Predicting band structures for 2D Photonic Crystals via Deep Learning

TL;DR

A supervised learning approach utilizing U-Net, along with transfer learning and Super-Resolution techniques, to forecast dispersion relations for 2D PhCs, which reduces computational expenses by producing high-resolution band structures from low-resolution data, eliminating the necessity for fine meshes throughout the Brillouin zone.

Abstract

Photonic crystals (PhCs) are periodic dielectric structures that exhibit unique electromagnetic properties, such as the creation of band gaps where electromagnetic wave propagation is inhibited. Accurately predicting dispersion relations, which describe the frequency and direction of wave propagation, is vital for designing innovative photonic devices. However, traditional numerical methods, like the Finite Element Method (FEM), can encounter significant computational challenges due to the multiple scales present in photonic crystals, especially when calculating band structures across the entire Brillouin zone. To address this, we propose a supervised learning approach utilizing U-Net, along with transfer learning and Super-Resolution techniques, to forecast dispersion relations for 2D PhCs. Our model reduces computational expenses by producing high-resolution band structures from low-resolution data, eliminating the necessity for fine meshes throughout the Brillouin zone. The U-Net architecture enables the simultaneous prediction of multiple band functions, enhancing efficiency and accuracy compared to existing methods that handle each band function independently. Our findings demonstrate that the proposed model achieves high accuracy in predicting the initial band functions of 2D PhCs, while also significantly enhancing computational efficiency. This amalgamation of data-driven and traditional numerical techniques provides a robust framework for expediting the design and optimization of photonic crystals. The approach underscores the potential of integrating deep learning with established computational physics methods to tackle intricate multiscale problems, establishing a new benchmark for future PhC research and applications.

Figures (14)

• Figure 1: Illustration of unit cell $\Omega$ for a square lattice (left) and the corresponding Brillouin zone (right): In $\Omega$, blue stands for alumina with permittivity $8.9$ and white for air with permittivity $1$; In $\mathcal{B}_F$, the IBZ is the shaded triangle with vertices $\Gamma=(0,0)$, $X=\frac{1}{a}(\pi,0)$ and $M=\frac{1}{a}(\pi,\pi)$.
• Figure 2: Example of unit cell with $p4m$ plane symmetry
• Figure 3: The workflow of predicting dispersion relation with given unit cell.
• Figure 4: The workflow of predicting dispersion relation with given "low-resolution" band function.
• Figure 5: Illustration of labeled data $\left((M_{\Omega,64}, I_{64}^{(n)}),\omega_{\text{H},n}^{64}\right)$ and $\left((M_{\Omega,16}, I_{16}^{(n)}),\omega_{\text{L},n}^{16}\right)$ for learning task \ref{['F 1']} with $n=1,\cdots,10$: unit cell matrices $M_{\Omega,64}$ (top left), $M_{\Omega,16}$ (bottom left) and the first 10 band functions $\omega_{\text{H},n}^{64}$ (top right), $\omega_{\text{L},n}^{16}$ (bottom right), arranged from left to right and top to bottom. All figures are displayed using MATLAB built-in function imagesc.