Photonic Floquet topological insulators in a fractal lattice

We present Floquet fractal topological insulators: photonic topological insulators in a fractal-dimensional lattice consisting of helical waveguides. The helical modulation induces an artificial gauge field and leads to a trivial-to-topological phase transition. The quasi-energy spectrum shows the existence of topological edge states corresponding to real-space Chern number 1. We study the propagation of light along the outer edges of the fractal lattice and find that wavepackets move along the edges without penetrating into the bulk or backscattering even in the presence of disorder. In a similar vein, we find that the inner edges of the fractal lattice also exhibit robust transport when the fractal is of sufficiently high generation. Finally, we find topological edge states that span the circumference of a hybrid half-fractal, half-honeycomb lattice, passing from the edge of the honeycomb lattice to the edge of the fractal structure virtually without scattering, despite the transition from two dimensions to a fractal dimension. Our system offers a realizable experimental platform to study topological fractals and provides new directions for exploring topological physics. Topological photonics: fractal lattices Photonic topological insulators are currently a subject of great interest because they support edge states that can propagate without being affected by defects and disorder. All topological insulators discovered thus far have a bulk surrounded by edges. Now, Zhaoju Yang and coworkers from Technion in Israel, found theoretically that photonic topological insulators can also exist in fractal lattices, comprising only edges—with no bulk at all. They studied fractal lattices structured as Sierpinski gasket composed of an array of evanescently coupled helical waveguides. Despite the lack periodicity in such structures, tight-binding simulations and quasienergy analysis predict the existence of topological edge states, residing either on outer or on inner edges, exhibiting a Chern number of 1 and displaying scattering-free propagation. The fractal symmetries of such lattices are found to be crucial for the existence of the topological properties. Such fractal lattices could be fabricated by femtosecond laser writing technology.