  The complete measurement of the quantum state of two correlated photons
requires reconstructing the amplitude and phase of the biphoton wavefunction.
We show how, by means of spatially resolved single photon detection, one can
infer the spatial structure of bi-photons generated by spontaneous parametric
down conversion. In particular, a spatially resolved analysis of the
second-order correlations allows us to isolate the moduli of the pump and
phasematching contributions to the two-photon states. When carrying this
analysis on different propagation planes, the free space propagation of pump
and phasematching is observed. This result allows, in principle, to gain enough
information to reconstruct also the phase of pump and phasematching, and thus
the full biphoton wavefunction. We show this in different examples where the
pump is shaped as a superposition of orbital angular momentum modes or as a
smooth amplitude with a phase structure with no singularities. The
corresponding phase structure is retrieved employing maximum likelihood or
genetic algorithms. These findings have potential applications in fast,
efficient quantum state characterisation that does not require any control over
the source.
