Azimuthally polarized hollow beams in free space: Exact vector solutions to Maxwell's equations

We derive a set of azimuthally polarized hollow beam solutions in free space. These solutions are exact vector solutions to Maxwell's and Poynting's equations with the electric- and magnetic-field components expressed as linear combinations of the partial derivatives of a sinc function. They depict beams the intensity of which vanishes everywhere on the beam axis, forming dark hollow channels. There is one adjustable dimensionless parameter $Q$ in the formulation which governs the degree of focusing of a beam. In the weak focusing limit $(Q\ensuremath{\gg}1)$, a beam shows a doughnut-shaped intensity profile while a multiple-ring intensity distribution is found for the strong focusing case with $Q\ensuremath{\le}1$.