Nonlinear evolution of the plasma beat wave: Compressing the laser beat notes via electromagnetic cascading

The near-resonant beat wave 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 comoving index grating that induces a laser phase modulation at the difference frequency. As a result, the cascade of sidebands red and blue shifted by integer multiples of the beat frequency is generated in the laser spectrum. 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. When the beat frequency is lower than the electron plasma frequency, the redshifted spectral components are advanced in time with respect to the blueshifted ones near the center of each laser beat note. The group velocity dispersion of plasma compresses so chirped beat notes 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 beat wave and 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 bistability regime of the EPW excitation [G. Shvets, Phys. Rev. Lett. 93, 195004 (2004)] can be achieved with the initially subthreshold beat wave pulse.

Figure 1

Schematic of the two-stage cascade compressor. The frequency modulation (FM) occurs in a rarefied plasma. Denser plasma is used for the compression of beat notes.

Figure 2

Normalized distance between the points of the first maximum and half maximum of $J_M\left(z\right)$ (solid line) and the scaling function $M^{-2∕3}$ (dashed line).

Figure 3

The two-stage cascade compression. The physical quantities are shown at the entrance ($z=0$, light gray) and exit of the modulator ($z=z_8$, medium gray), and after the compressor ($z=z_8+z_C$, black). (a) The laser pulse intensity (the time window contains about 100 beat notes). (b) The normalized amplitude of the near-resonant EPW, $\delta n_{-1}∕n_0=N_e∕d$. (c) The beat note intensity near the laser pulse center; one beat period near $\xi =0$ is shown. (d) The laser spectra near $\xi =0$. The nonlinearities and GVD in both plasmas are included.

Figure 4

The single-stage compressor with concurrent EMC and compression. In plots (a)–(c), the physical quantities are shown at the entrance ($z=0$, gray) and exit of the plasma ($z=z_8$, black). (a) The laser pulse intensity (the time window contains about 500 beat notes). (b) The normalized amplitude of the near-resonant EPW, $\delta n_{-1}∕n_0=N_e∕d$. (c) The beat note intensity near the laser pulse center, $\xi =0$. (d) The laser spectra near $\xi =0$ with (black) and without (gray) GVD and all nonlinear frequency shifts.

Figure 5

The EMC of the two-color short-pulse $\left(90\mathrm{fs}\right)$ laser [28] in a dense $(1.35\times {10}^{19}{\mathrm{cm}}^{-3})$ plasma. Physical quantities are shown at the plasma entrance ($z=0$, gray lines) and exit ($z=1.8\mathrm{mm}$, black lines).

Figure 6

Electron density perturbation (a) and the temporal profile of laser intensity (b) for the parameters the same as of Fig. 3 except the laser duration reduced by a factor of 2.5, ${\omega }_p{\tau }_L\approx 100$. Gray color corresponds to $z=0$, black—to $z=z_8$.

Figure 7

Electron density perturbation (a) and the temporal profile of laser intensity (b) for the parameters the same as of Fig. 4 except the laser duration reduced to ${\omega }_p{\tau }_L\approx 440$, and $\mid \delta \omega {\tau }_L\mid =10$. Gray color corresponds to $z=0$, black—to $z=z_8$.

Figure 8

Relativistic bistability of the EPW. Magnitude of the near-resonant electron density perturbation [plot (a)] and the laser intensity [plot (b)] are shown at $z=0$ (light gray), $z=0.52\mathrm{mm}$ (medium gray), and $z=0.54\mathrm{mm}$ (black). Inset in plot (b): one beat note selected near $\xi =0$. Plot (c): the laser spectrum at $z=0.54\mathrm{mm}$. The RB threshold is crossed at $z\approx 0.53\mathrm{mm}$. Crossing the threshold increases the EPW amplitude by a factor of 2.3.