Observation of nonlinear fractal higher order topological insulator

Fractals are self-similar structures characterized by a fractional Hausdorff dimension, such as the Sierpiński carpet (df = log3 8) and Sierpiński gasket (df = log2 3). Although aperiodic and often seen as lacking a conventional bulk due to multiple holes, inner edges, and corners, fractal systems in photonics can enable unusual light manipulation, including anomalous transport and flat bands. A key question is whether topological phases, particularly higher-order topological insulators (HOTIs) with corner-localized states, can arise in fractal structures where bulk properties are unconventional. Photonics has realized many topological phases (Chern, Floquet, valley-Hall, HOTIs), largely on periodic lattices. Recent works established that topological edge states can exist in fractal waveguide arrays, indicating that the absence of a conventional insulating bulk is not a barrier to topology and that bulk-edge correspondence can remain meaningful. While fractal HOTIs were proposed for electronics and demonstrated in acoustics and circuits, higher-order topological states in photonic fractals had not been observed. Nonlinearity in photonics provides powerful control over topological states, enabling effects such as lasing, harmonic generation, switching, and the formation of topological edge and corner solitons, as well as self-induced topological phases. In aperiodic topological systems, nonlinearity can tune propagation constants and internal modal structure with power, causing complex transitions across gaps and bands beyond simple localization changes. This study reports the first experimental realization of a photonic fractal HOTI using fs-laser-written Sierpiński gasket waveguide arrays and investigates the interplay of topology and nonlinearity leading to thresholdless topological corner solitons. The system allows corner states at both outer and inner corners and supports topology across wider parameter ranges than conventional periodic HOTIs. Real-space polarization index is used to characterize topology. The unusual spectral features and solitons stem from the internal fractal structure while diffraction is governed by a conventional 2D Laplacian.

• Prior photonic topological phases have been realized in systems such as Chern, Floquet, valley-Hall, and higher-order topological insulators, typically relying on periodic bulk lattices.
• Topological edge states have been shown to arise in fractal photonic systems (e.g., waveguide arrays with helical channels and fractal Haldane models), suggesting that the lack of a conventional bulk in fractals does not preclude topological phases and that bulk–edge correspondence can still apply.
• Fractal HOTIs have been proposed in electronic platforms and realized experimentally in acoustics and circuits, with unique features like localization at multiple inner corners and parameter ranges that depend strongly on fractal generation order, distinguishing them from conventional HOTIs.
• Nonlinearity in topological photonics has enabled a variety of phenomena: topological edge solitons, self-induced topological phases, and corner solitons in HOTIs, as well as nonlinear control of mobility and effects of geometrical frustration.
• There has been growing theoretical interest in nonlinear aperiodic topological photonic systems, but few predictions and no prior experimental observations for photonic fractal HOTIs before this work.

• Platform: Femtosecond-laser-written Sierpiński gasket waveguide arrays in fused silica, constructed in two types (case-1 and case-2). The arrays are self-similar with generations Gn; case-2 shares three common sites between previous-generation subgaskets, leading to a reduced site count compared to case-1 and multiple holes, inner corners, and edges. Effective Hausdorff dimension dH = log2 3.
• Structural control: A distortion parameter r is introduced by shifting neighboring waveguides in opposite directions while keeping next-nearest-neighbor spacing a constant, enabling control of intra- vs inter-cell couplings. Undistorted r = 0.5a; distorted examples r = 0.3a and 0.6a.
• Experimental parameters (for modeling): Array depth p = 5.7, lattice spacing a = 6.0 (normalized units), elliptical single-mode waveguides with widths dx = 0.25, dy = 0.75.
• Modeling framework: Continuous 2D nonlinear Schrödinger equation governing the envelope ψ(x,y,z): ∂ψ/∂z = (1/2)(∂2ψ/∂x2 + ∂2ψ/∂y2) − R(x,y)|ψ|2ψ, where R(x,y) describes the array as a sum of elliptical waveguide potentials of depth p at the Sierpiński gasket nodes. This model accounts for exact waveguide shapes, long-range coupling, radiation, and intra-site modal variations under nonlinearity. Time-reversal symmetry is present; disregarding waveguide ellipticity, the array has C3 rotational symmetry.
• Linear spectrum: Nonlinearity omitted to compute eigenmodes ψ = u(x,y)e^{ibz} via plane-wave expansion; b is the propagation constant. Spectra vs r were obtained for G3 and G4 fractals and compared to a non-fractal array of the same size.
• Tight-binding auxiliary model: Derived from the continuous model to facilitate topological characterization.
• Topological characterization: Real-space polarization index suited for aperiodic structures. Two Sierpiński gaskets are glued to form a rhombic structure; couplings to missing sites are set to zero to emulate the fractal. The quantized index identifies topological corner states and their existence domains.
• Comparative dynamics: Single-site excitation dynamics in fractal and non-fractal G1 structures compared to assess robustness of the HOTI phase (details in Supporting Information).
• Nonlinear analysis: Families of spatial (temporal dynamics neglected due to long pulses) topological corner solitons studied, bifurcating from linear corner states into spectral gaps, with localization controlled by input power.

