Plasma and Fusion Research: Letters

Volume 7, 1301086 (2012)

Observation of Electromagnetically Induced Transparency in
Numerical Magnetized Plasma ExperimentEiichirou KAWAMORI
Institute of Space, Astrophysical and Plasma Sciences, National Cheng Kung University, Tainan, 70101, Taiwan(Received 19 January 2012 / Accepted 1 May 2012)
We report the observation, in a particle-in-cell simulation, of a phenomenon in which a magnetized plasma
absorbing an electron cyclotron wave is rendered transparent by a pump wave, which is a classical analog toelectromagnetically induced transparency (EIT) in quantum physics. Evident mode coupling between the probe
electron cyclotron wave and the pump wave driving the longitudinal plasma oscillation, which is required for therealization of EIT, is identified in addition to an observation of the electron cyclotron wave recapturing propagation in the magnetized plasma.
c 2012 The Japan Society of Plasma Science and Nuclear Fusion ResearchKeywords: electromagnetically induced transparency, electron cyclotron wave, mode coupling
DOI: 10.1585/pfr.7.1301086
Conventional quantum electromagnetically induced
transparency (EIT) is a phenomenon in which an opticallythick medium is rendered transparent by controlled laser
light [1]. A cold three-level atomic gas is typically employed as the medium. In EIT in three-level atomic systems, a probe light having the same frequency as an eigen
frequency corresponding to the transition between two energy levels of the atom loses resonance with the atoms if
another appropriately tuned control light (pump light) isinjected. This transparency phenomenon is not a saturation or hole-burning phenomenon. Quantum EIT resultsfrom destructive interference (cancelation of dipole transitions) between eigen states of the atom and the lights, thatis, a type of mode-coupling. EIT has many applications
including the widely known light stopping [2], informationtransfer and lasing without inversion.
It is natural to consider expansion of this theoreticalidea to classical oscillator systems such as plasma. EIT
in plasma is a classical analog to quantum EIT in whichplasma is assumed to be the medium. In the magnetizedplasma EIT, the quantities equivalent to the eigen frequencies in three-level atomic EIT systems are considered to be
the characteristic frequencies of a magnetized plasma, suchas the plasma and cyclotron frequencies. EIT based on
this idea was proposed for magnetized and unmagnetizedplasma by Harris [3] and Litvak and Tokman [4], respectively. Although experimental studies of EIT in plasma
were not conducted immediately when Harris first suggested the idea of plasma EIT in 1996 [3], the first experimental identification of EIT in magnetized plasma was recently reported [5].
This EIT in magnetized plasma creates a new branchat the electron cyclotron frequency when the frequency ofauthor’s e-mail: kawamori@pssc.ncku.edu.twthe pump wave satisfies a matching condition ωprobe ≈ωpe + ωpump , where ωprobe , ωpe and ωpump are the angular frequencies of the probe-wave, plasma wave andpump-wave, respectively. A plasma immersed in a background axial magnetic field absorbs the electromagneticwave (probe wave), which is right-hand circularly polarized (RHCP) at the electron cyclotron frequency. Coldmagnetized plasma is normally opaque to RHCP wave inthe frequency band between ω = ωce and ω = ωRH (thestop band), where ωce and ωRH are the electron cyclotron
angular frequency and the right-hand cutoﬀ angular frequency, respectively. If another appropriate electromagnetic wave (pump wave) is simultaneously injected, theelectron cyclotron resonance (ECR) disappears, and theplasma becomes transparent to RHCP wave [6]. This disappearance of ECR is induced by cancelation of the polarization current of the ECR wave by mode coupling between the probe and pump waves [4, 6, 7].
Plasma EIT has various potential applications including energy compression of electromagnetic microwaves [8], ion acceleration by the ponderomotive force,
control of the spatial absorption profile of electron cyclotron heating waves in fusion plasmas [9] and free electron laser [10].
In this paper, we attempt to realize magnetized plasma
EIT by a numerical simulation. The simulation code we
employ is a full-relativistic particle-in-cell (PIC) code thatis spatially one-dimensional and follows three componentsof the particle velocities. Therefore, the code enables us
to examine plane-wave propagation in magnetized plasma.
The parameters of the simulation are as follows: The time
step Δt, grid size Δz, number of particles (ions and electrons) per grid ΔN and ratio between ion mass and electron mass mi /me are 0.1ω−1ce , cΔt, 10 × 2 and 1833.0, respectively, where c is the speed of light in vacuum. Fig-1301086-1

