Manipulation of photonic topological edge and corner states via trivial claddings

Technical Summary: Manipulation of Photonic Topological Edge and Corner States via Trivial Claddings

Problem Statement
While crystalline symmetry has established a robust framework for realizing photonic topological phases, such as topological crystalline insulators (TCIs), the role of "trivial claddings" in these systems is often underappreciated. In photonic TCIs, trivial claddings are typically employed merely to confine topological boundary states that exist above the light cone, preventing them from leaking into free space. However, the potential of these claddings to actively manipulate topological waves, influence the dispersion of interface states, and control the emergence of higher-order corner states has been largely overlooked. Furthermore, the reliance of corner states on specific boundary configurations, rather than solely on bulk topology, challenges the conventional understanding of bulk-boundary correspondence in photonic systems.

Methodology
The authors investigate two-dimensional kagome photonic crystals (KPCs) composed of dielectric rods with a permittivity of $\varepsilon = 11.7$ and a radius of $r = 0.1a$. The study focuses on three distinct primitive cell configurations based on their center positions within the unit cell:

1. H-KPCs: $C_6$-symmetric cells centered at the "1a" Wyckoff position.
2. U-KPCs and D-KPCs: $C_3$-symmetric cells centered at "1b" and "1c" Wyckoff positions, respectively.

The authors identify H-KPCs as photonic analogs of quantum spin Hall insulators (non-trivial topology) and utilize U-KPCs or D-KPCs as trivial claddings. The topological properties are characterized using symmetry indicators ($\chi^{(6)}$ and $\chi^{(3)}$) derived from rotation eigenvalues at high-symmetry points.

The study employs both numerical simulations (COMSOL MULTIPHYSICS) and microwave experiments. Key experimental techniques include Fourier-transformed electromagnetic field scanning to measure edge dispersions and the injection of point sources to observe spatial trapping. The researchers systematically tune the geometric parameter $d$ (the distance between the rod center and the primitive cell center) in the trivial cladding (D-KPCs) while keeping the topological core (H-KPCs) fixed. This approach allows for the observation of phase transitions in interface states and the modulation of corner states in finite hexagonal supercells.

Key Contributions and Results

1. Geometry-Induced Phase Transition in Interface States:
   By interfacing $C_6$-symmetric H-KPCs with $C_3$-symmetric D-KPCs, the authors demonstrate the existence of pseudospin-momentum locked topological interface states. They show that continuously tuning the geometric parameter $d$ of the trivial D-KPC cladding induces a topological phase transition in these interface states. Specifically, the frequency gap between the even- and odd-parity interface states at $k_x=0$ closes and reopens as $d$ increases. This transition is characterized by a sign change in the Dirac mass of the interface states, effectively altering the dispersion without changing the bulk topology of the core.

2. Topological Rainbow Trapping:
   Leveraging the tunability of the interface states, the authors design a graded structure where the geometric parameter $d$ of the D-KPC cladding varies spatially. This gradient reshapes the dispersion relation of the interface states, causing the group velocity to vary with frequency. Experimental results confirm that electromagnetic waves of different frequencies are spatially separated and trapped at distinct positions along the interface, demonstrating a "topological rainbow trapping" effect. This design utilizes two tunable interface bands, extending the working frequency range compared to previous single-band approaches.

3. Cladding-Dependent Higher-Order Topology:
   The study constructs a finite hexagonal supercell with an H-KPC core surrounded by D-KPC cladding. While the bulk H-KPC possesses higher-order topology, the authors find that the existence of corner states within the bandgap is critically dependent on the trivial cladding. By tuning the parameter $d$ of the cladding, the corner states can be made to merge into the bulk states or disappear entirely, even when the bulk topology and corner charge of the core remain unchanged. This confirms that corner states are not solely determined by the bulk but are sensitive to the boundary configuration and the nature of the surrounding trivial medium.

Significance
The paper establishes that trivial claddings are not passive confinement layers but active components in the manipulation of photonic topological states. The work highlights that:

• The dispersion and existence of topological interface states can be effectively tuned by modifying the geometry of the outer cladding.
• Higher-order topological features, specifically corner states, rely on the interplay between the topological core and the trivial cladding, challenging the notion that they are solely a manifestation of bulk topology.
• This geometry-induced control offers a novel approach for manipulating topological waves, with demonstrated potential for applications such as rainbow trapping, without requiring changes to the intrinsic bulk material properties.

The findings underscore the necessity of considering boundary configurations and cladding properties when designing and interpreting photonic topological devices.