  Achieving a truly accurate wavelength calibration of high-dispersion echelle
spectrographs is a challenging task but crucially needed for certain science
cases, e.g. to test for a possible variation of the fine-structure constant in
quasar spectra. One of the spectrographs best suited for this mission is
VLT/ESPRESSO. Nevertheless, previous studies have identified significant
discrepancies between the classical wavelength solutions and the one derived
independently from the laser frequency comb. The dominant parts of these
systematics were intra-order distortions, most-likely related to a deviation of
the instrumental line-spread function from the assumed Gaussian shape. Here, we
therefore present a study focused on a detailed modeling of the ESPRESSO
instrumental line-spread function. We demonstrate that it is strongly
asymmetric, non-Gaussian, different for the two slices and fibers, and varies
significantly along the spectral orders. Incorporating the determined
non-parametric model in the wavelength calibration process drastically improves
the wavelength calibration accuracy, reducing the discrepancies between the two
independent wavelength solutions from 50m/s to about 10m/s. The most striking
success is, however, that the different fibers and slices now provide fully
consistent measurements with a scatter of just a couple m/s. This demonstrates
that the instrument-related systematics can be nearly eliminated over most of
the spectral range by properly taking into account the complex shape of the
instrumental line-spread function and paves the way for further optimizations
of the wavelength calibration process.
