We derive focused laser pulse solutions to the electromagnetic wave equation in vacuum. After reproducing beam and pulse expressions for the well-known paraxial Gaussian and axicon cases, we apply the method to analyse a laser beam with Lorentzian transverse momentum distribution. Whilst a paraxial approach has some success close to the focal axis and within a Rayleigh range of the focal spot, we find that it incorrectly predicts the transverse fall-off typical of a Lorentzian. Our vector-potential approach is particularly relevant to calculation of quantum electrodynamical processes in weak laser pulse backgrounds.