  We propose and demonstrate numerically a measurement scheme for complete
reconstruction of the 2D quantum wave function of a Bose-Einstein condensate,
amplitude and phase, from a time-of-flight measurement. We identify a
fundamental ambiguity present in the measurement of phase structures of
high-symmetry excitations (e.g., vortices) and show how to overcome it by
allowing for different expansion durations in different directions. We
demonstrate this approach with the reconstruction of matter-wave vortices and
arrays of vortices.
