Topological flowers and spider webs in 3D vector fields

Topological structures currently are of special interest in the integration of singular optics and topological photonics. In this article, the topological flowers and spider webs, which got less attention comparing to the lemon-, star- structures of the same category, are investigated in 3D vector fields. We show that by strongly focusing higher-order singular beams, both the spin density (SD) vectors and the polarization states of the transverse fields on the focal plane exhibit flowers and spider webs structures in topology with 2| m  − 1| folds/sectors ( m is the beam order), and the topological structures of the SD vectors are demonstrated to have a 90°/| m  − 1| rotation. On the other hand, the topological theory also needs to be developed according to the rapid growth of topological photonics. Here, by defining a ‘relative topological charge’, we have observed and analyzed the topological reactions of the loops (composed of SD singularities) rather than the ‘point-type’ singularities in conventional reactions. More specially, the ‘radial index’ and ‘azimuthal index’ are proposed to characterize the topological features of the flowers and spider webs, and have been verified that the ‘radial index’ is peculiar to 3D vector fields and proportional to 1/| m  − 1| in general. Our work provides a way to describe the topological behaviors of groups of singularities and supplies new parameters for measuring the topological patterns in 3D vector fields, which will rich the topological theory and may have applications in topological photonics.