Focusing of Optical Vector-vortex Beams

Theoretical formalism using vectorial Rayleigh diffraction integrals is developed to calculate the electric field components Ex, Ey, Ez of generalized vector-vortex (VV) beams of different phase and polarization characteristics as a function of propagation distance ‘z’ in the focal region of an axicon. This formalism is used to generate sub-wavelength spot-size (0.43λ) ultra-long length (80λ) longitudinally-polarized optical needle beam by appropriately selecting the phase and polarization characteristics of the input VV beam. The formalism is further extended to also generate purely transverse polarized beam with similar characteristics. The focusing process leads to interference between different field components of the beam resulting in the formation of C-point polarization singularities of index Ic = ±1 whose transverse characteristics evolve with propagation distance. Experimental results to support our theoretical calculations are presented along with lens focus comparison results.