Amplification of external EM-wave by nonlinear wake waves in cold plasma

The interaction of external monochromatic, linearly polarized, plane electromagnetic (EM) wave with the nonlinear one-dimensional wake-wave, generated by relativistic electron bunch, moving in cold plasma, is considered. At definite conditions on parameters of plasma and electron bunch, nonlinear plasma electron density wake wave has a pronounced spikes, where density value is nearing the wave breaking limit and plasma electron velocities, being directed along the bunch velocity reach, their maximum. Presented calculations exhibit, that external EM-wave, propagating through plasma at such a conditions normal to the bunch velocity direction and with polarization along the bunch velocity can be amplified inside the spikes. The EM-wave, Thomson scattered on the spikes, is also amplified. Amplification factors are obtained for the both cases at different conditions for system plasma-electron bunch-external EM-wave parameters. It is shown that amplification factor can be larger, especially at the resonance condition, when frequency of external field is nearing to plasma frequency at spikes. Presented results are obtained using perturbative approach, when dimensionless external EM-wave amplitude is used as a small parameter; the known exact nonlinear one-dimensional solution, obtained previously for cold plasma-relativistic electron bunch system, is taken as a zero order approximation. Considered amplification process can serve as a physical ground for research and development of powerful klystron type amplifiers for future linear colliders