  Earth-like, potentially habitable exoplanets are prime targets in the search
for extraterrestrial life. Information about their atmosphere and surface can
be derived by analyzing light of the parent star reflected by the planet. We
investigate the influence of the surface albedo $A_{\rm s}$, the optical
thickness $b_{\rm cloud}$ and altitude of water clouds, and the mixing ratio
$\eta$ of biosignature O$_2$ on the strength of the O$_2$ A-band (around 760
nm) in flux and polarization spectra of starlight reflected by Earth-like
exoplanets. Our computations for horizontally homogeneous planets show that
small mixing ratios ($\eta$ < 0.4) will yield moderately deep bands in flux and
moderate to small band strengths in polarization, and that clouds will usually
decrease the band depth in flux and the band strength in polarization. However,
cloud influence will be strongly dependent on their properties such as optical
thickness, top altitude, particle phase, coverage fraction, horizontal
distribution. Depending on the surface albedo, and cloud properties, different
O$_2$ mixing ratios $\eta$ can give similar absorption band depths in flux and
band strengths in polarization, in particular if the clouds have moderate to
high optical thicknesses. Measuring both the flux and the polarization is
essential to reduce the degeneracies, although it will not solve them, in
particular not for horizontally inhomogeneous planets. Observations at a wide
range of phase angles and with a high temporal resolution could help to derive
cloud properties and, once those are known, the mixing ratio of O$_2$ or any
other absorbing gas.
