Topological invariants and topological charges in photonic systems

TL;DR

This paper introduces a real-space Hamiltonian framework that unifies global and local topological invariants in photonic systems, enabling the design of structures with customizable topological properties.

Contribution

It presents a symmetry-based, deformable Hamiltonian model that links far-field defects and global invariants, applicable beyond photonics.

Findings

Unified description of topological invariants in photonics.

Framework allows design of structures with tailored topological features.

Applicable to systems with engineered real-space couplings.

Abstract

Topology in photonics comes in two distinct flavors: global and local. Global topology considers invariants that are obtained by integrating over the energy band, whereas local topology considers defects, typically vortices, in the far-field emission. These topologies are described by a wide range of models built in both real and momentum space, which are connected only by computationally expensive numerical methods that lack physical intuition. Here we propose a general framework based on a real-space Hamiltonian capable of describing electric field as a vector in both near- and far fields, allowing us to bridge between topological defects in the far-field and global topological invariants. The proposed Hamiltonian is constructed from the symmetry-representations of the lattice, is deformable to both atomic localized-mode (tight-binding) and photonic delocalized-mode (long-range) limits, and allows for independent control over the energies of eigenmodes of different symmetries at high-symmetry points of the Brillouin zone. This symmetry-based approach enables the design of structures with almost arbitrary topological properties and is not limited to photonic systems, but could apply to any system with engineered real-space couplings.