Fractal Kagome Photonic Crystal

• Fractal Kagome photonic crystals are platforms that merge Kagome lattice connectivity with Sierpiński-gasket self-similarity to induce nontrivial topology.
• They use isospectral reduction to map uniform nearest-neighbor couplings onto an effective breathing Kagome model with energy-dependent parameters.
• Experimental realizations in waveguide arrays and microwave setups confirm the emergence and robustness of higher-order topological corner states driven solely by fractal geometry.

Searching arXiv for the cited works and closely related context. {"query":"(Zhang et al., 22 Jun 2026)", "max_results": 5} I’m checking the specific arXiv record for (Zhang et al., 22 Jun 2026). {"query":"(Zhang et al., 22 Jun 2026)","max_results":5} A fractal Kagome photonic crystal is a photonic platform in which Kagome-lattice physics is realized within a self-similar Sierpiński-gasket geometry, so that nontrivial topology emerges from fractality rather than from explicit dimerization, magnetic bias, spin-orbit coupling, long-range interactions, or temporal modulation. In the 2026 literature, this concept appears in two closely related experimental forms: a tight-binding-like microwave photonic crystal obtained by recursively decorating a Kagome lattice with a Sierpiński-gasket rule, and a femtosecond-laser-written Sierpiński-gasket waveguide array that is mapped, by isospectral reduction, onto an effective breathing Kagome model supporting higher-order topological corner states (Yan et al., 23 Jun 2026, Zhang et al., 22 Jun 2026). In both cases, the defining feature is that the parent photonic structure is uniform at the nearest-neighbor level, while the effective Kagome “breathing” and the associated corner localization are induced solely by the fractal geometry.

1. Definition and geometric construction

The underlying geometric idea is the integration of Kagome connectivity with Sierpiński self-similarity. In the microwave realization, the system is described as a tight-binding-like photonic crystal derived from a Kagome lattice and recursively decorated by a Sierpiński-gasket rule so that the lattice becomes self-similar and of non-integer dimension; fractality itself lifts the Kagome Dirac degeneracy at $K$, opens topological bandgaps, and produces higher-order topological corner states (Yan et al., 23 Jun 2026). The base Kagome lattice is a triangular Bravais lattice with a three-site basis forming corner-sharing triangles, and with only nearest-neighbor hopping and $v=w$ its bulk spectrum has a flat band and a Dirac cone at $K$ (Yan et al., 23 Jun 2026).

In the waveguide-array realization, the starting point is instead a third-generation Sierpiński-gasket waveguide array fabricated by femtosecond-laser direct writing in glass. The gasket has Hausdorff dimension $d_H=\log_2 3 \approx 1.585$ and is generated by recursively removing the central site of each upward-pointing triangle, producing a self-similar hierarchy of triangular motifs with nearest-neighbor links along the triangle edges (Zhang et al., 22 Jun 2026). The paraxial light dynamics are modeled by a tight-binding equation with uniform nearest-neighbor coupling $c_{m,n}=c_0=2\ \mathrm{cm}^{-1}$ and vanishing onsite detunings $\epsilon_m=0\ \mathrm{cm}^{-1}$, so the parent lattice is topologically trivial at the level of explicit couplings (Zhang et al., 22 Jun 2026).

These two constructions differ in physical implementation but share the same central claim: self-similar fractal geometry is not merely decorative. It changes the connectivity graph in such a way that an effective breathing Kagome description emerges, and that effective description lies in a higher-order topological regime (Yan et al., 23 Jun 2026, Zhang et al., 22 Jun 2026).

2. Isospectral reduction and the emergence of breathing Kagome physics

The principal theoretical mechanism is isospectral reduction, formulated as a Schur complement on a retained subspace. In the waveguide system, the full Hamiltonian is partitioned into a retained subset $S$ and an eliminated subset $\bar S$, yielding the reduced Hamiltonian

$$H_{\mathrm{eff}}(\beta)=H_{SS}+H_{S\bar S}(\beta I_{\bar S}-H_{\bar S\bar S})^{-1}H_{\bar S S},$$

with the property that an eigenpair $(\beta,\psi_S)$ of $v=w$0 corresponds to an eigenpair of the full $v=w$1, provided $v=w$2 is not a pole of $v=w$3 (Zhang et al., 22 Jun 2026). Recursive application along the Sierpiński hierarchy produces an effective Kagome-like graph with two inequivalent nearest-neighbor couplings: intercell couplings equal to the original $v=w$4 along the large downward triangles, and intracell couplings $v=w$5 along the small upward triangles generated by elimination of decorating sites. Uniform nearest-neighbor couplings on the fractal therefore map to a breathing Kagome model on $v=w$6, with energy-dependent breathing ratio $v=w$7, where $v=w$8 and $v=w$9 (Zhang et al., 22 Jun 2026).

