1. Introduction

Bound states in the continuum (BICs) as distinctive localized states that lie within the radiation background, have garnered tremendous attention in diverse wave disciplines such as hydrodynamics [1], acoustics [2], and electromagnetics [3–6]. Especially, the infinite quality (Q) factors and giant field confinement for optical BICs significantly boost the development of nanophotonic, ranging from sensing [7–10], waveguiding [11–13], lasing [14–17] to nonlinear frequency conversion [18–21]. For photonic crystal slabs (PCSs), BICs can be generally categorized as symmetry-protected (SP) BICs and tunable BICs [22]. The SP-BICs originate from symmetry mismatch and are pinned at high symmetric points in momentum space, while tunable BICs are strongly correlated to structural parameters, and can be further divided into accidental (single-resonance) BICs [23,24], Friedrich-Wintgen (FW) BICs [5,25,26] and Fabry-Pérot (FP) BICs [27,28], according to the distinct nature of mode coupling [22,29,30]. Despite of the different underlying physics, all the photonics BICs can be characterized as polarization singularities in momentum space, which are the vortex centers with integer topological charge in the far-field polarization [31–34]. The polarization singularities perspective not only provides a vivid picture for manipulating BICs, but also revitalizes the design of nonlocal optical elements in a topological way [35–47]. Within the context, a merging-BIC configuration empowered by the combination of ultrahigh-Q and topological nature of BICs is proposed [42–47], which can significantly boost Q factors in a broad range of wave vector (k) compared to an isolated BIC. The peculiar feature of merging BIC can substantially suppress the scattering loss caused by fabrication defects, and have potential to enhance light-matter interaction, such as high harmonic generation [46] and low threshold lasing [47].

To date, most works merge not more than two types of intrinsic BICs supported by an isolated PCS, such as merging SP-BIC and accidental BIC at Γ point [30,42,47], and FW-BIC and accidental BIC at off-Γ point [30,44], due to the limited degrees of freedom of structural parameters that are constrained by the symmetry requirements of BICs. However, the types of involved BICs have direct connection with the Q factor scaling rule [42], and more types of BICs benefit the Q factor enhancement. Moreover, the intrinsic BICs of isolated PCS are related to each other, and even to annihilate themselves when structural parameters deviate from the critical values [42,44], which would dramatically decrease the Q value of nearby resonances, especially for the off-Γ merging scenario. To better address the above challenge, more flexible and reliable structure beyond the isolated PCS should be devised.

In this study, we investigate diverse BICs merging including both intrinsic and extrinsic BICs in a PCS-multilayer system. We demonstrate that the intrinsic BICs supported by isolated one-dimensional (1D) PCS, and the extrinsic FP-BICs engineered by the multilayer substrate could be merged together without annihilation, leading to high-Q optical mode in a broaden k-domain. First, we consider an isolated 1D PCS and construct all the three intrinsic BICs, namely SP-BIC, accidental BIC, and FW-BICs on the hybrid photonic band, and achieve the steerable merging design of accidental BIC and FW-BIC at off-Γ points and all three types of intrinsic BICs at Γ point by varying the structural symmetry-independent parameters (i.e., slab thickness and bar width). Furthermore, we incorporate the extrinsic BICs into the dynamic processes, as demonstrated in a PCS-multilayer hybrid system. The gap thickness between PCS and multilayer can be utilized to flexibly manipulate extrinsic BICs, while the intrinsic BICs are immune to the gap tuning because of their non-radiation nature. In particular, we unveil a unique interaction between intrinsic and extrinsic BICs that is polarization singularity collision beyond annihilation, which only exchange the topological charge of BICs but not affect their existence, and all the manipulation processes are carried out in the framework of topological charge conservation. Finally, we demonstrate that the topological scaling rules of merging BIC formed by harnessing both intrinsic and extrinsic BICs are improved from $Q \propto k_x^{ - 4}$ to $Q \propto k_x^{ - 6}$ and $Q \propto k_x^{ - 6}$ to $Q \propto k_x^{ - 14}$ at the off-Γ points and Γ point respectively.

2. Merging the intrinsic BICs

2.1 Formation and dynamics of the intrinsic BICs

We start from an isolated 1D PCS consisting of TiO₂ grating (refractive index n = 2.58) with a lattice constant of a = 330 nm, bar width w = 0.5a, and thickness h = 506 nm. As illustrated in Fig. 1(a), the PCS is fabricated on a homogeneous SiO₂ substrate (n_a = 1.46) and immersed in a matched environment. Without loss of generality, transverse electric (TE)-like Bloch modes are studied hereafter, and the simulations are performed by finite element method (COMSOL Multiphysics 5.6). Figure 1(b) presents two hybrid photonics band with typical anticrossing behavior [5] in the PCS structure, resulting from the strong coupling between TE₀ and TE₂ modes [both of which have even parity in vertical direction, panel (i) and (ii) in Fig. 1(c)]. As shown in Fig. 1(b), the thickness of the band represents the radiation decay rate of the Bloch modes. The upper band (red strip, termed of TE-A) features low loss and even zero radiation loss at some specific wave vectors, such as SP-BICs at the Γ point, accidental BICs at off-Γ points owing to the accidental radiation cancellation [23], and FW-BIC nearby the anticrossing point owing to the destructive interference of the two modes, whereas the lower band (blue strip, termed of TE-B) remains high radiation loss by the constructive interference [5,26]. The field distribution of FW-BIC [panel (iii) in Fig. 1(c)] clearly indicates the superposition of the TE₀ and TE₂ modes. It can be seen that all types of intrinsic BICs coexist on the TE-A band by carefully engineering the PCS structure, as sketched in Fig. 1(b). The BICs can be further corroborated by the far-field polarization map for TE-A band [Fig. 1(d)], which is obtained by polarization projection [31] of radiation field from s-p plane (violet) to x-y plane (blue) for all directions [indicated by in plane wave vector (k_||)], as depicted in Fig. 1(a). The topological charge of BICs is defined as $q = {\textstyle{1 \over {2\pi }}}\oint_L {d{\boldsymbol{k}_{\boldsymbol{\parallel }}} \cdot } {\nabla _{{\boldsymbol{k}_{\boldsymbol{\parallel }}}}}\phi ({\boldsymbol{k}_{\boldsymbol{\parallel}}})$, where $\phi = {\textstyle{1 \over 2}}\textrm{Arg}({S_1} + i{S_2})$ is the orientation angle of polarization states obtained from the Stokes parameters S_i and L denotes a counterclockwise closed loop containing BICs [22,36]. As illustrated in Fig. 1(d), there are five polarization vortexes or anti-vortexes in momentum space for TE-A band, corresponding to the three types of intrinsic BICs. According to the topological nature, BICs move in momentum space without disappearance or splitting suddenly, unless the variation of structure reduces symmetries. All interactions of BIC comply with local topological charge conservation [31–33], and total topological charge in each bounded face of polarization graph is strictly zero [34].

