Properties of galaxies at the faint end of the H$\alpha$ luminosity
  function at $z\sim0.62$
  Studies measuring the star formation rate density, luminosity function, and
properties of star-forming galaxies are numerous. However, it exists a gap at
$0.5<z<0.8$ in H$\alpha$-based studies. Our main goal is to study the
properties of a sample of faint H$\alpha$ emitters at $z\sim0.62$. We focus on
their contribution to the faint end of the luminosity function and derived star
formation rate density, characterising their morphologies and basic photometric
and spectroscopic properties. We use a narrow-band technique in the
near-infrared, with a filter centred at 1.06 $\mu$m. The data come from
ultra-deep VLT/HAWK-I observations in the GOODS-S field with a total of 31.9 h
in the narrow-band filter. We perform a visual classification of the sample and
study their morphologies from structural parameters available in CANDELS. Our
28 H$\alpha$-selected sample of faint star-forming galaxies reveals a robust
faint-end slope of the luminosity function $\alpha=-1.46_{-0.08}^{+0.16}$. The
derived star formation rate density at $z\sim0.62$ is $\rho_\mathrm{SFR} =
0.036_{-0.008}^{+0.012} M_{\odot}~\mathrm{yr^{-1}~Mpc^{-3}}$. The sample is
mainly composed of disks, but an important contribution of compact galaxies
with S\'ersic indexes $n\sim2$ display the highest specific star formation
rates. The luminosity function at $z\sim0.62$ from our ultra-deep data points
towards a steeper $\alpha$ when an individual extinction correction for each
object is applied. Compact galaxies are low-mass, low-luminosity, and
starburst-dominated objects with a light profile in an intermediate stage from
early to late types.
