  We study the non-linear Breit-Wheeler process $\vec \gamma' + \vec L \to e^+
+ e^-$ in the interaction of linearly polarized probe photons ($\vec \gamma'$)
with a linearly polarized laser beam ($\vec L$). In particular, we consider the
asymmetry of the total cross section and the azimuthal electron distributions
when the polarizations of the photon and laser beams in the initial state are
mutually perpendicular or parallel. Considering intense laser beams and the
strong field asymptotic we explore essentially the multi-photon dynamics. The
asymmetry exhibits some non-monotonic behavior depending on initial kinematic
conditions; it depends sensitively on the laser pulse duration. Our results
provide additional knowledge for studying non-linear multi-photon effects in
quantum electrodynamics and may be used in planning experiments in upcoming
laser facilities.
