  A unified description of the free-space cylindrical vector beams is
presented, which is an integral transformation solution to the vector Helmholtz
equation and the transversality condition. The amplitude 2-form of the angular
spectrum involved in this solution can be arbitrarily chosen. When one of the
two elements is zero, we arrive at either transverse-electric or
transverse-magnetic beam mode. In the paraxial condition, this solution not
only includes the known $J_1$ Bessel-Gaussian vector beam and the axisymmetric
Laguerre-Gaussian vector beam that were obtained by solving the paraxial wave
equations, but also predicts two new kinds of vector beam, called the
modified-Bessel-Gaussian vector beam.
