The unimproved experiment seeks to detect axions by shining a laser down the bore of a powerful superconducting magnet. A wall in the middle stops the laser cold, with the theoretical axions continuing through the wall and into the other side of the magnet. There, the magnet reconverts them into photons, or particles of light.
The detection of this light “reappearing” on the other side of the wall is what gives the experiment its iconic name.
Researchers in the U.S. and Europe are in various stages of conducting the experiment. The activity has been stimulated by a recent Italian experiment that claims to have discovered axion-like particles. The hope is to confirm the Legnaro National Laboratories’ results or take them a step further.
Sikivie, UF physics professor David Tanner and Karl van Bibber, a physicist at the Lawrence Livermore National Laboratory, propose a redesign of the “shining light through walls” experiment to make it, in their words, “vastly more sensitive.”
I'll try to update this later with the exact citation of the paper. So far, it doesn't show up yet on the PRL website. The only reference to anything similar is a paper by almost the same group of people published in 2005. There is, however, an ArXiv preprint on this, and presumably, this is the paper.
 Phys. Rev. Lett. 95, 091304 (2005).