Electromagnetically Induced Transparency in a Plasma beyond the Ideal-Plasma Model: Damping, Finite Temperature, Geometry, Instabilities, and Experimental Feasibility of the Sub-Cutoff Passband

S. E. Harris [Phys. Rev. Lett. 77, 5357 (1996)] showed that in an ideal (cold, collisionless, homogeneous, unmagnetized) plasma, two transverse electromagnetic waves whose frequency difference is near, but not equal to, the electron plasma frequency resonantly drive a longitudinal plasma oscillation whose nonlinear contribution to the current density opposes the linear plasma current of either wave alone, opening a transparency passband at frequencies below the plasma cutoff. The analysis was explicitly restricted to infinite plane waves, with damping, finite temperature, inhomogeneity, competing plasma nonlinearities and instabilities, and finite beam and pulse effects neglected. The open problem is the quantitative resolution of these idealizations: whether, and under what parameter conditions, the predicted passband (its dispersion relations, pole and critical detuning, bandwidth, and pulse-energy thresholds) survives in realistic plasmas, and whether the effect is experimentally demonstrable. Later work resolved parts of the question—relativistic corrections and particle-in-cell confirmation in an infinite plasma, particle simulations of bounded slabs with a cautionary result, intense-wave relativistic-fluid theory, and finite-temperature theory plus experiments for magnetized (electron-cyclotron) analogs—but no treatment of the unmagnetized two-wave scheme with collisions, finite temperature, inhomogeneity, or transverse geometry exists, and the original scheme has no experimental demonstration.

Background

Electromagnetically induced transparency (EIT) is a quantum-interference effect in atoms in which two coherent fields drive a two-photon resonance that cancels the medium's resonant absorption for a weak probe. Harris's 1996 proposal transplanted this idea to a classical plasma: in a cold, collisionless, homogeneous, unmagnetized plasma of electron density $n_e$ , the electron plasma frequency $\omega_p=\sqrt{n_e e^2/(\epsilon_0 m)}$ defines a cutoff—an electromagnetic wave with frequency below $\omega_p$ is evanescent (the plasma is overdense for that wave and reflects it). Harris showed that two co-propagating transverse waves, a strong wave at $\omega_a$ and a weak Stokes wave at $\omega_s$ , with frequency difference $\omega_a-\omega_s$ detuned slightly from $\omega_p$ , beat at that difference frequency and resonantly drive a longitudinal charge-density (Langmuir) oscillation with slowly varying envelope $\rho$ . The nonlinear cross-term in the transverse current density, $-q\rho V/2$ , is phased to oppose the linear plasma current at the Stokes frequency, so the effective transverse dielectric response is modified. Eliminating the driven oscillation in the steady state yields a modified dispersion relation, characterized by a pump-induced pole frequency $\omega_{\text{pole}}=\frac{1}{2}(\mathcal{E}_a/mc^2)\omega_p$ and a critical detuning $\omega_{\text{crit}}$ , where $\mathcal{E}_a=q^2|E_a|^2/(4m\omega_a^2)$ is the ponderomotive (quiver) energy of the pump wave. For a representative parameter set with $\mathcal{E}_a/(mc^2)=0.02$ , $\omega_p=1$ , $\omega_s=0.75$ , $\omega_a=1.75$ , the predicted passband width is about $5.7\%$ of $\omega_p$ between zero detuning and $\omega_{\text{crit}}$ , where the Stokes wave would otherwise be evanescent with imaginary propagation constant. The same analysis gives lower bounds on the pump pulse energy per unit area: roughly $\frac{4\omega_s^2}{\omega_p^3}(n_e c)(mc^2)$ for the pump-induced shift $|\omega_{\text{pole}}|$ to exceed the plasma-resonance linewidth, and $\gtrsim n_e L\,mc^2$ to avoid distortion by the steep two-photon dispersion (large $\partial n/\partial\omega$ ) over an interaction length $L$ . Harris compared this scaling—energy of order $mc^2$ per electron in the laser path, versus $\hbar\omega$ per resonant atom in atomic EIT—and emphasized that the entire derivation assumed a cold, collisionless, homogeneous plasma, infinite transverse plane waves, and neglect of relativistic and ponderomotive self-focusing, stimulated Raman and Brillouin scattering (parametric instabilities in which the light scatters off the electron plasma wave and the ion-acoustic wave, respectively), and other competing nonlinearities, leaving these effects open.

