Non-Paraxial Propagation of Cylindrical Vector Vortex Beams in the Far-Field

On the basis of the vectorial Rayleigh diffraction integral, the analytical expressions for the electric fields of cylindrical vector vortex beams (CVVBs) with an arbitrary polarization order in the far-field are derived, which helps us investigate the far-field properties of the CVVBs in the non-paraxial and paraxial regimes. Some detailed comparisons of the non-paraxial results with the paraxial results are made, which show that the polarization order and the ratio of beam waist width to wavelength play an important role in determining the non-paraxiality of CVVBs. In addition, the polarization order and the ratio of beam waist width to wavelength have a great impact on the diffraction effect and the divergent behavior of CVVBs in the far-field.