Fractality-induced photonic topological insulators
Fractal lattices have recently emerged as a promising setting for topological wave physics, but in most realizations the topological character is inherited from externally engineered couplings, gauge fields, or temporal modulation rather than from the fractal geometry itself. Here, we experimentally realize a photonic higher-order topological insulator in which the topology is induced solely by the self-similar geometry of a Sierpiński-gasket lattice. Following the isospectral reduction method recently proposed by Eek \textit{et al.}~\cite{Eek2025}, we show that the fractal waveguide array with uniform nearest-neighbor couplings can be mapped onto an effective breathing Kagome model that supports corner states. We selectively excite these modes with a weakly coupled detuned auxiliary waveguide and directly observe robust corner localization in real space, whereas an otherwise equivalent uniform triangular lattice exhibits only bulk diffraction under the same protocol. Spectral analysis and open-boundary calculations associate the observed states with nontrivial $C_3$ rotational topology, and disorder measurements further show that the corner localization persists over a finite range of random and symmetry-preserving disorder. Our results establish fractal geometry itself as a mechanism for generating topological boundary states in photonic lattices.