Line Singularities and Hopf Indices of Electromagnetic Multipoles

Abstract

AbstractElectromagnetic 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.