  The localization of waves in non-periodic media is a universal phenomenon,
occurring in a variety of different quantum and classical systems, including
condensed-matter, Bose-Einstein condensates in optical lattices, quantum
chaotic systems, sound waves and light. A localization phase transition is
expected to occur in three dimensional disordered systems as the strength of
disorder crosses a critical value. Recently, a crossover from an extended to a
localized phase has been observed in low-dimensional photonic lattices and
Bose-Einstein condensates. Other experiments studied the critical behaviour
near the transition in three dimensions via transmission measurements. However,
no direct observation of a localization transition for light has been reported.
In 1979 Aubry and Andre predicted that for a certain class of quasi-periodic
potentials, a localization phase transition can occur already in one-dimension.
This strongly correlated potential is markedly different from the disordered
case were the potential is uncorrelated. Here we report an experiment that
realizes the Aubry-Andre model in quasi-periodic photonic lattices. We observe
the signature of a localization phase transition by directly measuring the
expansion rates of initially narrow wave packets propagating in the lattice.
Below the transition point, all the modes of the system are extended and
therefore an initially narrow wave-packet eventually spreads across the entire
lattice. Above the critical point, all modes are localized and expansion is
suppressed. In addition, we study the effect of weak nonlinear interactions on
light propagation below and above the transition.
