Fig. 1. (a) Schematic of the proposed PhC slab with lattice constant a=1090  nm. The radius and height of the cylinder are r=265  nm and h=850  nm, respectively. (b) Simulated TE-like band structure. TE1, TE2, and TE3 bands are marked in black, red, and blue, respectively. The insets show the mode profiles (Hz) at Γ point on the z=0 mirror plane. (c) Simulated Q factors of modes for TE2 and TE3 (red and blue circles, respectively) bands. The Q factors of the TE2 band show five divergent points along ΓX direction, which represent symmetry-protected BICs and accidental BICs. For band TE3, there is only one symmetry-protected BIC at Γ point. (d) 3D band structure including TE1, TE2, and TE3. The colors show the Q factors of the eigenmodes, and these bright spots represent BIC points. Compared with lower band TE1, band TE2 has a very narrow bandwidth, which is a quasi-flatband.

Fig. 2. (a) Simulated polarization vectors (white arrows) around the BICs near Γ point with the Q factors as the background color for the TE2 band at different periodic constants a. The black arrows indicate the moving direction of accidental BICs. When a is tuned from 1090 to 1101 nm, four accidental BICs with topological charge −1 merge with a symmetry-protected BIC with charge +1. Further increasing a, accidental BICs will deflect in the ΓM direction. (b) Simulated Q factors (dotted lines) and the corresponding fitting curves (solid lines). Q factors before (a=1090  nm) and after (a=1261  nm) BICs merging are marked as blue and magenta, respectively. The intermediate transient process (a=1101  nm) is merging BICs, marked as red. The right panel shows the scaling rules along the ΓX direction. Compared with isolated BICs (Q∝k−2), the merging BICs significantly enhance the Q factors of the nearby states, which satisfy the rule Q∝k−6 in the vicinity of Γ.

Fig. 3. (a) Simulated polarization vectors (white arrow) around the accidental BICs with Q factors as the background color for the TE2 band at different periods a. The black arrows indicate the moving direction of accidental BICs. When a is tuned from 1090 to 1083 nm, the two accidental BICs with opposite topological charges approach each other and form merging BICs at an off-Γ point. Further decreasing a, the two BICs annihilate and evolve into a quasi-BIC. (b) Simulated Q factors (dotted lines) and fitting curves (solid lines). The Q factors before (a=1090  nm) and after (a=1082  nm) BICs merging are marked as blue and magenta, respectively. The intermediate transient process (a=1083  nm) is merging BICs, marked as red. The merging BICs case has considerably enhanced Q factors for nearby states compared with the isolated BICs case, as the dependence of Q factor on wave vectors near BICs changes from k−2 to k−4. Even if the two BICs collide and annihilate each other, the converted quasi-BIC still maintains a high Q factor.

Fig. 4. (a) Simulated polarization vectors (white arrows) around accidental BICs at different wave vectors with Q factors as the background color for the TE2 band. The wave vectors of merging BICs at 0.4π/a, 0.6π/a, and 0.8π/a correspond to a, h of (1048.7 nm, 845 nm), (1083.2 nm, 850 nm), and (1115 nm, 857 nm), respectively. By properly selecting parameters a and h, merging BICs can be realized at almost any wave vector position. (b) Simulated Q factors (dotted lines) and fitting curves (solid lines) corresponding to merging BICs at different wave vectors. The merging BICs located at arbitrary wave vectors follow the scaling rule as Q∝k−2(k2−kBIC2)−4.

Fig. 5. (a) Transmission spectra of the quasi-BIC1 originated from isolated BIC at TE3 under different incident angles θ. (b) Transmission spectra of quasi-BIC2 originated from isolated BIC at TE2 and ED mode at TE2. The structural parameters satisfy the condition of the above-mentioned merging BICs at Γ point, i.e., (a,h) are (1101 nm, 850 nm). With the increase of θ, the linewidth of quasi-BIC1 increases obviously, while quasi-BIC2 always keeps an extremely narrow linewidth. Owing to the existence of a quasi-flatband, the wavelength of quasi-BIC2 is almost unchanged. (c), (d) Magnetic field distributions with different incident angles at 1551 nm (top) and 1567 nm (bottom), respectively. Compared with ED mode, even at a large incident angle, the highly localized MD mode of the electromagnetic field is still stable at the wavelength of merging BIC. (e)–(g) Contributions of different multipolar excitations for (e) quasi-BIC1, (f) quasi-BIC2, and (g) ED mode of the PhC slab at θ=4°.

Fig. 6. (a), (b) Simulated transmission spectra of different quasi-BICs by sweeping θ from 0° to 20°. (c), (d) Efficiency of SHG enhanced by (c) quasi-BIC1 and (d) quasi-BIC2 at different incident angles θ. Compared with the quasi-BIC1, quasi-BIC2 has advantages in both efficiency and wavelength stability, especially for increased θ.

Fig. 7. (a) Simulated transmission spectra of merging BICs at off-Γ point by sweeping from 0° to 25°. (b)–(d) Efficiency of SHG enhanced by quasi-BIC2 at θ=16°, 20°, and 24°, respectively. Owing to the existence of merging BICs at off-Γ, high-efficiency SHG can still be achieved at a huge oblique incidence angle.

Fig. 8. Evolution of merging BICs at Γ point with different structural parameters. (a) Periodic constant a. (b) Height of cylinder h. The dotted circle represents the position of merging BICs. Other parameters are h=850  nm in (a) and a=1090  nm in (b).

Fig. 9. Evolution of merging BICs at off-Γ point with different structural parameters. (a) Periodic constant a. (b) Height of cylinder h. The dotted circle represents the position of merging BICs. Other parameters are h=850  nm in (a) and a=1090  nm in (b). The position of merging BICs can be steered by adjusting parameters a and h.

Fig. 10. Influence of fabrication defect on Q factor for BICs at Γ. Cylinder in the unit cell with (a) deviated radius, (b) different radii on the top and bottom, and (c) tilted angle. (d)–(f) Q factor evolution near merging BICs (solid lines) and isolated BIC (dashed lines) at Γ point under parameter variations shown in (a)–(c), respectively. Here, apart from the varying parameters, other parameters are the same as in Fig. 2(a), i.e., a=1101  nm for merging BICs.

Fig. 11. Influence of fabrication defect on Q factor for BICs at off-Γ. (a)–(c) Q factor evolution near merging BICs (solid lines) and isolated BIC (dashed lines) at off-Γ point under parameter variations resulting from the above three defects, respectively. Here, apart from the varying parameters, other parameters are the same as in Fig. 3, i.e., a=1083  nm for the merging BICs and a=1090  nm for the isolated BIC.

Fig. 12. Influence of intrinsic loss on merging BICs. (a) Simulation for the evolution of Q factors with different intrinsic losses of materials at merging BICs. The Q factors with intrinsic loss Im(n)=10−6 for merging BIC at (b) Γ point and (c) off-Γ point. Other structural parameters are the same as in Fig. 2 and Fig. 3 for a=1101  nm and a=1083  nm, respectively.