  The transverse spatial modes of light offer a large state-space with
interesting physical properties. For exploiting it in future long-distance
experiments, spatial modes will have to be transmitted over turbulent
free-space links. Numerous recent lab-scale experiments have found significant
degradation in the mode quality after transmission through simulated turbulence
and consecutive coherent detection. Here we experimentally analyze the
transmission of one prominent class of spatial modes, the orbital-angular
momentum (OAM) modes, through 3 km of strong turbulence over the city of
Vienna. Instead of performing a coherent phase-dependent measurement, we employ
an incoherent detection scheme which relies on the unambiguous intensity
patterns of the different spatial modes. We use a pattern recognition algorithm
(an artificial neural network) to identify the characteristic mode pattern
displayed on a screen at the receiver. We were able to distinguish between 16
different OAM mode superpositions with only ~1.7% error, and use them to encode
and transmit small grey-scale images. Moreover, we found that the relative
phase of the superposition modes is not affected by the atmosphere,
establishing the feasibility for performing long-distance quantum experiments
with the OAM of photons. Our detection method works for other classes of
spatial modes with unambiguous intensity patterns as well, and can further be
improved by modern techniques of pattern recognition.
