Discrete diffraction for analytical approach to tightly focused electric fields with radial polarization

Radially polarized laser beams in high-resolution microscopy provide an effective means to reduce the focus size below the diffraction limit. Unfortunately, their theoretical manipulation is usually limited to numerical methods. Here, we demonstrate an approach that leads to analytical expressions for the focused electric fields. The approach is based on the discretization of the continuous character of diffraction taking place at the microscope focus. Comparisons with fully numerical calculation are discussed. It results that the new approach accurately reflects the distribution of light within the focal volume with relative deviations that are between 10⁻⁴ and 10⁻¹³.