On production $e^+ e^-$-pairs by a high energy photon in collision with   photons of a laser wave
  We calculate the number of $e^+ e^-$-pairs produced by a Compton high energy photon in turn formed in the process $e + n \gamma_0 -> e + \gamma$ at simultaneous collision with several photons $\gamma_0$ of the same laser beam $\gamma + s \gamma_0 -> e^+ + e^-$. The case when polarization states of initial electron beam and laser photons are helicity is considered. It is shown that taking into account nonlinear effects in the Compton backscattering leads to decreasing the threshold of production of $e^+ e^-$-pairs and increasing their number. The consideration is based on the direct calculation of matrix elements in the formalism of diagonal spin basis. 
Mikhail Galynskii and Sergei Sikach