Focusing of a radially polarized, cylindrically symmetric, vector beam by a thin lens: Simplified analysis of the radial-to-axial polarization transformation and determination of the electromagnetic fields at and near focus

The radial-to-axial polarization transformation that accompanies the focusing of a cylindrically symmetric vector beam by a thin lens is analyzed by use of polarization ray tracing and wave optics. A simple expression for the axial electric field at focus is derived in terms of an integral of the incident radial electric field over the entrance pupil of the lens. Subsequently, from Maxwell's equations, the near-axis radial electric field and azimuthal magnetic field in the focal plane are determined. Finally, the concept of wave impedance is invoked to obtain the electric field on-axis near focus.