  We compute the polarization of the Lyman alpha line photons emerging from an
anisotropically expanding and optically thick medium, which is expected to
operate in many Lyman alpha emitting objects including the primeval galaxy
DLA~2233+131 and Lyman break galaxies. In the case of a highly optically thick
medium, the escape of resonance line photons is achieved by a large number of
resonant local scatterings followed by a small number of scatterings in the
damping wing. We show that some polarization can develop because the wing
scatterings are coupled with strong spatial diffusion which depends on the
scattering geometry and kinematics. The case of a slab with a finite scattering
optical depth and expansion velocity of ~ 100 kms^{-1} is investigated and it
is found that Lyman alpha photons are emergent with the linear degree of
polarization up to 10% when the typical scattering optical depth tau>=10^5. We
subsequently investigate the polarization of Lyman alpha photons emerging from
a spherical shell obscured partially by an opaque component and we obtain ~ 5 %
of polarization. It is proposed that a positive detection of polarized Lyman
alpha with with P-Cygni type profile from cosmological objects can be a strong
test of the expanding shell structure obscured by a disk-like component.
