Relativistic laser-plasma interactions in the quantum regime

We consider nonlinear interactions between a relativistically strong laser beam and a plasma in the quantum regime. The collective behavior of electrons is modeled by a Klein-Gordon equation, which is nonlinearly coupled with the electromagnetic wave through the Maxwell and Poisson equations. This allows us to study nonlinear interactions between arbitrarily large-amplitude electromagnetic waves and a quantum plasma. We have used our system of nonlinear equations to study theoretically the parametric instabilities involving stimulated Raman scattering and modulational instabilities. A model for quasi-steady-state propagating electromagnetic wave packets is also derived, and which shows possibility of localized solitary structures in a quantum plasma. Numerical simulations demonstrate collapse and acceleration of electrons in the nonlinear stage of the modulational instability, as well as possibility of the wake-field acceleration of electrons to relativistic speeds by short laser pulses at nanometer length scales. Our study is relevant for understanding the localization of intense electromagnetic pulses in a quantum plasma with extremely high electron densities and relatively low temperature.

Figure 1

Dispersion curves ($\Omega$ vs. $K$) for the ES oscillations for $H={10}^{-4}$ (solid curves) and $H=0.007$ (dashed curves), where $H=\hslash {\omega }_{\mathit{pe}}/m_e{c}^2$. For $K{\lambda }_e>1/H$, the particle motion turns from weakly relativistic to ultrarelativistic.

Figure 2

Growth rate (${\Omega }_I/{\omega }_{\mathit{pe}}$ vs. $K_{\left|\right|}$ and $K_{\perp }$) for stimulated Raman scattering in the presence of a large-amplitude CPEM wave, for the amplitudes $a_0=1$, $a_0=5$, and $a_0=20$ (left to right panels) where $a_0=e\left|{\mathrm{Â}}_0\right|/m_{e}c$ for $H={10}^{-4}$ (top panels) and $H=0.007$ (bottom panels).

Figure 3

The growth rate (${\Omega }_I/{\omega }_{\mathit{pe}}$ vs. $K_{\left|\right|}$ and $K_{\perp }$) for the modulational instability in the presence of a large-amplitude CPEM wave, for the amplitudes $a_0=1$ (left panel), $a_0=5$ (middle panel), and $a_0=20$ (right panel) where $a_0=e\left|{\mathrm{Â}}_0\right|/m_{e}c$. $H=0.007$ for all cases.

Figure 4

The spatial profiles of the vector potential, the electron charge density, and the electrostatic potential (top to bottom panels), for standing solitary waves ($v_0=0$) for (a) $H=0.007$ and $\lambda =-0.34$, (b) $H=0.002$ and $\lambda =-0.34$, and (c) for a moving solitary wave with $v_0=0.0059c$ for $H=0.002$ and $\lambda =-0.30$.

Figure 5

The nonlinear stage of a modulational instability, showing localized EM wave-packets associated with depletions in the electron charge density, and positive ES potentials. The electron pseudodistribution function shows that the electrons have been accelerated and form BGK-like modes that travel away from the collapsed wave packets. Parameters are $H=0.007$ and initially a dipole field $a_0=1$ with $k_0=0$.

Figure 6

Attosecond laser pulse propagation into an underdense quantum plasma, at time $t=28.125\hspace{0.5em}{\omega }_{\mathit{pe}}^{-1}$. Top to bottom panels show (a) the vector potential of the EM pulse (the arrows show the propagation directions of the pulses), (b) the electron charge density, (c) the ES potential, and (d) the distribution of electrons in phase space in a 10-logarithmic color scale. Parameters are $H=0.007$, amplitude $a_0=1$ and wavenumber $k_0=20\hspace{0.5em}{\lambda }_e^{-1}$. Localized EM pulses excite large-amplitude oscillatory potential wakes behind them, as they penetrate the plasma slab.

Figure 7

The same as in Fig. 6 at time $t=37.5\hspace{0.5em}{\omega }_{\mathit{pe}}^{-1}$. Groups of electrons are accelerated to ultrarelativistic speeds by the large-amplitude ES wake field.