In this work, we apply the group representation theory to systematically study polarization singularities in the in-plane components of the electric fields supported by a planar electromagnetic (EM) resonator with generic rotation and reflection symmetries. We reveal the intrinsic connections between the symmetries and the topological features, i.e., the spatial configuration of the in-plane fields and the associated polarization singularities. The connections are substantiated by a simple relation that links the topological charges of the singularities and the symmetries of the resonator. To verify, a microwave planar resonator with the D8$D_8$ group symmetries is designed and numerically simulated, which demonstrates the theoretical findings well. Our discussions can be applied to generic EM resonators working in a wide EM spectrum, such as circular antenna arrays, microring resonators, and photonic quasi-crystals, and provide a unique symmetry perspective on many effects in singular optics and topological photonics.

中文翻译：

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具有旋转和镜像对称性的平面电磁谐振器中的偏振奇点

在这项工作中，我们应用群表示理论系统地研究了由具有一般旋转和反射对称性的平面电磁 (EM) 谐振器支持的电场的面内分量中的极化奇点。我们揭示了对称性和拓扑特征之间的内在联系，即面内场的空间配置和相关的极化奇点。这些联系由一个简单的关系来证实，该关系将奇点的拓扑电荷与谐振器的对称性联系起来。为了验证，微波平面谐振器与D 8$D_8$设计了群对称性并进行了数值模拟，很好地证明了理论发现。我们的讨论可应用于在宽 EM 频谱中工作的通用 EM 谐振器，例如圆形天线阵列、微环谐振器和光子准晶体，并为奇异光学和拓扑光子学中的许多效应提供独特的对称视角。