Fig. 3: Numerical characterization of topology in fibre.

a, Schematic explanation of the Kitaev sum used to compute a real-space Chern marker. We define three equal regions (A, B and C) and project the eigenmodes into the states above a selected band gap β_c using operators (\({{\mathcal{P}}}_{A},{{\mathcal{P}}}_{B},{{\mathcal{P}}}_{C}\)), respectively. The chiral difference \({{\mathcal{P}}}_{\circlearrowright }-{{\mathcal{P}}}_{\circlearrowleft }={{\mathcal{P}}}_{A}{{\mathcal{P}}}_{B}{{\mathcal{P}}}_{C}-{{\mathcal{P}}}_{C}{{\mathcal{P}}}_{B}{{\mathcal{P}}}_{A}\) approximates the Chern number in the bulk of our fibre despite inhomogeneity and finite size (see text and Supplementary Section II for details). b, Calculated real-space Chern marker for different values of twist and coupling strength. We plot experimentally relevant parameters to guide topological fibre design and fabrication. We find that the topologically non-trivial region \({\mathcal{C}}=1\) corresponds to both high twist rates (greater than 600 rad m⁻¹) and high coupling strength (greater than 3,000 m⁻¹), and the triangle in the upper right corresponds to our experimentally fabricated fibre. c, Conceptual explanation of the topological transition that occurs as twist rate is increased at a fixed coupling strength, based on equation (4) for guided light in twisted fibre. When the twist is small compared with the coupling strength, the vector-potential (Coriolis) term dominates. Increasing the twist increases the on-site (centripetal) term. At high twist rates, the effect of the centripetal term is to destroy the topology, which we avoid by increasing the coupling strength.