TM modes, are also present but are not the focus of this study. In contrast, under s-polarized wave excitation, a significant flat resonant dip is absent. Instead, a highly dispersive resonant dip is observed near f=0.31a/λ, along with a second resonant dip with a narrower linewidth near f=0.34a/λ, which is also efficiently accessible only at larger incident angles. These two transmission trajectories correspond closely to the dipole I and monopole band structures shown in Fig. 2(c), further confirming their resonance origins. Due to the C4 symmetry of the supercell structure, the transmission spectra exhibit similar features along the −Y−Γ−Y direction. Thus, using plane waves, we successfully achieve selective resonant excitation of these topological defect modes. Potential experimental implementations are discussed in the Supplement Material.

While the quasi-flatband BICs demonstrate significantly higher Q-factors compared to the quasi-flatband resonances and have recently garnered attention for various applications^[42,43], their practical applicability is limited by their narrower working bandwidths, restricted accessible angle ranges, and angle-dependent radiative responses. In contrast, the quasi-flatband resonances offer stable resonant frequencies and efficient excitation across the entire range of incident angles, making them promising candidates for diverse practical applications—a potential that has been largely overlooked in previous studies.

One such application is angle-invariant optical force generation. Optical forces are critical for light–matter interactions at the mesoscale, and strong light-driven near-field forces in various micro- and nanostructures have been extensively studied^[49]. Here, we consider a periodic array of nanoparticles positioned 150 nm away from the slab, with each nanoparticle precisely aligned at the center of the supercell as shown in the inset of Fig. 4(a). In this and the subsequent discussion, the lattice constant of the unit cell is set to be a=1μm. The nanoparticles, with a diameter of 200 nm, are assumed to be lossless and possess a refractive index of 2.0. The time-averaged optical force acting on a nanoparticle due to resonant optical fields is calculated using the Maxwell stress tensor (MST). The net force is obtained by integrating the MST over a closed surface S surrounding the nanoparticle^[50], F=∮S⟨T↔⟩·n^dS,(4)where n^ represents the unit vector normal to the surface S, and ⟨·⟩ denotes the time-average operator. The elements of the MST for optical fields are expressed as ⟨Tij⟩=18πℜ[εEiEj*+μHiHj*−12δij(εE·E*+μH·H*)].(5)

Here, δij represents the Kronecker delta, and ε and μ are the permittivity and permeability of the medium, respectively. The magnitudes of the force acting on the nanoparticle under the p- and s-polarized wave incidences at variant angles are shown in Figs. 4(a) and 4(b), respectively. Here, the incident electric field amplitude is assumed to be 1 V/m. Considering the mirror symmetry of the spectral response with respect to the Γ point, we focus exclusively on the case of the positive-angle incidence. For the p-polarized wave incidence, the results demonstrate that the dRes II can generate a resonant optical force on the order of 10−3pN near f=0.309a/λ, three orders of magnitude higher than the background non-resonant optical force. Especially, this force remains stable in frequency and amplitude across the entire range of incident angles. Notably, the giant optical forces observed at θi=±25° arise near the TE-TM mode crossing point, resulting from the spatial overlap between the electric and magnetic fields^[51]. This phenomenon provides an intriguing pathway for enhancing optical forces. However, a detailed investigation of this mechanism lies beyond the scope of the present study and will be addressed in future work. In contrast, under the s-polarized wave incidence, the resonant optical force exhibits pronounced frequency dispersion. This comparison highlights that quasi-flatband resonances ensure consistent force performance under varying illumination angles. This angular robustness is particularly advantageous for applications in optical manipulation and trapping, where uniform force is essential. Furthermore, the enhanced field localization associated with quasi-flatband resonances amplifies the optical force, facilitating precise and efficient operation.

