The weak localisation is an effect that occurs when a conductor is taken to a low temperature. It is experimentally seen as the metal in question having an increase of resistivity when it is normally expected for resistivity to decrease at low temperatures and at the critical point, drop to zero. This is caused by the fact that at a classical level the electron can be seen as flying through the metal lattice, scattering off the atoms, being led on a random walk. The resistivity, on a quantum level, is the probability of an electron moving between two fixed points. When in regular conditions this is simply the sum of the probabilities of every path between the points. When cold, the quantum regime takes over and an interference term stops being ignorable as it grows in size. These interference terms can be imagined as an increase in the probability that the electron will loop in a circle rather than making any progress and so resistivity increases.
Anderson localisation is an extension of a similar idea presented by American physicist Philip Anderson. It is where crystals, which are conductive in material, but incredibly structurally disordered, don’t just see a drop in conductivity, they lose it all together. It is actually true for many different waves propagating through chaotic media. The cause of this effect is that, like the weak localisation, the interference terms for the waves have grown strong, in this case so strong as to actually halt the wave altogether. For anyone with a good background in maths (university level) you might want to read this document by a Dirk Hundertmark which goes into a lot more of the background than I can.
Recently work has been done to experimentally investigate how structural disorder affects pulses of light passing through an artificial photonic crystal (a crystal that scatters photons) made from two loops of optical fibre which should demonstrate the Anderson effect in one dimension. Even though the disorder was not great it appears that at least weak Anderson localisation was observed in this system and the exact magnitude of the localisation was tested at various strengths of disorder. The results gathered perfectly matched current theories with the techniques for achieving localisation being able to be applied elsewhere also.
Paper links: Anderson localization in synthetic photonic lattices