 

Fig. 1. (a) Schematic illustration of an isolated 1D PCS and the far-field polarization state of its eigen Bloch modes. (b) Band structure of the PCS, wherein the red (blue) strip denotes the TE-A (TE-B) hybrid band, and the strip thickness denotes the decay rate. The disk with brown (blue, violet) rim represents SP (accidental, FW) BIC, and the plus (minus) signs therein indicate the topological charges. (c) The electric field distributions for TE₀ mode, TE₂ mode, and FW-BIC in (b) respectively. (d) Far-field polarization map and topological charge distribution of the TE-A hybrid band. Blue (red) region denotes the right (left)-handed ellipse polarization, and black solid lines indicate the linear polarization. (e), (f) The Q factor versus both slab thickness h and wave vector k_x for TE-A and TE-B hybrid band, respectively.

Download Full Size | PDF

By tuning the thickness of PCS, we investigate the dynamic evolution of diverse intrinsic BICs in the photonic hybrid band. Figures 1(e) and 1(f) depict the Q factor distributions in the synthetic space consisted of the slab thickness h and wave vector k_x for TE-A and TE-B band. The BICs are traced by the extremely bright curves, which is consistent with their ultra-high Q characteristic. It can be found that the SP-BIC exists robustly at the Γ point on the TE-A band because h is symmetry-independent, while FW-BICs and accidental BICs undergo different dynamic evolutions along with the variation of h. If the trajectories of intrinsic BICs are non-monotonic in the synthetic space, BICs would encounter at a specific thickness h, such as FW-BIC and accidental BIC on the TE-A band at h = 508.65 nm [green circle in Fig. 1(e)] or pairwise accidental BICs on the TE-B band at h = 530.75 nm [green circle in Fig. 1(f)], which enables the construction of the merging BIC configuration [42–47]. Because the formation of tunable intrinsic BICs is interrelated, BICs would annihilate each other after merging [44]. Herein, we focus on the merging process at the off-Γ point on the TE-A band, and provide more details in Appendix A.

2.2 Merging three types of intrinsic BICs at the Γ point

Given the existence of SP-BIC on the TE-A band, we take a further step to merge all three-types of intrinsic BICs at the Γ point. Because the FW-BIC and accidental BIC exhibit different dependences on structure parameters, we employ the bar widths w (symmetry-independent) in conjunction with h to simultaneously manipulate both types of BICs, and tune their merging wave vector. Figure 2(a) shows the slices of BIC evolution in the w-h-k_x synthetic space. In contrast with Fig. 1(e), the collection of TE₂ accidental BIC and FW-BIC [panel (i) of Fig. 2(a)] gradually move towards Γ point when w increases from 165 nm to 179.4 nm, and another intrinsic BIC pair exists from the opposite wave vector direction, due to the structural in-plane C₂ symmetry. When w = 183.5 nm, the two BIC pairs converge to the Γ point and encounter the SP- BIC [panel (ii) of Fig. 2(a)]. With further enlarging w from the critical value, the merging three-types BIC is destroyed [w = 198 nm, panel (iii) of Fig. 2(a)]. Since the generation of accidental BIC is mismatched with the thickness h where FW-BIC arrives the Γ point, only FW-BIC and SP-BIC are merged.

 

Fig. 2. (a) The Q factor versus both slab thickness h and wave vector k_x of the TE-A hybrid band, when bar width w is 174.9 nm (premerging, panel (i)), 183.5 nm (merging, panel (ii)), and 198 nm (postmerging, panel (iii)). The white dashed lines in panels (i)-(iii) indicate the critical slab thicknesses of 519.98 nm, 529.806 nm, and 541.67 nm where the merging of two off-centered, five centered, and three centered BICs happen. (b) The far-field polarization maps of the TE-A band in momentum space for the PCSs with structural parameters correspondingly indicated by the white dashed lines in panels (i)-(iii) of (a). (c) The Q factor as a function of wave vector k_x for the cases of panel (i) (blue), (ii) (red), and (iii) (violet) in both (b) and (c), wherein the circles and triangles denote the simulated values and the solid lines represent the fitting results.

Download Full Size | PDF

Figure 2(b) shows the distributions of polarization orientation angle in momentum space for the premerging, merging, and postmerging regimes, corresponding to the dashed lines depicted in Fig. 2(a). Before merging, the two pairs of charge ($q ={\pm} 1$) have inversion symmetry about the central charge ($q ={+} 1$). Wider region of ultra-high Q is observed around the off-Γ BIC pair than the isolated SP-BIC, and Q factor varies as $Q \propto k_x^{ - 2}{(k_x^2 - k_{\textrm{BIC}}^2)^{ - 4}}$ for the BIC configuration, as the blue line shown in Fig. 2(c). The simulation results agree well with the fitting curve. For merging scenario (k_BIC= 0), all five intrinsic BICs originated from three diverse mechanisms meet at the Γ point, and Q factors are boosted for nearby resonances following the scaling rule of $Q \propto k_x^{ - 10}$[red line in Fig. 2(c)]. Notably, the peculiar scaling rule can be analytically derived by effective non-Hermitian Hamiltonian [48,49], and more details are presented in Appendix C. Topological transition occurs after merging and the charge of SP-BIC is reversed accompanied by the annihilation of accidental BICs. Therefore, only two FW-BICs with $q ={+} 1$ merge with SP-BIC with $q ={-} 1$ in the postmerging regime [43]. The Q factors decrease and follow the reduced scaling rule of $Q \propto k_x^{ - 6}$[violet line in Fig. 2(c)].

