Efficient narrowband teraherz radiation from electrostatic wakefields in non-uniform plasmas
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Efficient narrowband teraherz radiation
from electrostatic wakefields in non-uniform plasmas
Alexander Pukhov
Institut fuer Theoretische Physik I, Universitaet Duesseldorf, 40225 Germany
Anton Golovanov and Igor Kostyukov
IAP RAS, Nizhni Novgorod, Russia
It is shown that electrostatic plasma wakefields can efficiently radiate at harmonics of the plasma
frequency when the plasma has a positive density gradient along the propagation direction of a
driver. The driver propagating at a sub-luminal group velocity excites the plasma wakefield with
arXiv:2107.00452v2 [physics.plasm-ph] 2 Jul 2021
the same phase velocity. However, due to the positive density gradient, the wake phase velocity
steadily increases behind the driver. As soon as the phase velocity becomes super-luminal, the
electrostatic wakefield couples efficiently to radiative electromagnetic modes. The period of time
when the phase velocity stays above the speed of light depends on the density gradient scale length.
The wake radiates at well-defined harmonics of the plasma frequency in the teraherz (THz) band.
The angle of emission depends on the gradient scale and the time passed behind the driver. For
appropriate plasma and driver parameters, the wake can radiate away nearly all its energy, which
potentially results in an efficient, narrow band and tunable source of THz radiation.
PACS numbers: PACS1
The teraherz (THz) band of electromagnetic radiation energy per pulse) are free electron lasers [10] which are
spans the frequency range 3 × 1011 Hz to 3 × 1013 Hz [1]. expensive and not compact.
From the engineering point of view, there are currently One of the possibilities to generate a narrowband THz
few practical radiation sources in this THz gap [2]. The radiation is to exploit plasma oscillations - or wakefields -
standart vacuum devices (gyrotrones, magnetrones, syn- excited by a relativistic driver. The driver can be a short
chrotrones, free electron lasers, etc) could in principle pulse laser or short bunch of charged particles. When
be modified to work in this range. However, these de- the driver propagates through plasma, it displaces the
vices are still in prototype form, are not compact, or plasma electrons from their equilibrium positions. This
may require cryogenic temperatures to operate. The is accomplished either by the laser ponderomotive force,
laser technology operates at the higher frequencies, with or by the transverse fields of the charged particles bunch.
the wavelengths below 10 µm. At the same time, radia- The plasma electrons continue to oscillate behind the
tion sources in the THz band are required to study rota- driver at the local plasma frequency. This plasma wave
tional energy levels in complex molecules, oscillations in is called the wakefield. The wakefield phase velocity vph
solid crystals, etc [3]. If the THz field is strong enough simply equals the group velocity vg of the driver. The
(above 100 MV/m), nonlinear interaction with solid state wakefield is an electrostatic plasma oscillation that nor-
materials becomes possible including excitation of a di- mally does not couple to electromagnetic waves.
verse zoo of oscillatory degrees of freedom (spin waves,
Yet, it is possible to cause the plasma waves to radiate.
phonons, magnons, excitons, etc) [4–6]. New physics re-
One of the options is to collide two plasma wakefields
lated to control of nonequilibrium processes in solid state,
by using counter-propagating drivers [11, 12]. In the
initiation of surface chemical reactions, security, location,
year 1958 Ginzburg and Zheleznyakov [13] first guessed
etc. require powerful sources of narrow band THz radi-
that the radioemission from colliding electrostatic plasma
ation with a tunable central frequency [7]. The tera-
waves is responsible for solar radio bursts. Tsytovich de-
hertz radiation also has a significant potential in medi-
veloped a kinetic theory of nonlinear waves coupling in
cal diagnosis and treatment because its frequency range
plasmas [14]. Later, this mechanism was widely accepted
corresponds to the characteristic energy of biomolecular
in astrophysics [15–17]. It is also known that wakes gen-
motion. Advantageously, terahertz-specific low photon
erated by a laser pulse at an angle to plasma density gra-
energy does not cause the ionization of biomolecules [8]
dients can emit broadband radiation via mode conversion
Recently, there was a lot of activity on laser- and [18]. This mechanism corresponds to the inverse resonant
accelerator-based schemes of THz generation, which lead absorption. Another known way to cause the wakefields
to creation of powerful THz sources [9]. Most of these radiating is to apply an external magnetic field [19].
generate just a single period of high amplitude THz emis- In this paper, we suggest a different coupling mecha-
sion. It is still a challenge to create a narrowband THz nism between the electrostatic wakefield and electromag-
source with the pulse energy beyond 1 µJ. Presently, the netic radiation. The non-linear wakefield current con-
most powerful sources of THz radiation (1-10 MW, 10 µJ tains a non-vanishing curl. When the plasma has a posi-
2
´ z −1
with the wake phase ϕ(t, z) = −ωp (z) t − vg (ζ)dζ .
The wake phase velocity simply equals the driver group
velocity vg (z) ≈ c in the tenuous plasma. The full for-
mula gives only a small correction, while complicates the
analytics. For this reason, we omit the small difference
and assume vg = c in the following formulas.
