A quasi-rigorous solution for focusing radially polarized vector beams by an arbitrary opening paraboloid mirror

For focusing of radially polarized vector beams using an arbitrary opening paraboloid mirror, a quasi-rigorous integral representation of light field is derived based on a rigorous vectorial diffraction theory of electromagnetic waves. Under far-field approximation conditions, the derived integrals reduce to the results from vectorial Debye-Wolf diffraction integral. Numerical calculations show that a marked deviation, as represented by evident focal shift, appears for low-aperture paraboloid mirrors, e.g. the maximum focusing semi-angle α below π/9 with the focal length f being 10³, or below π/6 when f reduces to 10². The radial light spot sharpens to be 0.39 (FWHM) and purely longitudinally polarized when α is close to 115°, while the axial light spot continuously sharpens to be about half wavelength with α approaching π. For an annular paraboloid mirror, it is possible to further compress the radial light spot down to 0.36 (FWHM).