The microwave photonic-crystal realization expresses the same mechanism in a more explicit Kagome language. Partitioning the first-generation fractal Kagome cell into a three-site subset $K$0 and a six-site complement $K$1, the isospectral reduction of the full Hamiltonian $K$2 gives an effective Hamiltonian of the same Schur-complement form,

$K$3

with effective couplings

$K$4

At the corner-state eigenenergy $K$5, one finds $K$6 and $K$7, so the fractalized lattice is isospectral to a breathing Kagome lattice in its topological regime despite the absence of explicit dimerization in the physical design (Yan et al., 23 Jun 2026).

A common implication of both formulations is that the effective nonuniformity of the Kagome model is not externally engineered. It is generated internally by fractality through energy-dependent couplings and onsite terms. This distinguishes the fractal Kagome platform from conventional Kagome higher-order photonic structures, where breathing is usually implemented directly in the geometry or couplings (Zhang et al., 22 Jun 2026, Yan et al., 23 Jun 2026).

3. Topological characterization and corner-state conditions

The effective Kagome description is that of a breathing Kagome higher-order topological insulator. In the waveguide realization, the reduced Hamiltonian at fixed spectral parameter $K$8 is

$K$9

where $d_H=\log_2 3 \approx 1.585$0 and $d_H=\log_2 3 \approx 1.585$1 denote the small upward and large downward triangles, respectively, $d_H=\log_2 3 \approx 1.585$2, and all nonuniformity arises from fractality via the energy dependence of $d_H=\log_2 3 \approx 1.585$3 and $d_H=\log_2 3 \approx 1.585$4 (Zhang et al., 22 Jun 2026). Higher-order corner modes appear when the intracell coupling is weaker than the intercell coupling, $d_H=\log_2 3 \approx 1.585$5, and the topological–trivial transition occurs at $d_H=\log_2 3 \approx 1.585$6, the uniform Kagome limit where the corner-state gap collapses (Zhang et al., 22 Jun 2026).

The corner-state confinement condition in the same system is more restrictive. The geometry induces corner states in the reduced model when two conditions are simultaneously satisfied: first, $d_H=\log_2 3 \approx 1.585$7, so that the corner-state propagation constant matches the effective onsite energy at the corner site; second, $d_H=\log_2 3 \approx 1.585$8, placing the breathing Kagome in its higher-order topological regime (Zhang et al., 22 Jun 2026). Open-boundary calculations verify these conditions for the corner states appearing in the nontrivial gaps.

Topological diagnosis is performed through $d_H=\log_2 3 \approx 1.585$9 rotational symmetry. In the waveguide case, the bulk band topology is characterized by a $c_{m,n}=c_0=2\ \mathrm{cm}^{-1}$0 symmetry indicator $c_{m,n}=c_0=2\ \mathrm{cm}^{-1}$1, evaluated from rotation eigenvalues of the occupied bands at the high-symmetry points $c_{m,n}=c_0=2\ \mathrm{cm}^{-1}$2, $c_{m,n}=c_0=2\ \mathrm{cm}^{-1}$3, and $c_{m,n}=c_0=2\ \mathrm{cm}^{-1}$4 under periodic boundary conditions. The nontrivial gaps hosting corner states exhibit $c_{m,n}=c_0=2\ \mathrm{cm}^{-1}$5, while trivial gaps have $c_{m,n}=c_0=2\ \mathrm{cm}^{-1}$6 (Zhang et al., 22 Jun 2026). In the microwave photonic-crystal study, the corresponding rotational invariant is written as $c_{m,n}=c_0=2\ \mathrm{cm}^{-1}$7, with $c_{m,n}=c_0=2\ \mathrm{cm}^{-1}$8, and nonzero $c_{m,n}=c_0=2\ \mathrm{cm}^{-1}$9 indicates a topological gap; the associated nominal electronic corner charge is $\epsilon_m=0\ \mathrm{cm}^{-1}$0 (Yan et al., 23 Jun 2026).

