Electromagnetically Induced Transparency in the Strongly Relativistic Regime

Stable transport of laser beams in highly overdense plasmas is of significance in the fast ignition of inertial confinement fusion, relativistic electron generation, and powerful electromagnetic emission, but hard to realize. Early in 1996, Harris proposed an electromagnetically induced transparency (EIT) mechanism, analogous to the concept in atomic physics, to transport a low-frequency (LF) laser in overdense plasmas aided by a high-frequency pump laser. However, subsequent investigations show that EIT cannot occur in real plasmas with boundaries. Here, our particle-in-cell simulations show that EIT can occur in the strongly relativistic regime and result in stable propagation of a LF laser in bounded plasmas with tens of its critical density. A relativistic three-wave coupling model is developed, and the criteria and frequency passband for EIT occurrence are presented. The passband is sufficiently wide in the strongly relativistic regime, allowing EIT to work sustainably. Nevertheless, it is narrowed to nearly an isolated point in the weakly relativistic regime, which can explain the quenching of EIT in bounded plasmas found in previous investigations.

Figure 1

[(a),(b)] Spectra of the laser fields collected in the right vacuum and [(c),(d)] laser field waveforms filtered with the frequency range $\omega /{\omega }_0\in (0.35,0.42)$, where different curves correspond to the mixing-wave, pump-wave, and LF-wave incidence cases. In plots (a) and (b), the black-dashed and purple-dotted lines represent the frequencies of ${\omega }_p^{\prime }={\omega }_p/\sqrt{{\gamma }_0}$ and ${\omega }_0-{\omega }_p^{\prime }$. [(e),(f)] Laser field waveforms filtered with $\omega /{\omega }_0\in (0.35,0.42)$, where the mixing waves are incident with the $p$-polarized LF wave and the pump wave of $p$ polarization (blue-solid line) and $s$ polarization (red-dashed line), respectively. The upper and lower rows correspond to high-density and low-density cases.

Figure 2

Dispersion relationship of Stokes-dominant branches with $\ln (|a_2^{*}/a_1|)<0$ calculated from Eq. (3) in (a) the low-density case with ${\omega }_p={\omega }_0/1.5$ and (b) the high-density case with ${\omega }_p={\omega }_0/0.6$, where the black-dashed and purple-dotted lines represent the frequencies of ${\omega }_p^{\prime }={\omega }_p/\sqrt{{\gamma }_0}$ and ${\omega }_0-{\omega }_p^{\prime }$. $\mathrm{Re}(k_1)$ (red curve), located in the orange area, means forward propagation. The bright yellow area in (b) highlights the passband of ${\omega }_1$ within $[0.377,0.446]{\omega }_0$. (c) Passband versus $a_0$, where the green and orange bars correspond to the analytical and PIC results ($a_0\ge 2$). The initial plasma density keeps $n_{\mathrm{ini}}/n_{\mathrm{cr}0}^{\prime }=0.389$, corresponding to the high-density case in (b). Passband width at $a_0=0.1$ is magnified by 10 times. (d) Passband obtained by PIC results versus the density.

Figure 3

Amplitudes of the Stokes and anti-Stokes waves (left axis) as well as the pump wave (right axis), where $a_0=10$ and $a_1=1$ are taken in (a)–(c), and $a_0=0.3$ and $a_1=0.03$ are taken in (d). The plasma density $n_{\mathrm{ini}}$ is adopted as $2.78n_{\mathrm{cr}0}$ (${\omega }_0=0.6{\omega }_{p,\mathrm{ini}}$) in (a), $3.31n_{\mathrm{cr}0}$ (${\omega }_0=0.55{\omega }_{p,\mathrm{ini}}$) in (b), and $0.44n_{\mathrm{cr}0}$ (${\omega }_0=1.5{\omega }_{p,\mathrm{ini}}$) in (c) and (d). We take ${\omega }_1=0.4{\omega }_0$ in (a)–(c) and ${\omega }_1={\omega }_0/3$ in (d). Note that the plasma is located within $10{\lambda }_0<x<30{\lambda }_0$.