Generation of multiple bound states in the continuum through doubly degenerate quasi-guided modes

We present a detailed theoretical analysis of a peculiar generation of multiple bound states in the continuum (BICs) in two-dimensional periodic arrays of dielectric spheres. They emerge in high-symmetry lattices with the $C_{6v}$ and $C_{4v}$ point groups and involve doubly degenerate quasi-guided modes at the $\mathrm{\Gamma }$ point that can couple to external radiation. By tuning a system parameter, the doubly degenerate modes can exhibit accidental BICs at a critical parameter. In the vicinity of the critical parameter, the two bands originating from the degenerate mode exhibit multiple off-$\mathrm{\Gamma }$ BICs. They move and annihilate across the two bands by changing the parameter around the critical one. A ring-like high $Q$ channel pinned with multiple BICs emerges particularly for the $C_{6v}$ case. Across the critical coupling, the total vorticity of the multiple BICs is conserved. The $\text{vb}\ast k\cdot \text{vb}\ast p$ perturbation theory explains some features of the phenomena reasonably well.