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We study linear and nonlinear higher-order topological insulators (HOTIs) based on waveguide arrays arranged into Sierpiński gasket and Sierpiński carpet structures, both of which have non-integer effective Hausdorff dimensionality. Such fractal structures possess different discrete rotational symmetries, but both lack transverse periodicity. Their characteristic feature is the existence of multiple internal edges and corners in their optical potential landscape, and the formal absence of an insulating bulk. Nevertheless, we show that a systematic geometric shift of the waveguides in the first generation of such fractal arrays, which affects the coupling strengths between sites of this building block as well as in subsequent structure generations, enables the formation of corner states of topological origin at the outer corners of the array. We find that, in contrast to HOTIs based on periodic arrays, Sierpiński gasket arrays always support topological corner states, irrespective of the direction of the shift of the waveguides, while in Sierpiński carpet structures, corner states emerge only for one direction of the waveguide shift. We also find families of corner solitons bifurcating from linear corner states of fractal structures that remain stable practically in the entire gap in which they form. These corner states can be efficiently excited by injecting Gaussian beams into the outer corner sites of the fractal arrays. Our results pave the way toward the investigation of nonlinear effects in topological insulators with non-integer dimensionality and enrich the variety of higher-order topological states.

Theory of nonlinear corner states in photonic fractal lattices