The structure of a nonparaxial Gaussian beam near the focus: II. Optical Vortices

Exact analytical solutions of Maxwell's equations describing the behavior of a nonparaxial optical vortex in the vicinity of a focal waist are obtained using the Whittaker method of scalar potentials, the point complex source method, and approximate Davis boundary conditions. It is shown that nonparaxial optical vortices in free space fall into three large groups: even and odd vortices with preferential circular polarization and azimuthally symmetric TE and TM vortices. The fields of even and odd nonparaxial vortices agree well with the fields of guided homogeneous and inhomogeneous vortices of a weakly guiding fiber. In the paraxial approximation, the expressions obtained for the fields are transformed to the fields of paraxial optical vortices. In the focal region, a nonparaxial beam experiences elliptic deformation of the cross section. This elliptic deformation is shown to result from the asymmetric location of regions with negative energy flows. The reversal of sign of the topological charge and the helicity of a combination of even and odd vortices causes both rotation of the dislocation axis through π/2 and longitudinal displacement of the focal spot, which are the transverse and the longitudinal optical Magnus effects.