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The interaction of an intense coherent photon beam with free electrons is discussed. The photon beam is treated as a classical external electromagnetic field. The discussion is exact within the approximation of neglecting radiative corrections and the restriction to the case of a plane-wave field or arbitrary spectral composition and polarization properties. The scattering of a single photon out of a monochromatic beam by an isolated free electron is considered in detail. The cross sections corresponding to the scattering of the various harmonics of the incident beam are evaluated. These cross sections display a complicated dependence upon the intensity of the incident beam, at least for very intense beams. It is found that a mass change induced in the electron by the external field shifts the wavelength of the scattered photons by an amount depending on the intensity of the incident beam. Other processes involving free electrons in the final state are also considered briefly, and a discussion of the magnitude of the effects depending upon the intensity is given. Two Appendices are concerned with the electron Green's function and the vacuum-vacuum transformation function in the presence of a plane-wave field. In the course of the discussion of the latter, the problem of the correct definition of the vacuum current is encountered, and it is shown that a very careful procedure is necessary to obtain a covariant result.
Brown et al. (Mon,) studied this question.