Focus engineering based on analytical formulae for tightly focused polarized beams with arbitrary geometric configurations of linear polarization.

Highly confined electromagnetic fields with controllable intensity profiles and polarization orientations are greatly desired in many fields. In this paper, we report on the generation of highly confined fields through tightly focused locally linearly polarized beams. Using the Richards-Wolf vectorial diffraction method, we derive and build integrated analytical formulae for calculating the tightly focused field of polarized beams with arbitrary geometric configurations of linear polarization. Based on the analytical model, the focusing properties of four types of polarized light beams, i.e., linearly, azimuthally, radially, and spatially variant polarized beams, with locally linear states of polarization are investigated numerically and discussed in detail. By manipulating the radial and azimuthal indices and initial phases, we obtain a tunable three-dimensional optical cage, multifoci, optical needles, and channels in the focal volume of a high-numerical-aperture objective lens. These peculiar properties may find applications in fields such as optical trapping and manipulation of nanoparticles and super-resolution microscopy imaging.