Interacting bound states in the continuum in Fabry-Pérot resonators: Merging, crossing, and avoided crossing

We study both analytically and numerically the interaction between bound states in the continuum (BICs) in 2D Fabry-P\'erot (FP) resonators in waveguides with waveguide expansions playing the role of the ``mirrors.'' The derived analytical model illustratively captures recently proposed twin-BICs (TBICs) stemmed from BICs in individual mirrors as well as the well-known FP BICs related to an FP resonance. We show that the formation of TBICs is a universal phenomenon, which occurs in different types of FP resonators, and is related to the sharp dependence of the phase of mirror reflection coefficient on system parameters near the BIC point in the parameter space of individual mirror. We also show explicitly that the complicated multimode interference taking place in the mirrors resulting in the coalescence of two perfect transmission Fano antiresonances can explain the merging of corresponding FP BICs without invoking topological concepts. Moreover, the coalescence of antiresonances in an individual mirror can be continuously changed into the crossing or even the avoided crossing (repulsion) of antiresonances by an infinitesimal change of parameter. The behavior of FP BICs follows the antiresonances pretty closely and demonstrates a similar transformation (either merging or repulsion, respectively). The results of the analytical model are confirmed by numerical simulations of quantum-mechanical and optical waveguides.