Complete transport of intensity equation for phase retrieval of optical vortex beams

We derive the complete form of the transport of intensity equation (TIE) for an arbitrary wave field, in which additional terms describing the effect of optical vortices are included. The Poynting vector of optical vortices goes in a spiral around the vortex core through space. Consequently, the intensity variation is not observable in the manner of wave propagation, thereby precluding the possibility of phase retrieval via near-field intensity measurements. This difficulty is elegantly overcome by introducing a hard-edged aperture, as a common procedure to produce the required Neumann boundary signal, to break the azimuthal symmetry of the propagation, and visualize the rotation of the beam’s local Poynting vector around the aperture boundary. The basic principle, numerical solution, and practical implementation of how to use this new version of TIE for phase retrieval of arbitrary wave fields including optical vortices are developed. Based on the new approach, we present the first experimental demonstration of non-interferometric phase retrieval of Laguerre–Gaussian vortex beams of different orders using only two near-field intensity measurements. Our results consummate the basic theoretical framework of TIE and represent an important step towards phase retrieval of arbitrary wave fields carrying phase singularities.