• First experimental realization and theoretical description of a nonlinear photonic higher-order topological insulator with fractal origin, using Sierpiński gasket waveguide arrays.
• Topological corner states exist for both r < 0.5a and r > 0.5a in fractal arrays, unlike conventional periodic HOTIs (e.g., kagome, square SSH) where corner states typically exist only for one distortion regime.
• Multiple coexisting outer corner states with distinct internal structures appear (e.g., modes with one or two primary lobes near the corner), including cases where the corner waveguide is not directly excited. These branches are three-fold degenerate and persist across generations (G3, G4).
• Hybrid corner states unique to the fractal geometry emerge with simultaneous strong localization at multiple inner and outer corners; their number of intensity spots scales with generation N = (3^n + 3)/2. The effective dimensionality de of hybrid states approaches the fractal’s Hausdorff dimension df = log2 3, while outer corner states are zero-dimensional (de → 0).
• Linear spectral structure remains qualitatively similar across generations, with localized corner and hybrid branches clearly visible in G4. In contrast, a non-fractal array of the same size shows additional bulk bands (especially for r > 0.5a) that shrink the existence domains of certain corner branches (green, cyan), causing coupling to bulk, whereas the fractal retains strong corner localization.
• Real-space polarization index equals 0.5 for both hybrid and outer corner states within their gap domains, confirming their topological origin. The system is classified as a higher-order topological crystalline insulator protected by C3 rotational symmetry.
• Robustness to weak disorder: Corner states’ eigenvalues may fluctuate under small disorder in waveguide depths/positions, but they remain within the topological gap and localized as long as the gap is not closed by disorder.
• Nonlinear regime: Thresholdless spatial topological corner solitons bifurcate from linear corner states and reside in forbidden gaps. Their localization can be efficiently controlled by input power, and sharp differences in nonlinear localization between outer and multiple inner corners/edges are observed.
• Modeling parameters representative of experiments: p = 5.7, a = 6.0, dx = 0.25, dy = 0.75; arrays implemented via fs laser writing in fused silica. The diffraction operator is a standard 2D Laplacian; the unusual spectral and soliton properties arise from the fractal geometry.

The study addresses whether higher-order topological phases and their nonlinear counterparts can exist in fractal photonic structures that lack a conventional periodic bulk. By constructing Sierpiński gasket arrays and tuning couplings via a distortion parameter, the authors demonstrate corner-localized topological states in both outer and inner corners across a broader parameter range than in conventional HOTIs. The persistence of these states across generations and the quantized real-space polarization index (0.5) confirm their topological nature in an aperiodic setting governed by crystalline C3 symmetry. Comparisons with non-fractal arrays reveal that fractality not only preserves but can expand the parameter regime supporting topological corner states by avoiding band overlap that would otherwise hybridize corner and bulk modes. The nonlinear findings show that these corner states bifurcate into thresholdless spatial solitons whose localization and internal structure can be tuned by power, enabling dynamic, power-controlled manipulation of topological excitations. The emergence of hybrid corner states with effective dimensionality approaching the fractal’s df highlights a richer taxonomy of topological excitations than in periodic HOTIs. Overall, the results establish fractal HOTIs as robust, tunable platforms for topological light localization and transport, with potential for applications in topological lasing, frequency conversion, and switching where nonlinear enhancement at corners is advantageous.

This work realizes and characterizes a nonlinear photonic higher-order topological insulator based on Sierpiński gasket fractal waveguide arrays. It uncovers topological corner states—both outer and hybrid inner–outer—existing for a wide range of coupling distortions on both sides of r = 0.5a, persisting across fractal generations, and confirmed by a quantized real-space polarization index. Fractality broadens the existence domains of certain topological states compared with non-fractal counterparts. In the nonlinear regime, thresholdless topological corner solitons bifurcate from linear states, with localization controllable by input power and distinct behavior between outer and inner corners/edges. These findings introduce a new paradigm for nonlinear topological photonics in aperiodic media and open avenues for tunable topological devices using fractal geometries. Future directions include exploring higher fractal generations and other fractal families, systematic studies of disorder and fabrication tolerances, power-dependent modal transitions across spectral gaps, and potential active or gain-assisted implementations for topological lasing and switching.

• Observation of truly stationary linear hybrid corner states may be challenging, as they require simultaneous excitation of many sites including inner corners; weak nonlinearity can help suppress slow switching but changes the regime.
• Topological protection persists under small disorder, but sufficiently strong disorder that closes the topological gap would destroy localization.
• Symmetry classification relies on approximate C3 rotational symmetry; ellipticity of waveguides is disregarded for this purpose.
• The analysis neglects temporal dynamics (long pulses assumed), focusing on spatial solitons under a standard 2D Laplacian; effects involving fractional diffraction operators are not addressed.
• Full details for some constructions (case-1 arrays, certain dynamics) are provided in Supporting Information and are not fully elaborated in the main text provided here.