c 2012 The Japan Society of Plasma
Science and Nuclear Fusion Research

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Fig. 2 Dispersion relation of waves propagating parallel tobackground magnetic field obtained from the PIC code
simulation (crosses, open squares and closed circles) andtheory of the CPDR.

Fig. 1 Spatial (in z-direction) distributions of (a) ions and electrons with x component of four-velocity γv x /c, whereγ ≡ (1 − v2 /c2 )−1/2 , (b) x components of the normalized electric field E x /(cBz0 ), and (c) ωce /ωprobe , that is,the background magnetic field strength Bz0 .

ure 1 shows the configuration and a typical snapshot of
the numerical experiment, specifically, the spatial distributions (in z direction) of (a) the ions and electrons withan x component of four-velocity γv x /c, where γ ≡ (1 −v2 /c2 )−1/2 , (b) the x components of the normalized electricfield E x /(cBz0 ), and (c) ωce /ωprobe , that is, the normalized
background magnetic field strength Bz0 . The plasma particles are confined by perfect reflection end-walls betweenz = 50cΔt and 500cΔt. The leftmost region of the computational domain is a vacuum region from which the probe
and pump waves are launched rightward. Both the rightgoing probe wave and a pump wave whose polarization
is right-hand circular are injected from the vacuum region
into a plasma along the background magnetic field (alongthe z-axis).
To achieve better impedance matching for the injectedprobe wave at the vacuum–plasma interface, the equilibrium magnetic field ωce /ωprobe is set higher (= 1.4) at thevacuum–plasma interface (z = 50cΔt) and decreases to 1.0
at z = 100cΔt. Although divB0 = 0 (B0 : the equilibrium magnetic field) is not satisfied in the region whereωce /ωprobe is not constant (z = 50−100cΔt), the wave fields
are properly treated in the entire region. That is, divB = 0for the wave magnetic field is guaranteed in the entire re-gion. The only eﬀect of divB0  0 is that the mirror force isneglected for the particles in the region (z = 50 − 100cΔt).
This neglect of the mirror force makes it possible to prepare a uniform equilibrium electron (and ion) density along
the z-axis.
First, we checked the validity of the PIC code by comparing the dispersion relation obtained from the PIC simulation with the cold plasma dispersion relation (CPDR).
For this test, we launched only a probe wave into a magnetized plasma without pump-wave injection. Here, ωpe /ωce0
is 0.5, where ωce0 is the electron cyclotron frequency inthe flat magnetic field region (z = 100 − 500cΔt). Figure 2 shows the dispersion relation (ω/ωce vs. ck|| /ωce ) in
the electron cyclotron frequency band. Blue crosses andgreen squares represent the PIC simulation results in themagnetized plasma for right-hand and left-hand polarizedwaves, respectively. Closed black circles represent wave
propagation in the vacuum. Solid curves are calculated
from CPDR. The PIC results agree well with the theoretical curves obtained by CPDR. By another simulation,
we confirmed that the PIC code accurately demonstrates
an electrostatic mode, that is, the longitudinal plasma oscillation, although the result is not shown in this paper.
The following three wave-injection methods were applied in an attempt to identify EIT by switching the probe
wave or pump wave on or oﬀ: (i) probe-wave injection only, (ii) pump-wave injection only and (iii) injection of both probe and pump waves. The amplitudes ofthe probe wave and pump wave |E|/(cBz0 ) were 0.03 and0.1, respectively. The angular frequencies of the probe

