# Line Singularities and Hopf Indices of Electromagnetic Multipoles

## TL;DR

This paper investigates the topology of electromagnetic multipoles through line singularities and Hopf indices, revealing their role in the structure of photonic crystal modes and establishing connections between topology and electromagnetic phenomena.

## Contribution

It introduces a comprehensive topological characterization of complex electromagnetic multipoles using line singularities and Hopf indices, linking these to photonic Bloch modes.

## Key findings

- Line singularities in electromagnetic multipoles are classified by Hopf indices.
- The index sum of all line singularities is always 2, regardless of multipolar composition.
- Topological structures of Bloch modes are explained via line singularities and their indices.

## Abstract

Electromagnetic multipoles can be continuously mapped to tangent vectors on the momentum sphere, the topology of which guarantees the existence of isolated singularities. For pure (real or imaginary) vectors, those singularities correspond to zeros of tangent fields, which can be classified by integer Poincar\'{e} indices. Nevertheless, electromagnetic fields are generally complex vectors, a comprehensive characterization of which requires the introduction of line fields and line singularities categorized by half-integer Hopf indices. Here we explore complex vectorial electromagnetic multipoles from the perspective of line singularities, focusing on the special case of polarization line field. Similar to the case of pure vectors, the Poincar\'{e}-Hopf theorem forces the index sum of all line singularities to be $2$, irrespective of the specific multipolar compositions. With this multipolar insight, we further unveil the underlying structures of radiative circularly-polarized Bloch modes of photonic crystal slabs, revealing their topological origins with line singularities of Hopf indices. Our work has established subtle connections between three seemingly unrelated but sweeping physical entities (line singularities of Hopf indices, electromagnetic multipoles, and Bloch modes), which can nourish new frames of visions and applications fertilizing many related fields.