Numerical methods for dispersion relation calculation for photonic crystals

Abstract

This thesis develops efficient numerical methods for computing the dispersion relation in photonic crystals (PhCs), which governs the frequency–wave vector dependence and underlies the formation of photonic band gaps. Accurate and efficient computation of dispersion relations is essential for designing photonic devices such as waveguides and resonators.

We begin by formulating the parameterized Helmholtz eigenvalue problem arising from Maxwell equations via the Bloch-Floquet theorem. A regularity analysis is carried out for the associated band functions, establishing their piecewise analyticity and identifying singularities. These results provide a theoretical foundation for designing effective approximation strategies.

Two numerical approaches are proposed. The first is a global polynomial interpolation method over the irreducible Brillouin zone, based on carefully selected sampling points. The second is a local hp-adaptive sampling algorithm, which refines the mesh near singularities and assigns polynomial degrees adaptively. This method achieves exponential convergence when singularities are finite, and first-order convergence otherwise, thus capturing fine-scale spectral features while maintaining computational efficiency. To demonstrate practical effectiveness, we apply the proposed methods to shape optimization problems aimed at maximizing band gaps in 3D photonic crystals. The adaptive method is extended to 3D settings and validated through numerical experiments, showing improved performance over traditional approaches.

We further explore deep learning techniques for band structure prediction. A supervised learning framework based on the U-Net architecture is introduced, incorporating transfer learning and super-resolution to reconstruct high-resolution band diagrams from coarse data. Numerical results show that the learning-based approach can accurately approximate dispersion relations with significantly reduced computational cost.

Overall, this thesis integrates rigorous mathematical analysis, numerical algorithms, and machine learning techniques, providing a unified and efficient framework for PhC band structure computation. The methods developed here lay the groundwork for future advances in high-precision photonic device design.