3. Merging the extrinsic and intrinsic BICs

3.1 Formation of extrinsic BICs

In the foregoing sections, we have investigated the manipulation of intrinsic BICs supported by the isolated 1D PCS, and the steerable merging two-types-BICs at off-Γ point or merging three-types-BICs at the Γ point. However, the intercoupled movements and the annihilation between intrinsic BICs (see Appendix A) hinder the Q factor improvement of merging BIC. Considering the present limitations of intrinsic BICs, we introduce extrinsic FP-BIC (for simplicity, we refer it as extrinsic BIC hereafter) as flexible degrees of freedom to broaden the scope of merging BIC configuration in momentum space. Figure 3(a) shows a schematic of our proposed hybrid system, which consists of the PCS we discussed above and a multilayered substrate underneath as the perfect reflector. The multilayer consists of a binary 1D photonic crystal (1DPC) containing 30 pairs of Si₃N₄ layers (refractive index of 2.02) with thickness d_b and SiO₂ layers with thickness d_a, covered by an additional SiO₂-gap layer with thickness d and deposited on semi-infinite substrate. Here, we focus on the situation of only far-field interaction between the PCS and the multilayer, when the gap thickness exceeds the attenuation length of Bloch components of the resonances. The multilayer totally reflects the downward radiation and generates the reflected optical field with controllable phase delay determined by the gap thickness [27]. Such far-field interference mechanism offers an extrinsic pathway to tune the polarization states of radiation and arrange the BICs beyond intrinsic scenario [50,51].

 

Fig. 3. (a) Schematic illustration of a PCS-multilayer hybrid system. The inset of (a) plots the diagram of the TCMT with two ports. (b) Band structure of hybrid system. The TE-A (red strip) and TE-B (blue strip) band within the band structure of the multilayered substrate for the gap thickness of 1930 nm, wherein the strip thickness represents the decay rate, the lavender regions denote the TE or TM photonic band, and the white region represents both the TE and TM band gap. The violet disc (ii) in (b) indicates a FP-BIC. (c) Electric field distributions for the SP-BIC of isolated 1D PCS (panel (i)) and ordinary modes (panel (iii)), and the FP-BIC mode at (panel (ii)) which correspond to the points that marked in (b). In calculation, the PCS is set with w of 165 nm and h of 506 nm. The multilayer is designed with da(b) = 166(24) nm.

Download Full Size | PDF

To start, we discuss the construction of extrinsic BICs for the hybrid system. Figure 3(b) plots the dispersion relations of the TE-A and TE-B Bloch modes supported in the hybrid structure, both of which lie within the polarization-independent photonic band gap of the 1DPC (white region) to guarantee the total reflection of the downward radiation. The extrinsic BICs emerge from the ordinary leaky modes at a specific condition where the upward radiation has destructive interference with the reflected optical field, such as the mode (${{{k_x}a} / {(2\pi )}} = 0.02$) indicated by violet dot (ii) in Fig. 3(b). To illustrate the distinct feature of extrinsic BICs intuitively, we plot a comparison of optical field distributions in Fig. 3(c). In contrast with the neighboring ordinary modes [panels (iii) in Fig. 3(c)], the upward radiation is totally canceled for the extrinsic BICs [panel (ii) in Fig. 3(c)], and the standing waves can be observed inside the gap. While the optical field is confined by the PCS for the intrinsic SP-BIC [panel (i) in Fig. 3(c)], indicating their distinct physical origins. In addition, we also further elucidate that the extrinsic BIC of the hybrid system is FP-BIC by the resonant properties and electric field distributions of modes, more details are provided in Appendix B. Notably, the extrinsic BIC can be manipulated by the gap thickness, but the intrinsic BICs are immune to gap thickness (the multilayer) because of their non-radiation nature.

Next, we employ the temporal coupled mode theory (TCMT) to investigate the formation of extrinsic BICs [51,52]. Considering a general two-port model, as sketched in the inset of Fig. 3(a), we can obtain

Here A is the amplitude of the resonance (with resonant frequency ${\omega _{\textrm{in}}}$ and radiation rate ${\gamma _{\textrm{in}}}$) in PCS, $\alpha = {({{\alpha_u},{\alpha_d}} )^\textrm{T}}$ and $\beta = {({{\beta_u},{\beta_d}} )^\textrm{T}}$ represent incoming and outgoing waves respectively, ${\textbf K} = {({{k_u},{k_d}} )^\textrm{T}}$ and ${\bf D} = {({{d_u},{d_d}} )^\textrm{T}}$ are coupling matrix for excitation and radiation processes, and C is the direct scattering matrix, wherein the superscript u(d) denotes the upper (lower) port. Owing to the ${\sigma _z}$ symmetry of the PCS, C can be express as

3.2 Collision between extrinsic BICs and intrinsic BICs

For the multilayer-assisted PCS, another symmetry-independent parameter of gap thickness d is unlocked for BIC manipulation. Figure 4(a) depicts the corresponding Q factor distribution in the d-k_x synthetic space, wherein the bright curve indicates the trajectory of extrinsic BICs, and the three vertical lines represent three intrinsic BICs (there are SP-BIC, accidental BIC, and FW-BIC from left to right in sequence). The extrinsic BICs can be continuously moved, while the intrinsic BICs are immune from the gap tuning. It is worth noting that the Q factor tracks exhibit crossing without interplay when an extrinsic BIC encounter other types of intrinsic BICs, which are distinct from the anticrossing-like feature in Fig. 1(e). In fact, the annihilation of extrinsic BICs occurs only through the interaction with another extrinsic one, as the scenario at the white circle in Fig. 4(a).

 

Fig. 4. (a) The Q factor versus both gap thickness d and wave vector k_x, wherein the white circle denotes the annihilation point of a pair of extrinsic BICs, and the dashed box indicates the crossing behavior happening when an extrinsic BIC meets an accidental BIC. (b) The evolution of Q factor distribution and (c) far-field polarization map of the TE-A hybrid band in momentum space for the crossing process shown in the dashed box of (a). Here the paired singularities with opposite charges elastically collide with each other instead of annihilation.

Download Full Size | PDF

To better demonstrate the crossing behavior, we simulate the relevant evolutions of Q factor and topological charges in momentum space corresponding to the white box region in Fig. 4(a), as plotted in Figs. 4(b) and 4(c) respectively. When d is 1958 nm [panel (i) in Figs. 4(b) and 4(c)], an extrinsic BIC (right, $q ={+} 1$) and intrinsic (accidental) BIC (left, $q ={-} 1$) are prepared for merging. As d decreases to 1942.5 nm, the extrinsic BIC approaches and merges with the intrinsic BIC pinned at the regional center [panel (ii) in Fig. 4(c)]. Although carrying opposite charges, they bounce off each other instead of annihilation, and meanwhile exchange their charges. After the merging [panel (iii) in Fig. 4(c)], the extrinsic BIC (left, $q ={-} 1$) moves away and intrinsic BIC (right, $q ={+} 1$) remains stationary. In analogy to elastic collision between two rigid spheres of equal mass in classical mechanics, we define the new interaction between extrinsic and intrinsic BICs as polarization singularities collision.

