Generations of Dirac cones with fractal quantum frequencies in the two-dimensional optical waveguide networks

The two-dimensional optical waveguide network (OWN) proposed in this work not only supports Dirac cones at the Brillouin zone center and their high-symmetry points but also enables the formation of fractal quantized Dirac cones. By leveraging the inherent topological translational periodicity, we have derived analytical relationships that govern the frequencies, the number, and the taper angles of the network's quantized Dirac cones. Moreover, we demonstrate that the fundamental properties of these Dirac cones are solely determined by two parameters (p and q). This discovery enables unprecedented on-demand control over Dirac cone characteristics through simple parameter tuning. Furthermore, when approaching the Dirac cone frequencies, the system exhibits two typical photonic behaviors: pseudo-diffusion phenomenon and defect-immune transmission. The essence of this phenomenon lies in the topologically protected transport capability conferred by Dirac cones with fractional-order eigenvalues, which enables robust optical transmission in defective environments. Unlike the topological transport mechanisms in traditional systems such as graphene, metamaterials, and photonic/phononic crystals, the unique fractional topological control transport mechanism in this square photonic network is fundamentally distinct from the physical nature of traditional integer-order topological protection.