Fine characteristics of polarization singularities in a three-dimensional electromagnetic field and their properties in the near field of a metallic nanospheroid

Isotropy parameters of truly linear (${L}^{T}$) and truly circular (${C}^{T}$) polarization singularities in a three-dimensional monochromatic electromagnetic field are defined. It is demonstrated that these parameters characterize the distribution of polarization ellipses near the singular points in a more detailed way than a topological index. A numerical algorithm which tracks ${C}^{T}$ and ${L}^{T}$ lines and visualizes the changes of isotropy parameters along the lines is proposed. The algorithm is applied to visualize the polarization singularity lines and their properties in the vicinity of a metallic nanospheroid that is irradiated by a plane elliptically polarized wave. It is shown that both ${C}^{T}$ and ${L}^{T}$ lines appear close to the surface of the nanoparticle and they are three-dimensional closed loops or arcs that start and end at the surface of the nanospheroid. In most cases the singular lines of different types are topologically linked.