Tightly focusing of linearly polarized vortex beams through a dielectric interface

Based on vectorial Debye theory, the tightly focusing properties of linearly polarized vortex beams through a dielectric interface are studied. Then the intensity distribution in the focal region is investigated in detail by numerical calculations. The focal shifts induced by the mismatch of the refractive indices across the interface has been found. And the dependence of focal shifts and spot size of the focused field on the probe depth and numerical aperture of the objective are presented. Furthermore, the characteristic of intensity pattern of the cross section of the focused field is illustrated, finding that the intensity pattern rotates with the propagation of beams.