Leveraging Machine-Learning Representations to Design Photonic Band Gaps Crystals

Abstract

The design of crystallographic structures for photonic band gaps (PBGs) is an active area of research. While topology, local environment, and symmetry are known to influ- ence band structure, a comprehensive understanding of these design rules remains elusive. This research investigates the design principles of 3D photonic crystals (PhCs) using ma- chine learning (ML) to analyze a large dataset of PhC structures and their corresponding band gap properties. The study focuses on evaluating the effectiveness of two structural descriptors: Steinhardt Order Parameters (SOP) and Betti numbers. SOP quantifies local symmetry, while Betti numbers provide a topological measure of connectivity within the PhC. Key Findings: Betti numbers demonstrate superior performance compared to SOP in predicting the min- imum and maximum frequencies of the photonic bands, indicating the dominant role of topology in governing PBG characteristics. This aligns with the understanding that topo- logical invariants significantly influence the presence, size, and location of band gaps. The study reveals that both SOP and Betti numbers outperform traditional symmetry- based descriptors, further emphasizing the importance of local environment and connec- tivity in PBG design. This is consistent with previous research highlighting the impact of lattice symmetry, shape and orientation of dielectric scatterers on photonic gap parame- ters. Analysis of Photonic Density of States (PDOS) reveals distinct clustering patterns for structures with and without band gaps, offering insights into the spectral properties and potential for designing robust PhCs. This approach aligns with the research on the di- versity of 3D photonic crystals, emphasizing the correlation between lattice geometry,

connectivity, and band gap size. The findings suggest that incorporating topological and local environment descriptors, such as Betti numbers and SOP, into the design process could lead to more effective and robust PhC structures for applications in waveguides, solar panels, and other photonic devices. Future research could explore combinations of Betti numbers that strongly correlate with PBG characteristics and investigate the appli- cation of these descriptors to disordered PhCs. Additionally, experimental validation of the predicted structures would be essential to assess their real-world performance.