An hp-Adaptive Sampling Algorithm for Dispersion Relation Reconstruction of 3D Photonic Crystals

TL;DR

This work tackles efficient reconstruction of dispersion relations for 3D photonic crystals by solving parameterized Maxwell eigenproblems with Bloch boundary conditions in the Brillouin zone $\mathcal{B}$. It introduces an $hp$-adaptive sampling strategy that refines the parameter-domain mesh near singularities, constructs conforming elementwise polynomial spaces, and employs an elementwise interpolation of band functions, with rigorous convergence results showing exponential decay when branch points are finite and algebraic decay otherwise. The method is coupled with a stability-preserving $\mathbf{H}(\text{curl})$-conforming, $\mathbf{k}$-modified Nédélec discretization and a gradient formula for $\partial_j\lambda_n$ to enable efficient band-structure computations. Numerical experiments on two 3D PhC models demonstrate significant gains in band-gap design efficiency, achieved via Bayesian optimization guided by adaptive sampling, and show faster convergence than uniform refinement. Overall, the approach provides a principled, scalable framework for accurate dispersion relation reconstruction and optimized photonic-band-gap engineering in 3D PhCs.

Abstract

In this work we investigate the computation of dispersion relation (i.e., band functions) for three-dimensional photonic crystals, formulated as a parameterized Maxwell eigenvalue problem, using a novel hp-adaptive sampling algorithm. We develop an adaptive sampling algorithm in the parameter domain such that local elements with singular points are refined at each iteration, construct a conforming element-wise polynomial space on the adaptive mesh such that the distribution of the local polynomial spaces reflects the regularity of the band functions, and define an element-wise Lagrange interpolation operator to approximate the band functions. We rigorously prove the convergence of the algorithm. To illustrate the significant potential of the algorithm, we present two numerical tests with band gap optimization.

Figures (8)

• Figure 1: Unit cells and their corresponding first Brillouin zones for Model 1 and Model 2.
• Figure 2: Uniform discretization of the unit cell $\Omega$. Here, only the mesh inside the silicon blocks and around the interior ball is depicted, and the same mesh size is utilized in the background.
• Figure 3: Band structures along the high symmetry points.
• Figure 4: Bayesian optimization.
• Figure 5: Model 1: Band structures along the high symmetry points.

Theorems & Definitions (25)

• Theorem 2.1: Piecewise analyticity and Lipschitz continuity of band functions
• Theorem 2.2: boffi2006modified
• Theorem 2.3
• proof
• Definition 3.1
• Remark 3.1
• Theorem 3.1: $\mathcal{T}_n^S\subset \mathcal{M}_n$
• proof
• Definition 3.2
• Definition 3.3: Element-wise, face-wise and edge-wise spaces of polynomial functions