Merging bound states in the continuum in the geometrical parameter space

Optical resonators can support bound states in the continuum (BICs) with infinite quality ($Q$) factors by eliminating radiation losses. However, practical optical resonators only support quasi-BICs with finite $Q$ factors due to the scattering losses caused by inevitable fabrication defects. Merging multiple BICs in momentum space can improve the $Q$ factors of resonators over a broad wavevector range. The dependence of a resonator on high-precision nanofabrication can also be decreased by improving the robustness of the $Q$ factors of quasi-BICs against asymmetric structural parameters, which is much easier to realize high $Q$ factors. Here, we propose an efficient method to merge multiple BICs in the geometrical parameter space by engineering a folded mode induced by Brillouin zone folding. Along with the topological charge evolution process in momentum space, this approach significantly improves the robustness of the $Q$ factor of resonators against perturbations caused by geometric symmetry breaking and wavevector. Compared with fundamental isolated BICs, the merged BICs are more immune to structural disorders. Our approach provides a path to achieve robust ultrahigh-$Q$ resonances, which holds immense potential in enhancing quantum and nonlinear effects, as well as improving the performance of optical sensors and nanolasers.