The proposal seeded a distinct literature on plasma analogs of EIT. Matsko and Rostovtsev [Phys. Rev. E 58, 7846 (1998)] found a regime of amplification of sub-cutoff electromagnetic waves in overdense homogeneous plasmas, with an application to free-electron lasers. Gordon, Mori and Joshi [Phys. Plasmas 7, 3145 (2000)] re-examined the theory in two parts: Part I derived a dispersion relation including relativistic mass effects in an infinite overdense plasma and confirmed the branches with particle-in-cell (PIC) simulations; Part II examined a bounded plasma slab, where PIC shows that short pulses are reflected even in the presence of an EIT passband, and that a two-frequency beatwave can cascade into a Stokes satellite below cutoff, producing an apparent EIT signature for certain parameters. Ersfeld and Jaroszynski [Laser Part. Beams 19 (2001); J. Mod. Opt. 49 (2002)] derived a relativistic-fluid dispersion relation for two intense waves (either or both below cutoff) including polarization and nonlinearities in both amplitudes, establishing conditions for induced transparency beyond the weak-probe limit. A parallel line developed the magnetized analog: Kryachko, Litvak and Tokman [JETP 95 (2002); Nucl. Fusion 44 (2004)] analyzed EIT for electron-cyclotron waves in finite-temperature magnetoactive plasma, showing that thermal motion radically changes the dispersion curves relative to the cold-plasma case, and later introduced 'parametrically induced transparency' in inhomogeneous media for extracting radiation from overdense tokamak plasmas; Hur, Wurtele and Shvets [Phys. Plasmas 10 (2003)] gave theory and 1D PIC simulations of electromagnetically and magnetically induced transparency in magnetized plasma. Experimental realizations exist only for magnetized variants: Kawamori's whistler-wave and electron-cyclotron EIT theory (whistler waves are right-hand circularly polarized waves below the electron cyclotron frequency in a magnetized plasma), linear-susceptibility and dark-state-polariton analyses (a dark-state polariton is a coupled light-medium excitation with strongly reduced group velocity formed when a pump renders the medium transparent), and experimental and numerical studies [Phys. Plasmas 17 (2010); Plasma Phys. Control. Fusion 53, 085015 (2011); Phys. Rev. Lett. 108, 075003 (2012)], and the first observation of magnetically induced transmission in a gaseous magnetized plasma by Gad, Leopold, Fisher, Fredkin and Ron [Phys. Rev. Lett. 108, 155003 (2012)], which used a transverse-periodic static magnetic field rather than Harris's two-frequency pump. All of these magnetized-variant results couple the waves to an electron cyclotron resonance that is absent in Harris's unmagnetized scheme. More recently, Qu and Fisch [Phys. Plasmas 24 (2017)] proposed a 'plasma shutter' for laser pulse sharpening that uses the EIT-like transparency window below cutoff, demonstrated by 1D simulations with broad pulses in high-density plasma, and Javan and Akbari [Commun. Theor. Phys. 61 (2014)] analyzed two intense circularly-polarized lasers in a cold plasma including a beat-wave second-harmonic effect. This body of work establishes the physics of the ideal model and its extensions, but leaves the robustness of the original unmagnetized scheme to realistic plasma effects largely untreated.

Problem Statement

For the unmagnetized, cold, collisionless, homogeneous, infinite-plane-wave plasma in which Harris predicted the electromagnetically-induced-transparency-like passband below the plasma cutoff, determine quantitatively whether and under what parameter conditions the effect survives the removal of the idealizations of the original analysis. Specifically: (1) compute how finite electron temperature, collisions, and density inhomogeneity—which give the plasma resonance a finite linewidth (including Landau damping, the collisionless damping of a plasma wave by electrons moving in resonance with it)—affect the pump-induced pole $\omega_{\text{pole}}$ (the pump-induced shift of the driven plasma resonance, of order $\frac{1}{2}(\mathcal{E}_a/mc^2)\omega_p$ in the ideal model, where $\mathcal{E}_a$ is the pump quiver energy and $\omega_p$ the electron plasma frequency) and can reduce or wash out the passband; (2) determine the role of the omitted plasma nonlinearities and instabilities (relativistic self-focusing, resonant ponderomotive self-focusing, stimulated Raman- and Brillouin-like scattering, i.e. parametric scattering of the light off the electron plasma wave and the ion-acoustic wave) in limiting the steady state or opening competing scattering channels, including the criteria that distinguish a genuine EIT passband from beatwave-cascade artifacts in bounded plasmas (cascading of the two-frequency beat into successively lower-frequency Stokes satellites, which simulations show can mimic a sub-cutoff passband); (3) determine the effects of finite transverse beam size and finite pulse shape (departure from infinite plane waves) on drive efficiency, phase matching, transverse stability, and pulse integrity against the steep two-photon dispersion (the large variation of the refractive index with frequency near the driven two-photon resonance, which disperses a pulse); (4) establish the experimental feasibility of the scheme given the derived pulse-energy-per-area requirements (of order $mc^2$ per electron in the laser path, and $\gtrsim n_e L\,mc^2$ over an interaction length $L$ , where $n_e$ is the electron density), identifying realistic parameter regimes and diagnostic signatures; and (5) provide the quantitatively modified dispersion relations, passband widths, and parameter thresholds in the presence of damping, inhomogeneity, and other broadening mechanisms. These five linked sub-questions share one objective: a validated prediction of whether the collective-excitation transparency window below the plasma cutoff is realizable in realistic unmagnetized plasmas, not merely in the ideal model. Sub-questions (1) and (5) form a single quantitative core rather than two independent sub-questions: (1) is the enabling pole/linewidth computation that (5) carries into the fully modified dispersion relations, passband widths, and parameter thresholds, so (5) is incomplete without the pole/linewidth step of (1), and (1) alone does not determine whether the passband survives. The survival question is determined by this (1)+(5) core together with the artifact-discrimination criteria of (2); (3) and (4) extend the prediction to finite transverse geometry and pulse shape and to experimental feasibility.

References

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