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

The O2 A-band in fluxes and polarization of starlight reflected by Earth-like exoplanets