The underlying principle of collision can be disclosed by calculating the eigenfrequency ω and decay rate γ of the resonance in the hybrid system. Combining Eqs. (2),(3) and the constraint of time-reversed process (i.e., ${\textbf C}{{\bf D}^ \ast } + {\bf D} = 0$ and ${{\bf D}^\dagger }{\bf D} = 2{\gamma _{\textrm{PCS}}}$), we can obtain

Because ${\omega _{\textrm{in}}} \gg {\gamma _{\textrm{in}}}$, eigenfrequency of the hybrid system approaches that of the isolated PCS according to Eq. (5), that is $\omega \approx {\omega _{\textrm{in}}}$. The band structures in Figs. 3(b) and 1(b) confirm that the eigenfrequencies of the two structures are nearly identical. Meanwhile, the Eq. (6) also verifies that the far-field interference mechanism (gap thickness d) does not affect the existence of intrinsic BICs ${\gamma _{\textrm{in}}} = 0$, owing to $\gamma $ remains 0. Combining Eqs. (5),(6) and using the above approximation, the corresponding Q factor is written as:

Moreover, we investigate the interaction between extrinsic BIC and the collection of tunable intrinsic BICs. At first, we calculate the evolutions of BICs in w-h-k_x synthetic space and construct the merging of FW-BIC and accidental BIC at ${{{k_x}a} / {(2\pi )}} = 0.039$ in the TE-A band, as mentioned in the Appendix A. Due to the independent manipulation with intrinsic BICs, extrinsic BICs can be introduced and manipulated by the gap tuning, eliminating the need of calculating the whole w-h-d-k_x synthetic space. This dramatically simplifies the design complexity of merging BIC configurations. Figure 5(a) sketches the topological charge distributions of premerging, merging and postmerging regimes between extrinsic BIC and merging (intrinsic) BIC. It can be clearly seen that the extrinsic BIC ($q ={+} 1$) approaches the collection of FW-BIC ($q ={-} 1$) and accidental BIC ($q ={+} 1$), and collides them in sequence accompanied by topological charge exchange, as d decreases from 1831 nm (top-most panel) to 1800 nm (middle panel). After collision (downmost panel), the extrinsic BIC departs, while the intrinsic BICs remain merged and exhibits the inversion of topological charge. These behaviors of singularities can be figuratively linked to the Newton’s Cradle.

 

Fig. 5. (a) The schematic diagram of topological charge evolution in momentum space, when gap thickness d is 1831 nm (premerging, panel (i)), 1813 nm (merging, panel (ii)), and 1800 nm (postmerging, panel (iii)). (b) Q factor versus wave vector k_x, for the cases of panel (i) (blue), (ii) (red), and (iii) (violet) in (a), wherein the circles and triangles denote the simulated results and the solid lines represent the corresponding fitting results. In calculation, the PCS is set with w of 174.9 nm and h of 519.98nm. The multilayer is designed with d_a(b) = 165(28) nm.

Download Full Size | PDF

Besides, we discuss the Q factors scaling rules during the collision process. By expanding the coefficient Q_ex into Taylor series near an extrinsic BIC, we can obtain the scaling rule of ${Q_{\textrm{ex}}} \propto {({k_x} - {k_{\textrm{BIC}}})^{ - 2}}$ (see Appendix B). When m intrinsic BICs merge with the extrinsic BIC, the coefficient Q_in satisfies ${Q_{\textrm{in}}} \propto {({k_x} - {k_{\textrm{BIC}}})^{ - 2m}}$. Thus, the Q factor increases nearby the merging point following the scaling rule of $Q \propto {({k_x} - {k_{\textrm{BIC}}})^{ - 2m - 2}}$, according to Eq. (7). Figure 5(b) presents the simulated Q factors and the fitting curves corresponding to the three scenarios shown in Fig. 5(a). The Q factor follows the scaling rule of $Q \propto {({k_x} - {k_{\textrm{BIC}}})^{ - 4}}$ in both the pre- and postmerging regimes (blue and violet lines). By harnessing extrinsic BIC (red line), the scaling rule is improved from $Q \propto {({k_x} - {k_{\textrm{BIC}}})^{ - 4}}$ to $Q \propto {({k_x} - {k_{\textrm{BIC}}})^{ - 6}}$ compared with the merging of only intrinsic BICs. The collision-based merging strategy can further enhance the Q factor whilst preserving the BICs from annihilation [42–44].

3.3 Super-BIC formed by merging four types of BICs

Within this context, we can further achieve the super-BIC, which consists of all four types of BICs (three types of intrinsic BICs and extrinsic BIC) merged at the Γ point. Figure 6(a) displays the distributions of the far-field polarization orientation angle in momentum space for three representative scenarios during the mering process. We initially set w (h) as 183.5 nm (529.8 nm), to merge all intrinsic BICs at the Γ point (for details, see section 2.2). Additionally, the gap d of 1862 nm is designed to generate extrinsic BICs near the Γ point. In the panel (i) of Fig. 6(a), we observed five intrinsic BICs with a total topological charge of -1 merging at the center and a pair of extrinsic BICs with charge +1 on their sides. When d decreases to 1848.5 nm, the extrinsic BICs approach the Γ point and form a super-BIC by merging seven BICs from four different types [panel (ii) of Fig. 6(a)]. With further decreases the gap thickness, two extrinsic BICs collide the five intrinsic BICs accompanied by topological charge exchange, and finally annihilate each other [panel (iii) of Fig. 6(a)].

 

Fig. 6. (a) The far-field polarization orientation angle map of TE-A band in momentum space for the hybrid systems with gap thickness d of 1862 nm (premerging, panel (i)), 1848.5 nm (merging, panel (ii)), and 1800 nm (postmerging, panel (iii)). (b) Q factor versus wave vector k_x, for the cases of panel (i) (blue), (ii) (red), and (iii) (violet) in (a), wherein the circles and triangles denote the simulated results and the solid lines represent he corresponding fitting results. The right panel of (b) shown the scaling rules in log₁₀-scale. In calculation, the PCS is set with w of 183.5 nm and h of 529.8nm. The multilayer is designed with d_a(b) = 165(30) nm.

