Analytical inversion for the joint axial and polarization-dependent transverse design of long-depth-of-focus fields

Based on the Richards-Wolf vectorial diffraction integral, we develop an analytical inversion method for jointly tailoring axial intensity and polarization-dependent transverse characteristics of long-depth-of-focus fields. For a target focal depth of 200λ, numerical results under radially polarized illumination demonstrate the generation of an ultra-long optical needle with tunable on-axis intensity and a stable transverse scale characterized by the full width at half-maximum (FWHM), as well as the joint design of on-axis intensity and FWHM. For azimuthally polarized illumination at the same focal depth, numerical results demonstrate the joint design of bright-ring peak intensity and inner FWHM for optical tubes. The performance limits of the proposed analytical design under a finite numerical aperture are also analyzed.