Fig. 1. (a) Unit cell of the photonic crystal slab possessing a C_4v symmetry. (b) Eigenfrequency evolution of the two involved symmetry-protected BICs at the Γ point with a varied distance g_t. The corresponding magnetic fields distribution of the BICs belonging to (c) the A1 and (d) the B1 representations. The band structure of the two symmetry-protected BICs around the Γ point in momentum space with three varied distances: (e) g_t/d ≃ 0.178, (f) g_t/d ≃ 0.158, and (g) g_t/d ≃ 0.133. The appearance of the Dirac point is indicated by the dotted circle. The parallel wave vector k_∥ = (k_x, k_y) is in the unit of 2π/d, and the eigenfrequency f is in the unit of c/d.

Fig. 2. (a) Eigenfrequency difference along the M-Γ-X direction of the two involved bands as a function of varied distance. The theoretical degeneracy prediction along the Γ-M direction is shown by the dashed line. (b) Quadratic and (c) linear fitting for each kind of degeneracy around its degenerate position. (d),(e) The logarithmically displayed Q factors for the two involved bands as a function of the varied distance and (f) at a selected distance g_t/d ≃ 0.133.

Fig. 3. Radiative polarization driven by the two symmetry-protected BICs at the Γ point when the varied distance is g_t/d ≃ 0.178. (a),(b) The radiative Stokes parameters −S₁/S₀ and (c),(d) the polarization vector distributions around each BIC for (a),(c) the lower and (b),(d) the higher bands.

Fig. 4. Phase profile ψ along different closed loops when the varied distance is g_t/d ≃ 0.178. The closed loops are schematically shown in the inset. The phase profile ψ for (a),(b) the lower band and (c),(d) the higher band along two different loops with (a),(c) radius k_r = 0.02 (2π/d) and (b),(d) k_r = 0.18 (2π/d).

Fig. 5. Radiative polarization driven by the two symmetry-protected BICs at the Γ point when the varied distance is g_t/d ≃ 0.133. (a),(b) The radiative Stokes parameter −S₁/S₀ and (c),(d) the polarization vector distributions for (a),(c) the lower and (b),(d) the higher bands.

Fig. 6. Phase profile ψ along different closed loops when the varied distance is g_t/d ≃ 0.133. The closed loops around each polarization singularity are schematically shown in the inset. The phase profile ψ for (a)–(d) the lower band and (e)–(h) the higher band along the loop with radius k_r = 0.02 (2π/d) around (a),(e) the Γ point, (b),(f) the Dirac point, (c),(g) the off-Γ BIC of the higher band, and (d),(h) along the loop with radius of 0.18 (2π/d).