Polarization of recoil photon in non-linear Compton process

The polarization of recoil photon ($\gamma'$) in the non-linear Compton process $e + \vec L \to \vec \gamma' +e'$ in the interaction of a relativistic electron with a linearly polarized laser beam ($\vec L$) is studied within the Furry picture in the lowest-order, tree-level S matrix element. In particular, we consider the asymmetry of differential cross sections ${\cal A}$ for two independent axes describing the Compton process equal to the intrinsic spin variable ${\xi}^f_3$, that determines the polarization properties of $\gamma'$. The sign and absolute value of the asymmetry determine the direction and degree of $\gamma'$ polarization. We have analyzed the process in a wide range of laser intensity that covers existing and future experiments. Our results provide additional knowledge for studying nonlinear multi-photon effects in quantum electrodynamics and can be used in planning experiments at envisaged laser facilities.