Nonlinear evolution of the plasma beatwave: Compressing the laser   beatnotes via electromagnetic cascading
  The near-resonant beatwave excitation of an electron plasma wave (EPW) can be employed for generating the trains of few-femtosecond electromagnetic (EM) pulses in rarefied plasmas. The EPW produces a co-moving index grating that induces a laser phase modulation at the difference frequency. The bandwidth of the phase-modulated laser is proportional to the product of the plasma length, laser wavelength, and amplitude of the electron density perturbation. The laser spectrum is composed of a cascade of red and blue sidebands shifted by integer multiples of the beat frequency. When the beat frequency is lower than the electron plasma frequency, the red-shifted spectral components are advanced in time with respect to the blue-shifted ones near the center of each laser beatnote. The group velocity dispersion of plasma compresses so chirped beatnotes to a few-laser-cycle duration thus creating a train of sharp EM spikes with the beat periodicity. Depending on the plasma and laser parameters, chirping and compression can be implemented either concurrently in the same, or sequentially in different plasmas. Evolution of the laser beatwave end electron density perturbations is described in time and one spatial dimension in a weakly relativistic approximation. Using the compression effect, we demonstrate that the relativistic bi-stability regime of the EPW excitation [G. Shvets, Phys. Rev. Lett. 93, 195004 (2004)] can be achieved with the initially sub-threshold beatwave pulse. 
Serguei Kalmykov and Gennady Shvets