Resonant scattering of a photon by an electron in the field of elliptic polarized electromagnetic wave

In this report we theoretically investigate the resonant Compton effect in the field of the elliptic polarized electromagnetic wave. Resonances arise when the Green function of an intermediate electron reaches the mass shell in the external field. We examine photon-electron resonance scattering under low intensities of the wave. Thus we limit the resonance approximation to the processes with two photons incident wave as higher-order multiphotons rate are much smaller. The calculation of the differential cross-section for the second order processes in the external fields is complicated by computational difficulties and a cumbersome form of the result. We propose a new method of analytical calculation of the differential cross-section this process under resonant conditions. Using this method, we have derived the differential cross-section of the Compton effect in the field of the elliptic polarized wave. It is shown that resonant cross-section may be several orders of the magnitude greater than the cross-section of the corresponding process in the absence of the external field.