Electromagnetic multipoles serve as one of the most ubiquitous languages in photonics, forming a complete expansion basis for radiations of arbitrary types. Electromagnetic multipoles from the perspective of line fields and singularities are explored. It is then discovered that, for arbitrary finite sources in a homogeneous background, there have to be isolated directions along which the radiation is either zero or circularly polarized. For such singular directions, half‐integer Hopf indices can be assigned and the index sum has to be a global invariant of 2. With this topological insight, the hidden dimensions of radiative circularly polarized Bloch modes of photonic crystal slabs are unveiled, revealing their topological origins of line singularities and indices. This work has established subtle connections between three seemingly unrelated but sweeping physical entities (line singularities of Hopf indices, electromagnetic multipoles, and Bloch modes), which can nourish new frames of visions and applications fertilizing many related fields.

中文翻译：

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电磁多极的线奇异性和Hopf指数

电磁多极子是光子学中最普遍的语言之一，为任意类型的辐射形成了完整的扩展基础。从线场和奇异性的角度研究了电磁多极。然后发现，对于均质背景中的任意有限源，必须存在辐射为零或圆偏振的孤立方向。对于这样的奇异方向，可以分配半整数霍普夫指数，并且指数总和必须为2的全局不变量。有了这种拓扑洞察力，揭示了光子晶体平板的辐射圆偏振布洛赫模的隐藏维度，从而揭示了它们的拓扑线奇点和索引的起源。