  The stochastic gravitational wave background (SGWB) is a rich resource of
cosmological information, encoded both in its source statistics and
anisotropies induced by propagation effects. We provide a theoretical
description of it, without employing theoretical tools which rely on the
geometric optics limit. Our formalism is based on the so-called {\it classical
matter approximation} and it is able to capture wave-optics effects, such as
interference and diffraction. We show that the interaction between the
gravitational waves and the cosmic structures along the line-of-sight produce
observable scalar and vector polarization modes, on top of modulating the
tensorial ones. We build the two point correlation function describing the
statistics of the SGWB, and introduce the intensity and gravitational Stokes
parameters for all its components. In the case of an unpolarized, Gaussian and
statistically homogeneous SGWB, we show that the interaction with matter
modulates its intensity and does not generate a net difference between left-
and right- helicity tensor and vector modes, as expected. We demonstrate that,
in order to produce $Q_T$- and $U_T$- tensor polarization modes the background
must posses an hexadecapole anisotropy, while a quandrupole anisotropy can
source the vector $Q_V$- and $U_V$- Stokes parameters.