The physical interpretation given in the waveguide work is that $\epsilon_m=0\ \mathrm{cm}^{-1}$1 encodes the mismatch of $\epsilon_m=0\ \mathrm{cm}^{-1}$2 rotation representations between $\epsilon_m=0\ \mathrm{cm}^{-1}$3 and $\epsilon_m=0\ \mathrm{cm}^{-1}$4 and acts as a symmetry-based indicator for a quantized bulk multipole moment that enforces higher-order boundary localization at $\epsilon_m=0\ \mathrm{cm}^{-1}$5-invariant corners (Zhang et al., 22 Jun 2026). In an electronic analog, this corresponds to a fractional corner charge $\epsilon_m=0\ \mathrm{cm}^{-1}$6; in photonics, it manifests as corner-confined mid-gap modes robust to symmetry-preserving perturbations (Zhang et al., 22 Jun 2026).

The corner-state wavefunction in the topological regime decays shell by shell away from a corner as

$\epsilon_m=0\ \mathrm{cm}^{-1}$7

so the localization length becomes short when the breathing ratio $\epsilon_m=0\ \mathrm{cm}^{-1}$8 (Zhang et al., 22 Jun 2026). This provides a direct geometric interpretation of how fractality controls confinement through the reduced Kagome parameters.

4. Experimental realizations and observed signatures

Two experimental platforms have established the phenomenon. The first is a femtosecond-laser-written waveguide system in glass operating at $\epsilon_m=0\ \mathrm{cm}^{-1}$9, with nearest-neighbor coupling $S$0. It includes both a third-generation Sierpiński-gasket array and an otherwise equivalent uniform triangular control with the same mean couplings (Zhang et al., 22 Jun 2026). A weakly coupled, detuned auxiliary waveguide is butt-coupled to the target corner site, with auxiliary coupling $S$1 and onsite detuning $S$2, tuned so that its propagation constant resonates with the corner-state energy $S$3 in gap II (Zhang et al., 22 Jun 2026). Light is injected into the auxiliary at $S$4, transferred into the lattice while on resonance, and the auxiliary is truncated at $S$5; a representative output facet is recorded at $S$6 (Zhang et al., 22 Jun 2026).

Under periodic boundary conditions, computed for the second-generation fractal, multiple band manifolds appear with alternating trivial and topological gaps. Under open boundaries for the third-generation lattice, discrete in-gap eigenvalues emerge within the nontrivial gaps, and their eigenvectors concentrate at the three $S$7 corners (Zhang et al., 22 Jun 2026). Simulated evolution $S$8 shows strong and persistent corner confinement in the fractal lattice for $S$9, while the uniform triangular control exhibits bulk diffraction (Zhang et al., 22 Jun 2026). Experimental output intensity distributions reproduce the simulations. Corner states also exist in gaps I and III, but gap I is too small for clean experimental isolation, and weakly localized gap-I states are not cleanly resolved due to the small gap and residual bulk overlap (Zhang et al., 22 Jun 2026).

The second platform is a 2D microwave tight-binding-like photonic crystal in a parallel-plate geometry. It uses dielectric rods that host Mie resonances and act as lattice sites, metallic rods acting as perfect electric conductors in simulations to laterally confine and disentangle resonances, and a triangular air-foam host made of ROHACELL 31 HF with $\bar S$0 and loss tangent $\bar S$1; copper-clad Teflon-glass laminates form the plates (Yan et al., 23 Jun 2026). A vector network analyzer, specifically a Keysight E5080, drives and probes the samples via two monopole antennas, and bulk dispersions are extracted by scanning $\bar S$2 fields and applying a spatial FFT (Yan et al., 23 Jun 2026).

