Interaction of a charge with a plane wave in a medium

The problem of simulating the energy state of a relativistic charge in the field of a circularly polarized plane wave and longitudinal magnetic field in a medium with the refractive index $k\neq 1$ is analyzed. The classical description of the interaction reduces to the integration of the Maxwell-Lorentz equation. The presence of a medium leads to qualitative changes in the charge dynamics. An exact analytic representation of the energy of an ensemble of relativistic charges as a function of the average flight coordinate $\bar{z}$ is obtained for $k=1$. Moreover, the asymptotic approximation for $k\neq 1$ is constructed in the first approximation in $\mu=1-1/k^2$ by the Van der Pol method.