The second application of quasi-flatband resonances introduced here is the enhancement of nonlinear optical effects, specifically second-harmonic generation (SHG) in 2D materials. As a proof of concept, we consider a 2D material of 1 nm thickness placed on the surface of the slab, as shown in the inset of Fig. 5(a). The 2D material is assumed to be lossless with a refractive index of 2.8 and a second-order nonlinear susceptibility χ(2)=1−10m/V. Under an incident electric field amplitude of 109V/m, the total SHG power is calculated by integrating the reflected and transmitted Poynting vector. The SHG powers under normal incidence, when the 2D material is placed on the PhC slab or a bare substrate (PhC slab without holes), are shown in Fig. 5(a). Compared to the substrate, the PhC slab induces a pronounced SHG peak at f=0.615a/λ, approximately twice the resonant frequency of dRes I. This enhancement is attributed to the near-field localization of the resonant optical fields, which in turn amplifies the nonlinear polarization of the 2D material^[52]. To quantify this effect, we define the SHG enhancement factor at the peak position as η=PPhC/Psub, where PPhC and Psub represent the SHG powers when the 2D material is on the PhC slab or substrate, respectively. Figures 5(b) and 5(c) further illustrate the SHG peak positions and enhancement factors for the p- and s-polarized wave incidences at varying angles. The results clearly demonstrate that quasi-flatband resonances, compared to non-flatband resonances, deliver more stable SHG peak positions. Meanwhile, quasi-flatband resonances yield an angle-averaged SHG enhancement factor of ∼56, representing a more than twofold increase over non-flatband resonances, which exhibit an average enhancement of ∼21. This stability and enhanced efficiency make quasi-flatband resonances particularly advantageous for nonlinear optical applications requiring robustness to variations in incident angles.

The final application of quasi-flatband resonances explored here is the angle-invariant strong coupling between photons and quasi-particles. Quasi-particles, such as excitons and phonons, are emergent entities that arise from interactions between particles, exhibiting unique properties in 2D materials due to the reduced dimensionality^[53]. Their strong coupling to photons in resonators enables efficient light–matter interaction, which is crucial for applications like polaritonic devices^[54]. Conventional strong coupling is typically characterized by observing Rabi splitting at one specific point in momentum space^[55]. In contrast, we demonstrate that quasi-flatband resonances can significantly extend the strong coupling region across momentum space. To illustrate the concept, we consider an ideal 2D material with quasi-particle responses deposited on the PhC slab. The in-plane permittivity of the 2D material is described by the Lorentz oscillator model^[56], εm(ω)=ε∞+fωm2ωm2−ω2−iΓmω,(6)where ε∞=2.9 represents the background permittivity, ωm is the collective excitation frequency (assumed to align with the resonant frequency of dRes at the Γ point), Γm=0.011ωm is the damping rate, and f=0.457 is the reduced oscillator strength. Figure 6(a) presents the transmission spectra of the suspended 2D material and the PhC slab under normal incidence. The spectral overlap between the eigenfrequency of the quasi-particle resonance and dRes facilitates coherent coupling between them. When the 2D material is placed atop the PhC slab, the transmission spectrum reveals spectral splitting, known as Rabi splitting, as shown in Fig. 6(a). This splitting is a hallmark of strong coupling between the quasi-particle resonance and dRes. The condition for strong coupling is given by^[57]C=Ω2Γd2/2+Γq2/2>1,(7)where C is the strong-coupling factor and Ω represents the Rabi splitting. Γd and Γq are the linewidths of the dRes and the quasi-particle resonance, respectively, which are extracted by fitting the transmission spectra with the Fano formula. In this case, we obtain C=3.96, which is significantly greater than 1. This clearly demonstrates that the strong coupling condition is fully satisfied at the Γ point.

The transmission spectra for both the p- and s-polarized wave incidences at varying angles are depicted in Figs. 6(b) and 6(c). Under the p-polarized wave incidence, the transmission peak splitting persists across the entire range of incident angles, indicative of robust strong coupling, as shown in Fig. 6(b). This phenomenon arises from the continuous spectral and spatial overlap between the dRes II and the quasi-particle resonance. In contrast, under the s-polarized wave incidence, a frequency mismatch between them occurs as the angle deviates from the Γ point, leading to the rapid disappearance of strong coupling, as shown in Fig. 6(c). The angle-invariant strong coupling facilitated by quasi-flatband resonances thus provides a robust light–matter interaction mechanism for highly efficient polaritonic devices with angle-independent operation.

In conclusion, the exploration of quasi-flatband resonances and quasi-flatband BICs supported by coupled topological defects in PhC slabs reveals exciting opportunities for light manipulation in photonic devices. This study uncovers topological defect modes originating from topological corner modes, characterized not only by quasi-flatband properties but also by distinctive resonant behaviors. These modes demonstrate remarkable angular and spectral stability under far-field excitation, making them highly suitable for diverse applications. Notably, the quasi-flatband resonances exhibit consistent and enhanced performance across varying incident angles, enabling advanced functionalities in light–matter interaction applications such as optical force manipulation, nonlinear enhancement, and strong coupling. This represents a novel pathway for the development of photonic devices with superior performance. By synergizing topological photonics, optical localization, and flatband physics, this work lays the foundation for the next generation of light-based technologies, including advanced optical tweezers, quantum light sources, and nanoscale polaritonic devices.