Download Full Size | PDF

Figure 6(b) plots the quantitative comparison of Q factor between these three cases, which are depicted in Fig. 6(a). For the premerging regime, the Q factor varies with wave vector following $Q \propto k_x^{ - 10}{({k_x}^2 - k_{\textrm{BIC}}^2)^{ - 2}}$ (blue line), wherein the former item stems from the centered merging BICs and the latter from the extrinsic BICs. The super-BIC emerges at the critical point of k_BIC = 0, resulting the special scaling rule of $Q \propto k_x^{ - 14}$ (red line). The analytical results through TCMT [51,52] in Appendix C match well with the simulation scaling rule for formation possesses of super-BIC. Then entering the postmerging regime, the scaling rule asymptotically transits to be $Q \propto k_x^{ - 10}$ (green line), along with the annihilation of extrinsic BICs. The fitting curves are in good agreement with simulated results. Noteworthily, an ultrahigh Q factor in a wide range of wave vector near the Γ point can be achieved by super-BIC, as shown explicitly in the log-scale plot (right panel of Fig. 6(b)), which also implies it has the potential to suppress the scattering losses. As a final remark, we simulated the stored energy for different scenarios. The super-BIC combining intrinsic and extrinsic BICs, exhibits a significant enhancement of local field compared with the merging design only involving intrinsic BICs (see Appendix D for more details). This may be useful for enhancing the high harmonic generation [46].

4. Conclusions

In summary, we have proposed a systematical route to merge diverse BICs in momentum space range from the intrinsic BICs supported by isolated PCS to extrinsic BICs empowered by the far-field interference. For the intrinsic scenario, we realize the steerable merging FW-BIC and accidental BIC at off-Γ points, and merging three-types of intrinsic BICs at the Γ point by carefully tuning the slab thickness h and bar width w of PCS. Then, the merging BICs design is extended into the extrinsic scenario where brings together extrinsic BICs with intrinsic BICs based on the 1DPC-assited PCS, and the gap thickness d plays a vital role in the processes. Notably, a unique singularity collision between extrinsic and intrinsic BICs is revealed, which does not affect their existence but only exchanges their topological charge. This collision can overcome the decrease in Q factor caused by the annihilation of BICs in the postmerging regime. Furthermore, the extrinsic BICs exhibit manipulation isolation with the intrinsic BICs, which provides an unprecedented degree of freedom for BIC arrangement and greatly reduces the complexity of constructing merging BIC configurations. Specifically, the merging BICs are formed by harnessing both extrinsic and intrinsic BICs to improve the topological scaling rules, which can be attained to $Q \propto k_x^{ - 6}$ and $Q \propto k_x^{ - 14}$ for the steerable merging BICs at off-Γ points (with three types) and at the Γ point (with four types, called super-BIC), respectively. Such strategy is also appliable to two dimensions PCS, and inspires a reconfigurable way to achieve more flexible manipulation of polarization singularities in momentum space. Our findings may facilitate some applications including beam steering, optical trapping and enhancing the light-matter interactions.

Appendix A: Merging tunable intrinsic BICs at off-Γ point

Following the dynamic process of intrinsic BICs, as shown in the Figs. 1(e) and 1(f), we can construct the merging tunable BICs at off-Γ point. Here, we focus on the merging process of TE-A band [nearby the circle in Fig. 1(e)]. When h is 506 nm, a pair of polarization orientation angle vortex (FW-BIC, $q ={+} 1$) and antivortex (accidental BIC, $q ={-} 1$) can be clearly observed in the panel (i) of Fig. 7(b), whose centers correspond to the bright spots shown in the panel (i) of Fig. 7(a). As h increases to 508.65 nm [panel (ii) in Figs. 7(a) and 7(b)], the two intrinsic BICs approach each other and merge together. Although global charge equals zero for BIC collection at the critical merging point, the corresponding Q factor diverges because the vortexes are still preserved. After merging (h = 511 nm), the paired tunable BICs possessing opposite charge annihilate each other [panel (iii) in Fig. 7(b)], which degrades the infinite Q peak and leads a plain in Q map [panel (iii) in Fig. 7(a)]. The Q factor of merging BIC at off-Γ point grows strictly as $Q \propto {({k_x} - {k_{\textrm{BIC}}})^{ - 4}}$ (red line), compared with the scaling rule of $Q \propto {({k_x} - {k_{\textrm{BIC}}})^{ - 2}}$ for isolated accidental and FW-BICs (blue lines), as shown in Fig. 7(c), the numerical results and fitting curves are in good agreement.

 

Fig. 7. (a) The Q factor distributions and (b) far-field polarization maps of the TE-A hybrid band in momentum space for the PCSs with h of 506 nm (premerging, panel (i)), 508.65 nm (merging, panel (ii)), and 511 nm (postmerging, panel (iii)). The black disk with blue (violet) rim denotes the accidental (FW) BICs. (c) The Q factor as a function of wave vector k_x for the cases of panel (i) (blue), (ii) (red), and (iii) (violet) in both (a) and (b), wherein the circles and triangles denote the simulated values and the solid lines represent the corresponding fitting results. In calculation, the bar width is fixed at 165 nm.

Download Full Size | PDF

Appendix B: Characterization of FP-BICs

For the 1DPC-assited PCS, we simulate the Q factor of mode at ${{{k_x}a} / {(2\pi )}} = 0.02$ in the TE-A hybrid band versus gap thickness d, as shown in Fig. 8(a). The Q factor of the resonance diverges periodically as d increases, indicating that the formation of different orders of extrinsic BICs. Noteworthily, the simulating results derive from TCMT for the 0th BIC, owing to the near-field coupling. In this article, we focus on the high order BICs, which only exist far-field coupling at large thickness. Besides, Fig. 8(b) counts the quantized resonant gap thicknesses for each order of BIC and fit with the line with slope of ${\pi / {{k_z}}}$. The resonant thickness increases linearly versus the BIC sequence, and the different order of BIC also corresponds to different integer numbers of half wavelength standing waves between two mirrors (i.e., 1DPC and PCS), as the field distributions shown in Fig. 8(c). Therefore, we refer the emerging BIC introduced by 1DPC-assited PCS as FP-BIC.

 

Fig. 8. Characterization of FP-BICs (a) The Q factor of mode in the TE-A hybrid band for 1DPC-assited PhC slab versus gap thickness d. (b) Resonant gap thickness versus BIC sequence. (c) Electric field distributions for different orders of FP-BIC.

