Phase-Flow Topology of Bound States in the Continuum
Control of topological charges of bound states in the continuum (BICs) is essential for advanced topological photonics. A rigorous theoretical framework for understanding the formation of these charges is therefore necessary for further progress in this field. However, conventional multipolar formalism often fail to predict the topological charge when no single multipole dominates the mode. To address this fundamental issue, we present a rigorous dynamical systems framework for describing formation of topological charges. This attitude provides a direct link between polarization vortex and the local structure of the vector polarization field. Within the proposed framework, we challenge the established identification of the topological charge with the Hopf index of the dominant multipole and demonstrate the significant role played by derivatives of the multipolar coefficients in determining the charge. Our theoretical framework is validated within semi-analytical multipolar decompositions and full-wave numerical simulations of periodic dielectric metasurfaces. Furthermore, we show how the symmetry of the unit cell influences the local structure of the polarization field around BIC and identify the necessary conditions for the formation of high topological charges. Our findings provide a more rigorous basis for analysing BICs' topological properties, paving the way for advanced applications in topological photonics.