Fig. 1. Symmetry-protected bound states. A photonic crystal slab (PhCS) with a 180° rotational symmetry around the z axis (C₂). At the Γ point, modes that are even under C₂ cannot radiate because plane waves in the normal direction are odd under C₂. Away from the normal direction, the bound states become leaky with finite quality factors. TE_1,2 are the first and second modes transverse-electric-like modes; TM_1,4 are the first and fourth transverse-magnetic-like modes

Fig. 2. Single-resonance BIC. (a) Diagram of the photonic crystal slab (PhCS); (b) The band structure of PhCS(The trapped state is marked with a red circle, and the TM₁ band is marked with a green line); (c) Distribution of quality factor in momentum space; (d) Electric-field profile of off-Γ point BIC

Fig. 3. Bound states in the continuum as vortex centers of polarization vectors. (a) Schematic diagram of polarization vector decomposition; (b) Distribution diagram of the polarization field near the BIC point; (c) Two vortex patterns of the polarization field near the BIC point, corresponding to topological charges of +1 and −1, respectively

Fig. 4. Merging BIC. (a) Topological charge distribution in the spatial Fourier transform before merging (left), pre-merging (middle), and after merging (right); (b) Radiation Q factor as a function of lattice constant for period number N = 15; (c) Inverse radiation Q factor values plotted as a function of lattice constant for N = 15 (black) and N = 21 (purple). Vertical red dashed lines indicate the merging point in the infinite-size domain; (d) Schematic diagram of a 1D photonic lattice and a homogeneous dielectric waveguide; (e) FEM-simulated dispersion curves near the second stopband for four different t values. Blue and red insets show the spatial distributions of the electric field (E_y) for band-edge modes on the y = 0 plane. Vertical dashed lines denote mirror planes in the computational cell，where M denotes the topological charge of associated with merging BICs, K = 2π/Λ represents the amplitude of the grating wavevector with Λ being the grating period

Fig. 5. Quality factor optimization. (a) 3D schematic of the photonic bandgap structure; (b) Process of merging BICs in the photonic bandgap structure and the corresponding change in the Q factor; (c) Merging multiple BICs by manipulating the phase state of PCMs. Distribution of the Q-factors calculated when the PCMs is in the amorphous and crystalline states, the topological charge of each BICs is marked on the plot. The polarization vectors around the BICs are indicated by the white-arrows; (d) Band structure for TE polarization (left), Q factor of the TE band structure along the ΓX direction (right); (e) Schematic of a photonic crystal slab and the factors contributing to loss; (f) Simulated Q values near the centre of the Brillouin zone

Fig. 6. Merging BIC (a) Band structures and Q-factors of the topological and nontrivial supercells under periodic boundary conditions. The gray shadows show the region that is below the light line. The supercell structures for the trivial and nontrivial domains are shown by the insets, respectively. The lattice constant a = 35 μm, and the radius of the hexagon r = a/\begin{document}$ \sqrt 3 $\end{document}. The topological and trivial supercells have geometric parameters of r₁ = 0.48316r, r₂ = 0.38316r, r₃ = 0.31316r and r₄ = 0.45316r, respectively; (b) The laser spectra at different pump current density as function of frequency. The measured SMSR is around 20 dB; (c) Comparison of the passive Q-factors between the topological bulk BIC cavity (two domain) and a regular BIC cavity that only has topological nontrivial domain