Three-dimensional intensity distribution of focused annular non-uniformly polarized beams

An approximate formula for evaluating the relative focal shift in an annular focusing system illuminated by non-uniformly polarized (NUP) light is derived. It is shown that the relative focal shift is determined by the effective Fresnel number, which is equal to the product of the Fresnel number of the annular focusing system and a parameter characterizing the polarization distribution of the NUP beam across the aperture of the annular focusing system. A three-dimensional intensity distribution of the focused annular NUP beam is presented. It is found that, even if the effective Fresnel numbers for different polarization distributions of the annular NUP beams are identical, i.e. the relative focal shifts are nearly the same, the corresponding threedimensional intensity distributions near the focus are still quite different.