The fundamental, or first, band gap is of unmatched importance in the study of photonic crystals. Here, we address precisely where this gap can be opened in the band structure of three-dimensional photonic crystals. Although strongly constrained by symmetry, this problem cannot be addressed directly with conventional band-symmetry analysis due to the existence of a photonic polarization vortex at zero frequency. We develop an approach for overcoming the associated symmetry singularity by incorporating fictitious, auxiliary longitudinal modes. Our strategy also enables us to extend recent developments in symmetry-based topological analysis to the fundamental gap of three-dimensional photonic crystals. Exploiting this, we systematically study the topology of the minimal fundamental gaps. This reveals the existence of topological gap obstructions that push the fundamental gap higher than what a conventional analysis would suggest. Our work demonstrates that topology can play a crucial role in the opening of the fundamental photonic gap and informs future theoretical and experimental searches for conventional and topological band gaps in three-dimensional photonic crystals.

中文翻译：

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光子晶体中基本能隙的位置和拓扑

基本的或第一的带隙在光子晶体的研究中具有无与伦比的重要性。在这里，我们精确地解决了在三维光子晶体的能带结构中可以打开这个间隙的位置。尽管受到对称性的强烈限制，但由于在零频率处存在光子偏振涡旋，因此无法通过常规带对称分析直接解决此问题。我们开发了一种通过结合虚构的辅助纵向模式来克服相关对称奇异性的方法。我们的策略还使我们能够将基于对称的拓扑分析的最新发展扩展到三维光子晶体的基本间隙。利用这一点，我们系统地研究了最小基本间隙的拓扑。这揭示了拓扑间隙障碍的存在，这些障碍将基本间隙推到了高于传统分析所暗示的水平。我们的工作表明，拓扑结构可以在打开基本光子间隙中发挥关键作用，并为未来对三维光子晶体中常规和拓扑带隙的理论和实验研究提供信息。