Non-linear scattering of light in the limit of ultra-strong fields

The non-linear interaction of two electromagnetic waves in a vacuum is treated by the introduction of the Lagrangian for a strong electromagnetic field. The quantum mechanical effect of the scattering phenomenon between plane electromagnetic waves can be simulated within the framework of classical theory under the assumption of an effective non-linear interaction between the electromagnetic waves. The promise of continuous, intense, and monochromatic polarized radiation from electron synchrotrons and storage rings motivates the study. The scattered fields are obtained and a ratio of the power in the scattered fields to that of the incident fields is estimated for a simple case.