Variation in polarization of high-energy $γ$-quanta traversing a bunch of polarized laser photons

The elastic light-light scattering below the threshold of the $e^+e^-$ pair production leads to a variation in polarization of hard $\gamma$-quanta traversing without loss a region where the laser light is focused. Equations are obtained which determine the variation of Stokes parameters of $\gamma$-quanta in this case, and their solutions are given. It is pointed out that this effect can be observed in the experiment E-144 at SLAC. It should be taken into account (and, perhaps, it can be used) in experiments at future $\gamma \gamma$ colliders.