Download Full Size | PDF

Appendix C: Q factor scaling rule for intrinsic and extrinsic scenarios

For the scenario of merging intrinsic BICs, we take the coupled mode theory (CMT) in the form of effective non-Hermitian Hamiltonian (H_eff) to obtain the analytical scaling rule for merging BIC [48,49]. Considering the existence of FW-BICs for the isolated 1D PCS, the Hamiltonian can be expressed as:

(8)$${\textrm{H}_{\textrm{eff}}} = \left( {\begin{array}{{cc}} {\varepsilon + ak_x^2}&u\\ u&{ - \varepsilon - ak_x^2} \end{array}} \right) - i\left( {\begin{array}{{cc}} {{G_1}}&{\sqrt {{G_1}{G_2}} }\\ {\sqrt {{G_1}{G_2}} }&{{G_2}} \end{array}} \right),$$

where 2ɛ is the gap of resonance frequencies at the Γ point, the two bands are roughly quadratic with respect to k_x [as shown in Fig. 1(b)]. G_1,2 is the decay rates of mode, and u and $\sqrt {{G_1}{G_2}} $ denote the near-field and far-field coupling. For the formalism, we focus on the imaginary part of the eigenfrequency. Based on the condition of FW-BICs nearby the Γ point [44,48], we can get $\varepsilon = {{u({G_1} - {G_2})} / {2\sqrt {{G_1}{G_2}} }}$. Taking the relationship into Eq. (8) and solving related secular equation, we obtain the decay rate of the band with FW-BICs:

(9)$${\gamma _{\textrm{in}}} = {\mathop{\rm Im}\nolimits} ({\tilde{\omega }_{\textrm{in}}}) = \frac{{{G_1}{G_2}{a^2}k_x^4}}{{({{|u |}^2} + {{|{{G_1}{G_2}} |}^2})({G_1} + {G_2})}}.$$

The decay rate scaling rule depends on the form of G₁ and G₂ from Eq. (9). We construct the coexistence of SP-BIC, FW-BICs and accidental BICs on the TE-A band, while the TE-B band remains high radiation loss. Therefore, the reasonable expressions of G₁ and G₂ can be set as: ${G_1} = {\gamma _1}k_x^2{({k_x} - {k_a})^2}{({k_x} + {k_a})^2}$, ${G_2} = {\gamma _2}$. Herein, ${\pm} {k_a}$ are the wave vector of accidental BIC, ${\gamma _1}$ and ${\gamma _2}$ are constants (and ${\gamma _2} \gg {\gamma _1}$). When the accidental BICs approach to the Γ point, and involve the merging process (i.e., ${k_a} = 0$), the decay rate scaling rule would transform into ${\gamma _{\textrm{in}}} \propto k_x^{10}$, according to the Eq. (9). Then the corresponding Q factor scaling rule is in the form of $Q \propto k_x^{ - 10}$ for merging three types of intrinsic BICs at the Γ point.

Next, the Q factor scaling rule of merging BIC for the extrinsic scenario by TCMT [51,52] is investigate. At first, we derived the scaling rule for extrinsic BIC. According to Eq. (6), when the coefficient ${\gamma _{\textrm{ex}}} = {\textrm{Re}} \left( {1 + \frac{{r + t}}{{{e^{ - i\delta }} - r}}} \right)$ is zero, the hybrid system supports extrinsic BICs. The coefficient ${\gamma _{\textrm{ex}}}$ can be expanded into Taylor series nearby the extrinsic BIC as:

(10)$${\gamma _{\textrm{ex}}} = \gamma _{\textrm{ex}}^\mathrm{\prime }(k - {k_{\textrm{BIC}}}) + \frac{{\gamma _{\textrm{ex}}^\mathrm{\prime\prime }}}{2}{(k - {k_{\textrm{BIC}}})^2} + o(k - {k_{\textrm{BIC}}}),$$

where ${k_{\textrm{BIC}}}$ is the wave vector for extrinsic BIC. We assume that $r = iR{e^{i\varphi }}$ and $t = T{e^{i\varphi }}$. Herein R, T, φ and δ are real, and the coefficients R and T fulfill the condition of ${R^2} + {T^2} = 1$ and $RR^{\prime} + TT^{\prime} = 0$ owing to the unitarity of direct scattering matrix C [52]. The item of first-order derivative can be expressed as:

(11)$$\gamma _{\textrm{ex}}^\mathrm{\prime } ={-} \frac{1}{{{T^2}}}{\textrm{Re}} ({i{\delta^\mathrm{\prime }}{R^2} + i{\delta^\mathrm{\prime }}{T^2} + R{R^\mathrm{\prime }} + T{T^\mathrm{\prime }} + i{\varphi^\mathrm{\prime }}} )={-} \frac{1}{{{T^2}}}{\textrm{Re}} (i{\delta ^\mathrm{\prime }} + i{\varphi ^\mathrm{\prime }}) = 0.$$

Combining Eq. (11), and ignoring the infinitesimal quantities, the coefficient can be written as:

(12)$${\gamma _{\textrm{ex}}} = \frac{{\gamma _{\textrm{ex}}^\mathrm{\prime\prime }}}{2}{({k_x} - {k_{\textrm{BIC}}})^2},$$

wherein $\gamma _{\textrm{ex}}^\mathrm{\prime\prime } = \frac{1}{{{T^2}}}({{\varphi^\mathrm{\prime }}^2 + {\delta^\mathrm{\prime }}^2 + 2{\varphi^\mathrm{\prime }}{\delta^\mathrm{\prime }} + {R^\mathrm{\prime }}^2 + {T^\mathrm{\prime }}^2} )+ \frac{2}{{{T^3}}}{R^\mathrm{\prime }}({{\varphi^\mathrm{\prime }} + {\delta^\mathrm{\prime }}} )$. Besides, Eq. (6) has another important feature, i.e., the decay rate can be regarded as the form of product: $\gamma = {\gamma _{\textrm{in}}}{\gamma _{\textrm{ex}}}$. This form is consistent with Eq. (7), which is able to analytically demonstrate the Q factor scaling rule of merging BICs at the off-Γ point, as shown in Fig. 5.

