  The geometric optics approximation traditionally used to study the
propagation of gravitational waves on a curved background, breaks down in the
vicinity of compact and extended astrophysical objects, where wave-like effects
like diffusion and generation of polarization occur. We provide a framework to
study the generation of polarization of a stochastic background of
gravitational waves propagating in an inhomogeneous universe. The framework is
general and can be applied to both cosmological and astrophysical gravitational
wave backgrounds in any frequency range. We derive an order of magnitude
estimate of the amount of polarization generated for cosmological and
astrophysical backgrounds, in the frequency range covered by present and
planned gravitational wave experiments. For an astrophysical background in the
PTA and LISA band, the amount of polarization generated is suppressed by a
factor $10^{-4}$ $(10^{-5})$ with respect to anisotropies. For a cosmological
background we get an additional $10^{-2}$ suppression. We speculate on using
our approach to map the distribution of (unresolvable) structures in the
Universe.