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Fig. 3 Spatial profiles of E x (E⊥ ) of probe wave for probe-wave
injection only (light blue) and injection of both probe andpump waves (red). These are obtained from the bandpass-filtered wave amplitude using a FIR filter having afrequency range of 0.99ωce − 1.01ωce .

wave ωprobe and pump wave ωpump were ωce0 and 0.7ωce0 ,respectively. The plasma angular frequency ωpe at thegiven plasma density satisfies the EIT matching condition
ωprobe ≈ ωpe + ωpump .
Figure 3 shows snapshots of spatial profiles of the normalized probe wave electric field E x /(cBz0 ) for (i) probewave injection only and (iii) injection of both probe andpump waves. A finite impulse response (FIR) filter extracts E x /(cBz0 ) whose angular frequency is ωce0 from thefull-band data. The electron cyclotron wave recaptures its
propagation when accompanied by the pump wave (case(iii)).
Figure 4 shows Fourier power spectra of the electric fields E x (E⊥ ) (top) and Ez (E|| ) (bottom) measured
at z = 110.0cΔt, where the axial magnetic field corresponds to the ECR field. The window size of the discrete
Fourier transformation is 4096, and the number of ensembles used to calculate the power spectrum is 4. As shownby the light blue curve in Fig. 4 (top), the spectrum of E⊥
in case (i), only the probe-wave injection, has no peak at
f = ωprobe /(2π) ∼ 0.16 because the probe wave is resonantly absorbed by the electrons. On the other hand, in thecase (iii), the pump-wave injection and probe-wave injections (red curve), two peaks in E⊥ are clearly observed atf ∼ 0.11 and f ∼ 0.16 which correspond to ωpump /(2π)
and ωprobe /(2π), respectively.
When the pump wave is injected in addition to theprobe wave, longitudinal plasma oscillation is excited asshown by the red curve in Fig. 4 (bottom). The frequencyof this longitudinal oscillation agrees well with ωpe /(2π).
These three frequencies, f ∼ 0.059, 0.107, and 0.163, observed in case (iii) accurately satisfy the EIT matching condition ωprobe ≈ ωpe + ωpump .
Figure 5 shows the phase diﬀerence Δφ between E xand Ey in case (iii), the injection of both probe and pumpwaves. At f ∼ 0.16, Δφ is −π/2, indicating that the electric field of the wave with f = ωprobe /(2π) is right-handpolarized.
We examine the possibility of stimulated Raman scat-

Fig. 4 Fourier power spectra of E x (E⊥ ) and Ez (E|| ) measured atz = 110.0. Curves show probe-wave injection only (lightblue), pump-wave injection only (black), and injection ofboth probe and pump waves (red).

Fig. 5 Phase diﬀerence Δφ between E x and Ey for injection ofboth probe and pump waves.

tering (SRS) of the pump wave. In case (ii), shown by thegray curve in Fig. 4 (top), the peak at f = ωprobe /(2π) ∼0.16 is absent, whereas a peak produced by the pump wave
appears. This result rules out the occurrence of SRS of thepump wave by the plasma wave to the electron cyclotron
wave in this simulation.

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Relativistic eﬀects make almost no contribution toEIT creation in the simulation. The average relativistic factor γ of the electrons is small (∼ 1.05) in the EIT demonstration.
These results indicate that pump-wave injection renders the plasma transparent to the electron cyclotron probe
wave as a result of coupling between the probe and pumpwaves which drives the plasma oscillation.
In summary, we have shown evidence of EIT in magnetized plasma in a PIC simulation. Plasma becomestransparent to the electron cyclotron wave when a tuned
pump wave is injected. Spectral analysis of the measured
wave electric field indicates the mode coupling between
the probe and pump waves that excites longitudinal plasma
oscillation. This is consistent with the theory reported elsewhere [4, 6].

Acknowledgments

This work was partly supported by Grants-in-Aid 97-

2112-M-006-016-MY3 from the National Science Councilof Taiwan.
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