For the super-BIC, we firstly construct the ensemble of merging three types of intrinsic BICs at the Γ point, and decay rate scaling rule complies with ${\gamma _{\textrm{in}}} \propto k_x^{10}$, according to the prior discussion. Then, the extrinsic BICs are introduced by the underlying 1DPC, and symmetrically locate the ensemble of intrinsic merging-BICs since in-plane C₂ symmetry of hybrid system. Combining Eq. (12), the scaling rule for the BIC configuration can be expressed as:

(13)$$\gamma \propto k_x^{10}{({k_x} - {k_{\textrm{BIC}}})^2}{({k_x} + {k_{\textrm{BIC}}})^2}.$$

Finally, when the extrinsic BICs move to the Γ point to form the super-BIC (i.e., ${k_{\textrm{BIC}}} = 0$), the decay scaling rule would evolve into $\gamma \propto k_x^{14}$. Accordingly, we can obtain the Q factor scaling rule is proportional to $k_x^{ - 14}$.

Appendix D: Stored energy for various merging BIC scenarios

To reveal the giant local-field enhancement for super-BIC by harassing the extrinsic BIC and intrinsic BICs, we compared the following four scenarios: the isolated intrinsic (SP) BIC, the conventional merging BICs (i.e., merging SP-BIC and accidental BICs), merging all three types of intrinsic BICs, and super-BIC. Herein, we calculated the stored energy by the volume integration of ${\varepsilon _0}{n^2}|E{|^2}$ in a unit cell [46], when the incident TE polarization waves along the direction of ${{{k_x}a} / {(2\pi )}} = 0.03$, as shown in Fig. 9. It can be observed that the stored energy on resonance increases significantly as the number involved BICs grows. Compared with the merging intrinsic BIC design, the super-BIC with the introduced extrinsic BIC shows a significant enhancement of local field, which is also verified by the (log-scale) normalized field distributions [insets in Fig. 9].

 

Fig. 9. The stored energy in one unit cell for (a) isolated intrinsic BIC (h = 420 nm, w = 165 nm), (b) conventional merging BIC consisted of the SP-BIC and accidental BICs (h = 456.52 nm, w = 138.6 nm), (c) the merging all the three types of intrinsic BICs (h = 529.8 nm, w = 183.5 nm), (d) the super-BIC (h = 529.8 nm, w = 183.5 nm, d = 1848.5 nm), when the excited TE polarization wave along the direction of ${{{k_x}a} / {(2\pi )}} = 0.03$. The insets indicate the normalized field distributions in log-scale for the four scenarios.

Download Full Size | PDF

Funding

National Key Research and Development Program of China (2020YFB2007501, 2021YFA1400700); National Natural Science Foundation of China (U20A20216); Postdoctoral Fellowship Program of China Postdoctoral Science Foundation (GZC20233531).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. P. J. Cobelli, V. Pagneux, A. Maurel, et al., “Experimental study on water-wave trapped modes,” J. Fluid Mech. 666, 445–476 (2011). [CrossRef]  

2. L. Huang, B. Jia, A. S. Pilipchuk, et al., “General framework of bound states in the continuum in an open acoustic resonator,” Phys. Rev. Applied 18(5), 054021 (2022). [CrossRef]  

3. D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Bound states in the continuum in photonics,” Phys. Rev. Lett. 100(18), 183902 (2008). [CrossRef]  

4. J. Lee, B. Zhen, S.-L. Chua, et al., “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. 109(6), 067401 (2012). [CrossRef]  

5. S. I. Azzam, V. M. Shalaev, A. Boltasseva, et al., “Formation of bound states in the continuum in hybrid plasmonic-photonic systems,” Phys. Rev. Lett. 121(25), 253901 (2018). [CrossRef]  

6. S. Han, P. Pitchappa, W. Wang, et al., “Extended bound states in the continuum with symmetry-broken terahertz dielectric metasurfaces,” Adv. Opt. Mater. 9(7), 2002001 (2021). [CrossRef]  

7. S. Romano, G. Zito, S. Torino, et al., “Label-free sensing of ultralow-weight molecules with all-dielectric metasurfaces supporting bound states in the continuum,” Photonics Res. 6(7), 726–733 (2018). [CrossRef]  

8. A. Tittl, A. Leitis, M. Liu, et al., “Imaging-based molecular barcoding with pixelated dielectric metasurfaces,” Science 360(6393), 1105–1109 (2018). [CrossRef]  

9. A. Leitis, A. Tittl, M. Liu, et al., “Angle-multiplexed all-dielectric metasurfaces for broadband molecular fingerprint retrieval,” Sci. Adv. 5(5), eaaw2871 (2019). [CrossRef]  

10. Y. Chen, C. Zhao, Y. Zhang, et al., “Integrated molar chiral sensing based on high-Q metasurface,” Nano Lett. 20(12), 8696–8703 (2020). [CrossRef]  

11. Z. Yu, X. Xi, J. Ma, et al., “Photonic integrated circuits with bound states in the continuum,” Optica 6(10), 1342–1348 (2019). [CrossRef]  

12. J. Li, S. Zhao, J. Chen, et al., “Manipulating leaky mode in silicon waveguides harnessing bound states in the continuum,” Opt. Lett. 48(9), 2249–2252 (2023). [CrossRef]  

13. Z. Feng and X. Sun, “Experimental observation of dissipatively coupled bound states in the continuum on an integrated photonic platform,” Laser Photonics Rev. 17(7), 2200961 (2023). [CrossRef]  

14. A. Kodigala, T. Lepetit, Q. Gu, et al., “Lasing action from photonic bound states in continuum,” Nature 541(7636), 196–199 (2017). [CrossRef]  

15. C. Huang, C. Zhang, S. Xiao, et al., “Ultrafast control of vortex microlasers,” Science 367(6481), 1018–1021 (2020). [CrossRef]  

16. X. Zhang, Y. Liu, J. Han, et al., “Chiral emission from resonant metasurfaces,” Science 377(6611), 1215–1218 (2022). [CrossRef]  

17. Z. Zhai, Z. Li, Y. Du, et al., “Multimode vortex lasing from dye-TiO2 lattices via bound states in the continuum,” ACS Photonics 10(2), 437–446 (2023). [CrossRef]  

18. K. Koshelev, Y. Tang, K. Li, et al., “Nonlinear metasurfaces governed by bound states in the continuum,” ACS Photonics 6(7), 1639–1644 (2019). [CrossRef]  

19. M. Minkov, D. Gerace, and S. Fan, “Doubly resonant χ(2) nonlinear photonic crystal cavity based on a bound state in the continuum,” Optica 6(8), 1039–1045 (2019). [CrossRef]  

20. G. Zograf, K. Koshelev, A. Zalogina, et al., “High-harmonic generation from resonant dielectric metasurfaces empowered by bound states in the continuum,” ACS Photonics 9(2), 567–574 (2022). [CrossRef]  

