  Aims. We study the 2D spectral line profile of HARPS (High Accuracy Radial
Velocity Planet Searcher), measuring its variation with position across the
detector and with changing line intensity. The characterization of the line
profile and its variations are important for achieving the precision of the
wavelength scales of 10^{-10} or 3.0 cm/s necessary to detect Earth-twins in
the habitable zone around solar-like stars. Methods. We used a laser frequency
comb (LFC) with unresolved and unblended lines to probe the instrument line
profile. We injected the LFC light (attenuated by various neutral density
filters) into both the object and the reference fibres of HARPS, and we studied
the variations of the line profiles with the line intensities. We applied
moment analysis to measure the line positions, widths, and skewness as well as
to characterize the line profile distortions induced by the spectrograph and
detectors. Based on this, we established a model to correct for point spread
function distortions by tracking the beam profiles in both fibres. Results. We
demonstrate that the line profile varies with the position on the detector and
as a function of line intensities. This is consistent with a charge transfer
inefficiency (CTI) effect on the HARPS detector. The estimate of the line
position depends critically on the line profile, and therefore a change in the
line amplitude effectively changes the measured position of the lines,
affecting the stability of the wavelength scale of the instrument. We deduce
and apply the correcting functions to re-calibrate and mitigate this effect,
reducing it to a level consistent with photon noise.
