# Singularities and Poincar\'{e} Indexes of Electromagnetic Multipoles

## TL;DR

This paper systematically maps the singularities and topological indexes of electromagnetic multipoles, revealing their connection to bound states in the continuum and unifying multipole theory with topology.

## Contribution

It provides a comprehensive analysis of multipolar singularities and their indexes, linking them to topological charges and the formation of bound states in the continuum.

## Key findings

- Index sum over the sphere is always 2, consistent with Poincaré-Hopf theorem.
- Bound states in the continuum are related to overlapping multipolar singularities.
- Multipolar singularities are connected to topological charges of bound states.

## Abstract

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 slipped into oblivion. Here we map all the singularities of multipolar radiations of different orders, identify their indexes, and show explicitly the index sum over the entire momentum sphere is always $2$, consistent with the Poincar\'{e}-Hopf theorem. Upon those revealed properties, we can attribute the formation of bound states in the continuum to overlapping of multipolar singularities with open radiation channels. This insight unveils a subtle connection between indexes of multipolar singularities and topological charges of those bound states, revealing that essentially they are the same. 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.