21. D. Liu, Y. Ren, Y. Huo, et al., “Second harmonic generation in plasmonic metasurfaces enhanced by symmetry-protected dual bound states in the continuum,” Opt. Express 31(14), 23127–23139 (2023). [CrossRef]  

22. C. W. Hsu, B. Zhen, A. D. Stone, et al., “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016). [CrossRef]  

23. C. W. Hsu, B. Zhen, J. Lee, et al., “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013). [CrossRef]  

24. Y. Yang, C. Peng, Y. Liang, et al., “Analytical perspective for bound states in the continuum in photonic crystal slabs,” Phys. Rev. Lett. 113(3), 037401 (2014). [CrossRef]  

25. S. Joseph, S. Sarkar, S. Khan, et al., “Exploring the optical bound state in the continuum in a dielectric grating coupled plasmonic hybrid system,” Adv. Opt. Mater. 9(8), 2001895 (2021). [CrossRef]  

26. S. G. Lee, S. H. Kim, and C. S. Kee, “Bound states in the continuum (BIC) accompanied by avoided crossings in leaky-mode photonic lattices,” Nanophotonics 9(14), 4373–4380 (2020). [CrossRef]  

27. C. Hsu, B. Zhen, S. Chua, et al., “Bloch surface eigenstates within the radiation continuum,” Light: Sci. Appl. 2(7), e84 (2013). [CrossRef]  

28. V. Liu, M. Povinelli, and S. Fan, “Resonance-enhanced optical forces between coupled photonic crystal slabs,” Opt. Express 17(24), 21897–21909 (2009). [CrossRef]  

29. S. Joseph, S. Pandey, S. Sarkar, et al., “Bound states in the continuum in resonant nanostructures: an overview of engineered materials for tailored applications,” Nanophotonics 10(17), 4175–4207 (2021). [CrossRef]  

30. P. Hu, J. Wang, Q. Jiang, et al., “Global phase diagram of bound states in the continuum,” Optica 9(12), 1353–1361 (2022). [CrossRef]  

31. B. Zhen, C. W. Hsu, L. Lu, et al., “Topological nature of optical bound states in the continuum,” Phys. Rev. Lett. 113(25), 257401 (2014). [CrossRef]  

32. Y. Zhang, A. Chen, W. Liu, et al., “Observation of polarization vortices in momentum space,” Phys. Rev. Lett. 120(18), 186103 (2018). [CrossRef]  

33. W. M. Ye, Y. Gao, and J. L. Liu, “Singular points of polarizations in the momentum space of photonic crystal slabs,” Phys. Rev. Lett. 124(15), 153904 (2020). [CrossRef]  

34. Q. Jiang, P. Hu, J. Wang, et al., “General bound states in the continuum in momentum space,” Phys. Rev. Lett. 131(1), 013801 (2023). [CrossRef]  

35. T. Yoda and M. Notomi, “Generation and annihilation of topologically protected bound states in the continuum and circularly polarized states by symmetry breaking,” Phys. Rev. Lett. 125(5), 053902 (2020). [CrossRef]  

36. Y. Zeng, G. Hu, K. Liu, et al., “Dynamics of topological polarization singularity in momentum space,” Phys. Rev. Lett. 127(17), 176101 (2021). [CrossRef]  

37. B. Wang, W. Liu, M. Zhao, et al., “Generating optical vortex beams by momentum-space polarization vortices centred at bound states in the continuum,” Nat. Photonics 14(10), 623–628 (2020). [CrossRef]  

38. H. Liu, K. Wang, H. Ma, et al., “Switchable optical vortex beam generator based on an all-dielectric metasurface governed by merging bound states in the continuum,” Opt. Express 31(12), 19159–19172 (2023). [CrossRef]  

39. W. Chen, Q. Yang, Y. Chen, et al., “Extremize optical chiralities through polarization singularities,” Phys. Rev. Lett. 126(25), 253901 (2021). [CrossRef]  

40. X. Yin, J. Jin, M. Soljačić, et al., “Observation of topologically enabled unidirectional guided resonances,” Nature 580(7804), 467–471 (2020). [CrossRef]  

41. Z. Zhang, F. Wang, H. Wang, et al., “All-pass phase shifting enabled by symmetric topological unidirectional guided resonances,” Opt. Lett. 47(11), 2875–2878 (2022). [CrossRef]  

42. J. Jin, X. Yin, L. Ni, et al., “Topologically enabled ultrahigh-Q guided resonances robust to out-of-plane scattering,” Nature 574(7779), 501–504 (2019). [CrossRef]  

43. N. D. Le, P. Bouteyre, A. K. Aldine, et al., “Super bound states in the continuum on photonic flatbands: concept, experimental realization, and optical trapping demonstration,” arXiv, arXiv:1905.00215v2 (2023). [CrossRef]  

44. M. Kang, S. Zhang, M. Xiao, et al., “Merging bound states in the continuum at off-high symmetry points,” Phys. Rev. Lett. 126(11), 117402 (2021). [CrossRef]  

45. M. Kang, L. Mao, S. Zhang, et al., “Merging bound states in the continuum by harnessing higher-order topological charges,” Light: Sci. Appl. 11(1), 228 (2022). [CrossRef]  

46. Q. Liu, L. Qu, Z. Gu, et al., “Boosting second harmonic generation by merging bound states in the continuum,” Phys. Rev. B 107(24), 245408 (2023). [CrossRef]  

47. M. S. Hwang, H. C. Lee, K. H. Kim, et al., “Ultralow-threshold laser using super-bound states in the continuum,” Nat. Commun. 12(1), 4135 (2021). [CrossRef]  

48. Z. Zhang, E. Bulgakov, K. Pichugin, et al., “Super quasibound state in the continuum,” Phys. Rev. Applied 20(1), L011003 (2023). [CrossRef]  

49. E. Bulgakov, G. Shadrina, A. Sadreev, et al., “Super-bound states in the continuum through merging in grating,” Phys. Rev. B 108(12), 125303 (2023). [CrossRef]  

50. L. Liu, H. Luo, Z. Xi, et al., “Ultrahigh-Q and angle-robust chiroptical resonances beyond BIC splitting,” Opt. Lett. 49(1), 153–156 (2024). [CrossRef]  

51. H. Luo, L. Liu, Z. Xi, et al., “Dynamics of diverse polarization singularities in momentum space with far-field interference,” Phys. Rev. A 107(1), 013504 (2023). [CrossRef]  

52. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20(3), 569–572 (2003). [CrossRef]