In this microwave implementation, the unfractaled Kagome photonic crystal shows a clear Dirac point at $\bar S$3, whereas first- and second-generation Sierpiński fractal Kagome photonic crystals exhibit multiple clean gaps in which the Dirac point is lifted (Yan et al., 23 Jun 2026). Simulations show three topological and one trivial bandgap for the first generation, and seven topological and four trivial bandgaps for the second generation (Yan et al., 23 Jun 2026). Corner-state transmission peaks are observed at $\bar S$4 and $\bar S$5 for the first generation, and at $\bar S$6, $\bar S$7, and $\bar S$8 for the second generation; in each topological gap, three nearly degenerate corner modes exist, one per corner of the triangular sample, and their $\bar S$9 field profiles are tightly confined to the corresponding corner sites (Yan et al., 23 Jun 2026).

5. Disorder, protection, and limitations

The observed corner localization is robust but not unconditionally immune to perturbation. In the waveguide-array realization, random onsite disorder is introduced by drawing onsite detunings $$H_{\mathrm{eff}}(\beta)=H_{SS}+H_{S\bar S}(\beta I_{\bar S}-H_{\bar S\bar S})^{-1}H_{\bar S S},$$0 uniformly from $$H_{\mathrm{eff}}(\beta)=H_{SS}+H_{S\bar S}(\beta I_{\bar S}-H_{\bar S\bar S})^{-1}H_{\bar S S},$$1, implemented experimentally via laser-writing speed variations (Zhang et al., 22 Jun 2026). Because the rotational indicator $$H_{\mathrm{eff}}(\beta)=H_{SS}+H_{S\bar S}(\beta I_{\bar S}-H_{\bar S\bar S})^{-1}H_{\bar S S},$$2 is not well-defined for arbitrary disorder, robustness is probed through open-boundary spectra and eigenstate profiles rather than through symmetry-indicator evaluation (Zhang et al., 22 Jun 2026).

Disorder-averaged spectra over $$H_{\mathrm{eff}}(\beta)=H_{SS}+H_{S\bar S}(\beta I_{\bar S}-H_{\bar S\bar S})^{-1}H_{\bar S S},$$3 realizations show that corner states remain spectrally isolated up to gap-closing thresholds $$H_{\mathrm{eff}}(\beta)=H_{SS}+H_{S\bar S}(\beta I_{\bar S}-H_{\bar S\bar S})^{-1}H_{\bar S S},$$4 for gap I, $$H_{\mathrm{eff}}(\beta)=H_{SS}+H_{S\bar S}(\beta I_{\bar S}-H_{\bar S\bar S})^{-1}H_{\bar S S},$$5 for gap II, and $$H_{\mathrm{eff}}(\beta)=H_{SS}+H_{S\bar S}(\beta I_{\bar S}-H_{\bar S\bar S})^{-1}H_{\bar S S},$$6 for gap III (Zhang et al., 22 Jun 2026). Below threshold, for example at $$H_{\mathrm{eff}}(\beta)=H_{SS}+H_{S\bar S}(\beta I_{\bar S}-H_{\bar S\bar S})^{-1}H_{\bar S S},$$7, corner-state intensity remains sharply localized; above threshold, at $$H_{\mathrm{eff}}(\beta)=H_{SS}+H_{S\bar S}(\beta I_{\bar S}-H_{\bar S\bar S})^{-1}H_{\bar S S},$$8, corner modes merge with the bulk continuum and delocalize (Zhang et al., 22 Jun 2026). Measurements at $$H_{\mathrm{eff}}(\beta)=H_{SS}+H_{S\bar S}(\beta I_{\bar S}-H_{\bar S\bar S})^{-1}H_{\bar S S},$$9 under a three-site excitation corroborate this transition.

A distinct role is played by symmetry-preserving perturbations. For disorder configurations that preserve $(\beta,\psi_S)$0 symmetry, the rotational indicator $(\beta,\psi_S)$1 remains well-defined and corner modes exhibit enhanced robustness compared with fully random onsite disorder, emphasizing the protective role of $(\beta,\psi_S)$2 (Zhang et al., 22 Jun 2026). The microwave photonic-crystal study expresses the same point in symmetry-indicator language: time-reversal symmetry is preserved, no chiral or inversion symmetry is invoked in the diagnosis, and the key protecting structure is $(\beta,\psi_S)$3 rotation together with the fractionalization of rotation eigenvalues across high-symmetry points (Yan et al., 23 Jun 2026).

