Inverse Design of the Topology Bandwidth Tradeoff in Valley Photonic Crystals

TL;DR

The paper addresses robust on-chip routing in valley photonic crystals (VPCs) by jointly maximizing usable bandwidth and valley topology. It introduces a six-parameter, mixed-integer inverse-design framework that uses a topology-inspired objective $\mathcal{T}(\boldsymbol{x})=\left(\frac{\Delta f}{f_0}\right)^2|C_v|$, with bulk bands computed via plane-wave expansion and Berry curvature evaluated through a gauge-invariant lattice discretization. Device performance is validated with full-wave FDTD simulations of domain-wall interfaces, demonstrating robust broadband transport through sharp bends. The results reveal a Pareto frontier between bandwidth and valley topology, highlighting that strong valley protection can persist even when valley-Chern numbers are not strictly quantized, thereby offering a practical route toward topology-aware photonic-device design.

Abstract

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.

Figures (4)

• Figure 1: Inverse-design workflow for topology-aware bandwidth optimization in valley photonic crystals.(a) Six-dimensional unit-cell parameterization using two symmetry-inequivalent regular polygonal air holes, with continuous variables (feature sizes and rotations) and discrete variables (polygon side counts). (b) Optimization objective combines a normalized bulk gap (bandwidth) with valley-chern number, a topological indicator computed from Berry curvature, forming a single figure of merit explored by a modified particle swarm optimization. (c) Device-level validation: optimized structures are interfaced to form domain-wall waveguides and benchmarked by full-wave simulations (transmission and field confinement across straight and sharp-bend routes).
• Figure 2: Topology-bandwidth tradeoff and Pareto frontier obtained from mixed-integer PSO. Each point corresponds to a candidate unit cell evaluated during the optimization, plotted as the valley topology indicator $|C_v|$ versus the relative band gap metric used in the objective, $(\Delta f/f_0)^2$. The Pareto-optimal set forms the region in the outermost middle part, illustrating that valley topology and gap fraction can be co-optimized but not simultaneously maximized within a fixed design family. The color encodes the composite topological figure of merit used internally by the optimizer (defined in Methods/SI), and is shown here to visualize how the optimization weights select designs along the tradeoff boundary.
• Figure 3: Optimized unit cell and its properties.(a) Unit cell geometry obtained by the mixed–integer PSO; inversion symmetry is broken by two symmetry-inequivalent inclusions. (b) Photonic band structure along $G\!-\!M\!-\!K\!-\!G$ with the optimized bulk gap highlighted (blue band) (c) Berry curvature $\Omega(\mathbf{k})$ over the Brillouin zone, showing opposite-sign localization at $K$ and $K'$. (d) Representative real-space phase profile of a valley Bloch eigenmode for the optimized unit cell, highlighting the inversion-broken modal texture. (e) Projected band diagram of the domain wall: interface (kink) modes span the bulk gap; the gray region denotes the bulk band region. (f) Magnetic Field Profile ($|H_z|$) of the domain wall edge modes (left one is for the top edge mode at $K$ and the right one is for the bottom edge mode at $K$.
• Figure 4: Full-wave waveguiding validation of the optimized VPC design.(a) Steady-state field intensity $\lvert H_z\rvert$ for a compact, sharp-bend domain-wall route and a straight domain-wall rotue built from the optimized unit cell and its inverted partner. (b) Poynting-vector maps illustrating energy flow along the interface for the bent and straight geometries in the boxed regions from part (a), respectively, confirming guided transport through corners with strongly suppressed back-reflection. (c) Transmission spectra (dB) for bent and straight waveguides; the gray shaded region highlights the operating window associated with the bulk valley-Hall gap where the interface mode supports broadband, low-backscattering transport.