Singularities and Poincaré Indices of Electromagnetic Multipoles

Electromagnetic multipoles have been broadly adopted as a fundamental language throughout photonics, of which general features such as radiation patterns and polarization distributions are generically known, while their singularities and topological properties have mostly been left unattended. Here we map all the singularities of multipolar radiations of different orders, identify their indices, and show explicitly that the index sum over the entire momentum sphere is always 2, consistent with the Poincaré-Hopf theorem. Upon those revealed properties, we attribute the formation of bound states in the continuum to the overlapping of multipolar singularities with open radiation channels. This insight unveils a subtle equivalence between indices of multipolar singularities and topological charges of those bound states. Our work has fused two fundamental and sweeping concepts of multipoles and topologies, which can potentially bring unforeseen opportunities for many multipole-related fields within and beyond photonics.