Inverse Design of the Topology Bandwidth Tradeoff in Valley Photonic Crystals

Integrated on-chip photonics increasingly relies on wave propagation that remains stable in the presence of fabrication imperfections, tight bends, and dense routing. Valley photonic crystals (VPCs) offer an attractive path: by opening a gap at the Dirac points of a hexagonal lattice, one can engineer guided modes confined to domain walls that thread around corners with reduced backreflection. We develop a design framework that co-optimizes the photonic bulk band gap and valley Chern number using a modified particle-swarm optimization (PSO), while evaluating the photonic band structure via plane-wave expansion and the topological characteristics using a gauge-invariant lattice discretization to compute the Berry-curvature. The optimized structures exhibit a clean valley-Hall gap with edge bands traversing the gap and high interface transmission in full-wave simulations. These results consolidate topology-aware geometry optimization for robust on-chip guiding.

💡 Research Summary

Integrated on‑chip photonics demands waveguides that remain functional despite fabrication tolerances, tight bends, and dense routing. Conventional dielectric waveguides suffer from reflections at discontinuities and high bend loss, prompting interest in topological photonics where guided states are protected by band‑structure topology. Valley photonic crystals (VPCs) exploit inversion‑symmetry breaking in a hexagonal lattice to open a gap at the Dirac points (K and K′). The resulting Berry curvature becomes sharply localized at these valleys, giving rise to domain‑wall “kink” edge modes that are immune to back‑scattering because of valley‑momentum mismatch and topological protection.

The authors present a systematic inverse‑design framework that simultaneously maximizes the bulk band‑gap (which determines usable bandwidth) and the valley Chern number (which quantifies topological robustness). They parameterize the unit cell with six variables: two independent regular polygons characterized by side count (N₁, N₂), normalized size (l₁, l₂), and in‑plane rotation (θ₁, θ₂). This mixed continuous‑discrete design space is explored with a modified particle‑swarm optimization (PSO). Continuous variables follow the standard velocity‑position update with a decaying inertia weight, while the discrete polygon‑type variables are mutated via a copy‑mutate operator after each continuous step. Low‑discrepancy sampling initializes the swarm, and feasibility is enforced by projection onto symmetry‑folded bounds.

For each candidate, the bulk TE‑like band structure is computed using plane‑wave expansion (PWE). The valley topology is quantified by a gauge‑invariant lattice formulation of Berry curvature (U(1) link‑variable plaquette method). Integrating the curvature over patches around K and K′ yields the valley Chern indicator Cᵥ, which may be non‑quantized in the large‑gap regime. The objective (figure of merit) is defined as

 T(x) = ((Δf / f₀)²) · |Cᵥ|,

where Δf is the minimum direct bulk gap within the valley window and f₀ is the mid‑gap frequency. This formulation balances bandwidth (Δf/f₀) against topological strength (|Cᵥ|).

The PSO generates a cloud of solutions; plotting |Cᵥ| versus (Δf/f₀)² reveals a Pareto front. The front shows that increasing the normalized gap inevitably spreads Berry curvature over a larger portion of the Brillouin zone, reducing |Cᵥ| from its ideal ±½ value. Nevertheless, the optimizer finds designs with |Cᵥ|≈0.4–0.45 while achieving Δf/f₀≈0.25–0.30, i.e., a substantial bulk gap together with a strong valley‑topological response. The optimized unit cell features two asymmetric polygons that break inversion symmetry but preserve lattice periodicity, thereby lifting the Dirac degeneracy and opening a sizable valley‑Hall gap.

To validate device performance, the optimized lattice is paired with its inverted counterpart to form a defect‑type domain wall. Full‑wave finite‑difference time‑domain (FDTD) simulations (Tidy3D) are performed on two configurations: a straight waveguide and a compact 90° bend containing multiple corners. Field intensity (|H_z|) maps show tight confinement of energy to the domain wall with negligible leakage into the bulk. Poynting‑vector visualizations confirm that power follows the interface around corners without forming standing‑wave patterns typical of back‑reflection‑dominated guides. Transmission spectra across the valley‑Hall gap indicate >90 % transmission for both straight and bent routes; loss is dominated by out‑of‑gap scattering rather than bend‑induced backscattering.

In summary, the paper makes three principal contributions: (1) a mixed‑integer inverse‑design methodology that couples bulk band‑gap and Berry‑curvature‑based topological metrics; (2) a quantitative Pareto analysis exposing the fundamental trade‑off between bandwidth and valley‑topology in VPCs; and (3) demonstration of high‑transmission, low‑loss domain‑wall waveguides that retain robust guiding even through sharp bends. These results provide a practical pathway for designing topologically protected photonic interconnects that are tolerant to fabrication imperfections and layout constraints, advancing the integration of valley photonic crystals into future on‑chip photonic systems.