In this article we show that in a three dimensional (3D) optical field there can exist two types of hidden singularities, one is spin density (SD) phase singularity and the other is SD vector singularity, which are both unique to 3D fields. The nature of these SD singularities is discussed and their connection with traditional optical singularities is also examined. Especially it is shown that in a 3D field with purely transverse spin density (`photonic wheels'), these two types of singularities exhibit very interesting behaviors: they are exactly mapped to each other regardless of their different physical meanings and different topological structures. Our work supplies a fundamental theory for the SD singularities and may provide a new way for further exploration of 3D optical fields.

中文翻译：

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3D 矢量场中的隐藏奇点

在本文中，我们展示了在三维 (3D) 光场中可以存在两种类型的隐藏奇点，一种是自旋密度 (SD) 相位奇点，另一种是 SD 矢量奇点，它们都是 3D 场独有的。讨论了这些 SD 奇点的性质，并检查了它们与传统光学奇点的联系。特别是在具有纯横向自旋密度（“光子轮”）的 3D 场中，这两种奇点表现出非常有趣的行为：它们完全映射到彼此，而不管它们的物理意义和拓扑结构不同。我们的工作为 SD 奇点提供了一个基本理论，并可能为进一步探索 3D 光场提供一种新方法。