Geometric phases are a universal concept that underpins numerous phenomena involving multi-component wave fields. These polarization-dependent phases are inherent in interference effects, spin-orbit interaction phenomena, and topological properties of vector wave fields. Geometric phases have been thoroughly studied in two-component fields, such as two-level quantum systems or paraxial optical waves. However, their description for fields with three or more components, such as generic nonparaxial optical fields routinely used in modern nano-optics, constitutes a nontrivial problem. Here we describe geometric, dynamical, and total phases calculated along a closed spatial contour in a multi-component complex field, with particular emphasis on 2D (paraxial) and 3D (nonparaxial) optical fields. We present several equivalent approaches: (i) an algebraic formalism, universal for any multi-component field; (ii) a dynamical approach using the Coriolis coupling between the spin angular momentum and reference-frame rotations; and (iii) a geometric representation, which unifies the Pancharatnam-Berry phase for the 2D polarization on the Poincaré sphere and the Majorana- sphere representation for the 3D polarized fields. Most importantly, we reveal close connections between geometric phases, angular-momentum properties of the field, and topological properties of polarization singularities in 2D and 3D fields, such as C-points and polarization Möbius strips.

中文翻译：

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2D 和 3D 极化场中的几何相位：几何、动力学和拓扑方面

几何相位是一个普遍的概念，它支持涉及多分量波场的众多现象。这些偏振相关相位是干涉效应、自旋轨道相互作用现象和矢量波场的拓扑特性所固有的。几何相位已经在双分量领域进行了彻底的研究，例如两能级量子系统或近轴光波。然而，他们对具有三个或更多分量的场的描述，例如现代纳米光学中经常使用的通用非近轴光学场，构成了一个重要的问题。在这里，我们描述了在多分量复杂场中沿封闭空间轮廓计算的几何、动力学和总相位，特别强调 2D（近轴）和 3D（非近轴）光场。我们提出了几种等效的方法：(i) 代数形式主义，适用于任何多分量领域；(ii) 在自旋角动量和参考系旋转之间使用科里奥利耦合的动力学方法；(iii) 几何表示，它统一了 Poincaré 球上 2D 极化的 Pancharatnam-Berry 相位和 3D 极化场的 Majorana-sphere 表示。最重要的是，我们揭示了几何相位、场的角动量特性以及 2D 和 3D 场中偏振奇点的拓扑特性（例如 C 点和偏振莫比乌斯带）之间的密切联系。它统一了庞加莱球上 2D 极化的 Pancharatnam-Berry 相位和 3D 极化场的 Majorana-sphere 表示。最重要的是，我们揭示了几何相位、场的角动量特性以及 2D 和 3D 场中偏振奇点的拓扑特性（例如 C 点和偏振莫比乌斯带）之间的密切联系。它统一了庞加莱球上 2D 极化的 Pancharatnam-Berry 相位和 3D 极化场的 Majorana-sphere 表示。最重要的是，我们揭示了几何相位、场的角动量特性以及 2D 和 3D 场中偏振奇点的拓扑特性（例如 C 点和偏振莫比乌斯带）之间的密切联系。