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A unified description of the free-space cylindrical vector beams ispresented, which is an integral transformation solution to the vector Helmholtzequation and the transversality condition. The amplitude 2-form of the angularspectrum involved in this solution can be arbitrarily chosen. When one of thetwo elements is zero, we arrive at either transverse-electric ortransverse-magnetic beam mode. In the paraxial condition, this solution notonly includes the known $J_1$ Bessel-Gaussian vector beam and the axisymmetricLaguerre-Gaussian vector beam that were obtained by solving the paraxial waveequations, but also predicts two new kinds of vector beam, called themodified-Bessel-Gaussian vector beam.

Integral transformation solution of free-space cylindrical vector beams and prediction of modified-Bessel-Gaussian vector beams