At larger amplitudes, the wakefield becomes anhar-
monic. In the 1D case, its potential satisfies the nonlinear
oscillator equation [21]
" #
d2 Φ mc2 1 1
=− 1− (3)
dϕ2 e 2 eΦ 2
1 + mc 2
" 2 3 #
mc2 eΦ
3 eΦ eΦ
Figure 1: (a)-(c) Snapshots of THz emission from a laser- =− − +2 + ... .
e mc2 2 mc2 mc2
plasma wakefield at the times t1,2,3 = 650fs,1450fs,2000fs.
(d)-(f) Evolution of the tongitudinal field Ez . The effective
wavelength of the wake increases with time. The angle of THz The quasi-linear solution contains higher harmonics of
emission changes correspondingly. the plasma frequency ωl = lωp , where l is the harmonic
number.
Our ansatz for the wake phase is ϕ(t, z) = −ωp (z)τ ,
tive density gradient along the driver propagation direc- where τ = t − z/c is the fast time. The wake phase fre-
tion, the wake phase velocity is continuously increasing quency is then ω = −∂ϕ/∂t = ωp (z) and the longitudi-
with time after the driver has passed over. At some point, nal wavenumber is kz = ∂ϕ/∂z = ωp (z)/c − τ ∂ωp (z)/∂z.
the wake phase velocity becomes superluminal and the The wake phase velocity along the driver propagation di-
non-linear curl of the plasma wave current can efficiently rection is not constant anymore, but changes with time
couple to electromagnetic modes. The wake phase veloc- after the driver passage
ity can remain above the vacuum speed of light for many
plasma periods. This time is defined by the plasma den-
ω c
sity gradient scale length and can be sufficiently long to vphz = = . (4)
radiate nearly all energy stored in the wake field. Thus, kz 1 − cτ /L
an efficient regime of energy conversion from the driver The phase velocity (4) stays superluminal for times
to the plasma wakefield and then to THz radiation can
be found.
In the following, we assume the plasma wakefield is L
03
that causes the mode mixing. We Fourier-transform pulse duration TL = 30 fs. This pulse duration is close
the wakefield
´P potential in plasma frequency harmonics to the “resonant” one [25]. The laser pulse has the wave-
Φ = l Φ lk⊥
exp (−ilωp t + ilkz z + ik⊥ r⊥ ) dk⊥ , with length λ0 = 800 nm and is focused to a focal spot with
|l| being the harmonic number, and obtain the radiation the FWHM diameter 14 µm. The focal plane position is
source (compare Eq. (12.11) in [14] and Eq. (1) in [16]): at z = 100 µm. This laser pulse has 40 mJ energy. The
laser pulse group velocity here is vg ≈ 1 − ωp2 /2ω02 c ≈
∞ ¨ 0.9995c. The three-dimensional simulation box was
iec X
Lx ×Ly ×Lz = 240 µm×240 µm×160 µm sampled by grid
Rl,k⊥ = δl1 +l2 ,l δ (k⊥ − k1⊥ − k2⊥ )
2mωp steps hz = 0.14 µm and hx = hy = 0.4 µm the time step
l1 ,l2 =−∞
(7) was ∆ = chz [26]. The boundary conditions were open
for both fields and particles on all sides of the simulation
k12 k22
− k1 × k2 Φl1 k1⊥ Φl2 k2⊥ dk1⊥ dk2⊥ box.
l2 l1 For simplicity and to save computational resources,
we simulate only the first 150 µm of the density ramp.
The radiated frequency is ω = (l1 + l2 )ωp and the wave
As the laser pulse propagates, it ionizes the background
vector k = k1 +k2 have to satisfy the electromagnetic dis-
gas within the focal spot and produces a plasma column.
persion relation ω 2 = ωp2 +c2 k 2 . The source (7) is propor-
Outside of the plasma column, the gas remains unionized.
tional to the wake field potenital squared. The wakefield
Fig. 1 shows 2D (z, x) snapshots of the THz emission (the
amplitude in turn scales linearly with the laser intensity
right-bound field Ex +By ) and the longitudinal wake field
IL . Assuming the wavenumbers scale with k ∼ ωp /c, we
Ez at the times t1,2,3 = 650fs,1450fs,2000fs. The excited
obtain that for weakly relativistic wakefields, the power
wakefield has the amplitude eΦ/mc2 ≈ 0.1 immediately
of emitted THz radiation at the second harmonic scales
behind the laser pulse [20]. This corresponds to the lon-
as PTHz ∼ |Rω,k |2 ∼ n2 IL4 . For higher harmonics with
gitudinal field of Ez ≈ 12 GeV/m.
|l| ≥ 4, the simple formula (7) requires relativistic cor-
Evolution of the on-axis longitudial plasma wakefield
rections, which will be published elsewhere.
is shown in Fig. 2. One sees as the wakefield period
The formula (7) tells us further that plane plasma
increases with time, thus increases the effective phase ve-
waves do not radiate. The radiation requires presence
locity. The continuous emission of electromagnetic waves
of non-collinear wave vectors k1 and k2 with k12 6= k22
leads to the steady drop of the wakefield amplitude.
in the wave. Thus, a transversely symmetric wakefield
Simultaneously, one sees as the plasma wakefield emits
never radiates exactly forward.
