Optical vortices in a vector field: the general definition based on the analogy with topological solitons in a 2D ferromagnet, and examples from the polarization transverse patterns in a laser

New approaches to the characterization of the vector vortical structure of the paraxial monochromatic field are introduced using the analogy between vector field singularities in a distribution of the unit vector of normalized Stokes parameters m(X, Y, Z), m(2) = 1 in the transverse plane and topological solitons in a 2D ferromagnet. In this approach the vector vortex of the first order appears as zero and pole of the complex homographic function = (X + iY)(1 - Z)(-1). The application of the developed method to the results of numerical simulations of a large aperture Zeeman laser is presented.