Evolution of merging bound states in the continuum at Γ point in parameter space

Bound states in the continuum (BICs) are resonances with infinite lifetimes, even though they are embedded in the continuous spectrum of free space. Merging multiple BICs can be a promising approach to further improve the Q factors of nearby states over a broad wave-vector range. Previous studies have shown that merging BICs can only appear for specific parameter values of a structure; thus, it is commonly believed that they do not have stability. Here, we analytically study the existence and stability of merging BICs in parameter space, including material and geometric parameters. Specifically, we derive the conditions for the existence of merging-BICs in a system with a periodic plasmonic chain and demonstrate that merging BICs can stably exist when the other parameters are varied. Furthermore, in the momentum-geometric-material space, a BIC surface can be obtained analytically without performing a tedious numerical search of diverging Q factors in a multiple parameter space. Based on this surface, BICs at the Gamma point that merge in the momentum space can also merge again in the parameter space by changing the material parameters. The merging BIC with improved Q factor not only covers a broad wave-vector range but also spans a wide geometric parameter space when compared with the original BIC. Our findings provide a different perspective in the investigation of ultrahigh Q factors that substantially enhance light-matter interaction and improve the performance of photonic devices.