THz radiation at an angle to the laser propagation direc-
The emission direction can be obtained from the res-
tion. The angle of emission θ can be estimated by the
onance condition. The angle of emission θq p inside the
inclination of wave phase fronts outside of the plasma col-
plasma column is cos θp = (1 − cτ /L) / 1 − ωp2 /ω 2 . umn. This angle increases with time as predicted by Eq.
This angle depends not only on the delay τ after the (8). Fig. 3 shows the angle of emission observed in the
driver has passed, but also on the radiated frequency ω. 3D PIC simulation (circles) as a function of time. The
However, when the radiation leaves the plasma column solid line gives the formula (8).
and passes the lateral plasma-vacuum boundary, it un- The radiated wave field has been recorded at the right
dergoes an additional refraction, so that the observed boundary at the point located 100 µm away from the laser
angle of emission θvacuum does not depend anymore on optical axis. The field itself and the spectrum are shown
the frequency and all harmonics exit plasma at the same in Fig. 4(a)-(b). We see a nearly monochromatic signal
angle at the second plasma frequency 2ωp , which lasts for cou-
ple picoseconds. The wakefield in this case is only weakly
cτ nonlinear, so the emission at the harmonics of plasma fre-
cos θvacuum = 1 − (8) quency is low. Yet, we see a weaker emission at 3ωp and
L
even at the plasma frequency itself. The electromagnetic
The expression (8) defines the angle of emission as a func- wave at ωp has an implicit transverse wavenumber k⊥ , so
tion of the delay τ after the driver passage. it is an evanescent wave inside plasma. However, as the
We simulate the THz emission from a plasma wakefield plasma column radius is comparable with the plasma skin
by the fully relativistic three-dimensional (3D) particle- depth c/ωp , the evanescent wave can reach the plasma-
in-cell code VLPL (Virtual Laser-Plasma Laboratory) vacuum boundary and escape. In addition, we see in
[24]. In the simulation, the background gas is assumed Fig. 4(b) a radiation at a very low frequency. This is
to be hydrogen (H2 ) with a linear density gradient so the Sommerfeld wave [27, 28] attached to the plasma fil-
that the molecular density rises from 0.85 × 1018 cm−3 to ament. The Sommerfeld wave is a slow wave with a sub-
1.7 × 1018 cm−3 over the distance of L = 0.5 mm. The luminal phase velocity. Because it has a very long wave-
hydrogen is dynamicall ionized by the short pulse laser length, it can be quasi-resonantly excited even without
with the intensity IL = 5 × 1017 W/cm2 and FWHM any density gradient.4
Figure 3: Angle of THz emission as a function of time. Circles
mark the simulation results. The solid line is given by the
formula (8).
Figure 2: (a) The on-axis wakefield Ez for the three times
t1,2,3 = 650fs,1450fs,2000fs. It is seen that the effective wave-
length increases. At the same time the wakefield amplitude
drops continuously due the radiation losses. (b) Fourier trans-
formation of the wakefields. At the earlier time, the wake
wavenumber is centered at kp = ωp /c. At the later times, the
peak is shifting to smaller wavenumbers.
Figure 4: THz signal and its spectra recorded at the right
boundary the radial position r = 100 µm outside of the
plasma column (a)-(b) for the laser intensity IL = 5 ×
To highlight the THz emission at harmonics of plasma 1017 W/cm2 and focal spot diameter 14 µm; (c)-(d) for the
frequency, we did another simulation with the same gas laser intensity IL = 2 × 1018 W/cm2 and focal spot diam-
target, but focused the laser pulse to a twice smaller spot eter 7 µm. Radiation emission at the harmonics of plasma
of 7 µm and peak intensity IL = 2 × 1018 W/cm2 . The frequency fp ≈ 12 THz are observed
plasma wave produced by the higher intensity laser pulse
has a larger amplitude and radiates more at the higher
harmonics. The THz signal recorded at the same posi- asymmetry of the THz field seen in Fig. 4(c).
tion is shown in Fig. 4(c). The field amplitude exceeds In conclusion, we have shown that a plasma wakefield
0.5 GV/m and lasts for about half a picosecond. During can efficiently radiate electromagnetic modes, when there
this time, the wakefield radiates away nearly all its en- is a positive density gradient along the driver propagation
ergy. The spectrum of radiation is shown in Fig. 4(d). direction. The density scale length L defines how long
Although the second plasma harmonic still dominates, we the wake phase velocity remains superluminal and the
see at least five harmonics of the plasma frequency. Men- coupling to the electromagnetic waves is possible. The
tion also the emission at the very low frequencies, around low amplitude linear wake emits a narrow band THz ra-
f ≈ 0. This is again the Sommerfeld mode (cylindrical diation at the second plasma harmonic, ω = 2ωp . At
plasma surface wave). This mode causes the significant higher amplitudes, the nonlinear plasma wave emits at5
the higher harmonics as well. The angle of emission is region of interaction of two short laser pulses in a rar-
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