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(Abridged). We aim at bridging the gap between absorption selected andemission selected galaxies at z~3 by probing the faint end of the luminosityfunction of star-forming galaxies at z~3. We have performed narrow-band imagingin three fields with intervening QSO absorbers (a damped Ly$\alpha$ absorberand two Lyman-limit systems) using the VLT. We target Ly-alpha at redshifts2.85, 3.15 and 3.20. We find a consistent surface density of about 10Ly-alpha-emitters per square arcmin per unit redshift in all three fields downto our detection limit of about 3x10^41 ergs s^-1. The luminosity function isconsistent 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 formost Lyman-break galaxy surveys. In none of the three fields do we detect theemission counterparts of the QSO absorbers. In particular we do not detect thecounterpart 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 endof the luminosity function at z~3. There is a very high surface density of thisclass of objects. This is consistent with a very steep slope of the faint endof the luminosity function as has been inferred by other studies. This faintpopulation of galaxies is playing a central role in the early Universe. Thereis evidence that this population is dominating the integrated star-formationactivity, responsible for the bulk of the ionizing photons at z~3 and likelyalso responsible for the bulk of the enrichment of the intergalactic medium.

The Building the Bridge survey for z=3 Ly-alpha emitting galaxies II: Completion of the survey