An analytical solution of the field in a fibre up-taper with a parabolic index profile is obtained by solving the scalar wave equation. The field is expressed in generalised Laguerre-Gaussian polynomials. With this description, the evolution of the spot size and the wavefront curvature are easily seen. The presence of spot size oscillations along a taper is predicted in the case of a mismatch between the exciting beam at the taper entrance and the fundamental mode.