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The concept of Chern number has been widely used to describe the topologicalproperties of periodic condensed matter and classical wave systems. It istypically defined in the momentum space and is closely related to the geometricphase of dispersion bands. Here, we introduce a new type of spin Chern numberdefined in the real space for optical near fields of finite-sized structures.This real-space spin Chern number is derived from the geometric phase of thenear fields and exhibits an intrinsic relationship with the topology of theoptical structures: the spin Chern number of optical near fields is quantizedand equal to the Euler characteristic of the optical structures. Thisrelationship is robust under continuous deformation of the structure's geometryand is independent of the specific material constituents or externalexcitation. Our work enriches topological physics by extending the concept ofChern number from the momentum space to the real space and opens excitingpossibilities for manipulating light based on the topology of opticalstructures, with potential applications in high-precision optical metrology,optical sensing and imaging.

Near-field Spin Chern Number Quantized by Real-Space Topology of Optical Structures