Nonlinear propagation of light in Dirac matter

The nonlinear interaction between intense laser light and a quantum plasma is modeled by a collective Dirac equation coupled with the Maxwell equations. The model is used to study the nonlinear propagation of relativistically intense laser light in a quantum plasma including the electron spin-1/2 effect. The relativistic effects due to the high-intensity laser light lead, in general, to a downshift of the laser frequency, similar to a classical plasma where the relativistic mass increase leads to self-induced transparency of laser light and other associated effects. The electron spin-1/2 effects lead to a frequency upshift or downshift of the electromagnetic (EM) wave, depending on the spin state of the plasma and the polarization of the EM wave. For laboratory solid density plasmas, the spin-1/2 effects on the propagation of light are small, but they may be significant in superdense plasma in the core of white dwarf stars. We also discuss extensions of the model to include kinetic effects of a distribution of the electrons on the nonlinear propagation of EM waves in a quantum plasma.

Figure 1

Dispersion curves for the linear and nonlinear propagation of light in Dirac matter for different values of $H=\hslash {\omega }_{pe}/m_e{c}^2$. For the linear cases in (a)–(c), the solid and dotted curves correspond to solutions using the upper sign in Eq. (37), and dashed curves correspond to solutions with the lower sign in Eq. (37). (a) shows a high-frequency branch, the EM branch shifted $\pm H/4$ compared to the plasma frequency at $k_0=0$, and a low-frequency branch. For $H>4/3\sqrt{3}\approx 0.77$ the pair branch merges with the upshifted EM branch, and there is an instability for waves with small wave numbers; the growth rate is indicated with the dotted curve for $H=1$. (d)–(f) show the dispersion curves for finite amplitude ($a_0=1$) EM waves.

Figure 2

The cutoff frequency ${\omega }_0$ at $K=k_0=0$ as a function of $H$ for different values of $a_0=e{A}_0/m_{e}c$. The cutoff frequency is upshifted for the spin up state (solid lines) and downshifted for spin down (dashed lines). In the classical limit $H\to 0$, we have ${\omega }_0\to {\omega }_{pe}/\sqrt{{\gamma }_0}$ for both spin states, where ${\gamma }_0=\sqrt{1+a_0^2}$.