Several limitations are explicitly identified. In the waveguide work, the reduced Hamiltonian is energy-dependent, so corner-state selection benefits from resonant excitation through the auxiliary waveguide; off-resonant probing can reduce confinement (Zhang et al., 22 Jun 2026). Finite-size and boundary truncation matter because the reduced Kagome is approximate and sensitive to fractal generation and boundary terminations; small gaps, particularly gap I, are more susceptible to leakage (Zhang et al., 22 Jun 2026). In the microwave work, some fractality-induced gaps are very narrow, so corner modes can spectrally overlap the bulk, complicating excitation and readout; increasing fractal generation creates more gaps and modes but tends to reduce some gap sizes (Yan et al., 23 Jun 2026). Practical constraints there include fabrication tolerances of rod positions and diameters, ohmic and dielectric losses at microwave frequencies, and finite-size effects that split nominal degeneracies (Yan et al., 23 Jun 2026).

6. Relation to earlier fractal–Kagome photonics and broader significance

Earlier work on fractal–Kagome photonics established a different, though related, paradigm. In a 2015 study of a one-dimensionally periodic Kagome waveguide strip whose elementary plaquettes are filled with finite-generation Sierpinski gasket waveguide meshes, the central phenomena were a hierarchy of dispersionless photonic bands, compact localized states with finite spatial support, and tunable onset length of localization produced by real-space renormalization of the fractal degrees of freedom (Nandy et al., 2015). In that setting, the scalar wave equation on the network is mapped to an energy-dependent tight-binding-like model with

$(\beta,\psi_S)$4

and the flat-band condition is obtained as $(\beta,\psi_S)$5 after renormalization (Nandy et al., 2015). The number of flat bands increases monotonically and rapidly with Sierpinski-gasket generation, the corresponding compact localized states occupy a hierarchy of localization areas, and slow-light operation is associated with flat bands and band edges (Nandy et al., 2015).

That 2015 line of work did not formulate the system as a fractality-induced higher-order topological insulator. Its focus was on flat bands, compact localization, staggered localization, slow light, and mode crossover in a fractal-decorated Kagome strip (Nandy et al., 2015). The 2026 developments extend the fractal–Kagome theme into higher-order topology by showing that fractality can lift Kagome degeneracies, open topological gaps, and produce corner modes whose existence is diagnosed by $(\beta,\psi_S)$6 symmetry indicators and explained by isospectral reduction to a breathing Kagome model (Yan et al., 23 Jun 2026, Zhang et al., 22 Jun 2026).

This suggests a broader conceptual distinction within the phrase “fractal Kagome photonic crystal.” One usage refers to a photonic crystal whose physical lattice is a Kagome structure recursively decorated by a Sierpiński rule; another refers to a uniform-coupling Sierpiński photonic lattice whose reduced description is an effective breathing Kagome. In both usages, fractality is the source of the effective Kagome hierarchy, but the first emphasizes explicit fractalization of a Kagome parent lattice, whereas the second emphasizes emergence of Kagome higher-order topology from a self-similar non-Kagome parent graph (Yan et al., 23 Jun 2026, Zhang et al., 22 Jun 2026).

The broader significance claimed in the 2026 studies is that fractal geometry itself can serve as a mechanism for generating topological boundary states in photonic lattices, without artificial gauge fields, long-range couplings, magnetic fields, staggered hopping, spin-orbit coupling, or temporal modulation (Zhang et al., 22 Jun 2026, Yan et al., 23 Jun 2026). Proposed application directions in the microwave photonic-crystal work include low-threshold corner-state lasing, sensing with enhanced local fields at corners, compact frequency-selective elements or robust switches, and reconfigurable topological routing by selecting corners as ports (Yan et al., 23 Jun 2026). The waveguide study further suggests that the same isospectral design principle could be transferred to dielectric photonic-crystal platforms by embedding Sierpiński-like fractal hole arrays whose band manifolds, under reduction, produce breathing Kagome physics, with design priorities including uniform nearest-neighbor coupling in the parent lattice, corner-access ports for resonant excitation, and preservation of $(\beta,\psi_S)$7 symmetry in the truncation (Zhang et al., 22 Jun 2026).