An optical beam with an e ilϕ phase structure carries an orbital angular momentum of lℏ per photon. For integer l values, the phase fronts of such beams form perfect helices with a single screw-phase dislocation, or vortex, on the beam axis. For non-integer l values, Berry (2004 J. Opt. A: Pure Appl. Opt. 6 259) predicts a complex-phase structure comprising many vortices at differing positions within the beam cross-section. Using a spatial light modulator we produce e ilϕ beams with varying l. We examine the phase structure of such beams after propagation through an interference-based phase-measurement technique. As predicted, we observe that for half-integer l values, a line of alternating charge vortices is formed near the radial dislocation.