The Building the Bridge survey for z=3 Ly-alpha emitting galaxies II:
  Completion of the survey
  (Abridged). We aim at bridging the gap between absorption selected and
emission selected galaxies at z~3 by probing the faint end of the luminosity
function of star-forming galaxies at z~3. We have performed narrow-band imaging
in three fields with intervening QSO absorbers (a damped Ly$\alpha$ absorber
and two Lyman-limit systems) using the VLT. We target Ly-alpha at redshifts
2.85, 3.15 and 3.20. We find a consistent surface density of about 10
Ly-alpha-emitters per square arcmin per unit redshift in all three fields down
to our detection limit of about 3x10^41 ergs s^-1. The luminosity function is
consistent with what has been found by other surveys at similar redshifts.
About 85% of the sources are fainter than the canonical limit of R=25.5 for
most Lyman-break galaxy surveys. In none of the three fields do we detect the
emission counterparts of the QSO absorbers. In particular we do not detect the
counterpart of the z=2.85 damped Ly-alpha absorber towards Q2138-4427.
Narrow-band surveys for Ly-alpha emitters are excellent to probe the faint end
of the luminosity function at z~3. There is a very high surface density of this
class of objects. This is consistent with a very steep slope of the faint end
of the luminosity function as has been inferred by other studies. This faint
population of galaxies is playing a central role in the early Universe. There
is evidence that this population is dominating the integrated star-formation
activity, responsible for the bulk of the ionizing photons at z~3 and likely
also responsible for the bulk of the enrichment of the intergalactic medium.
