# Femtosecond laser-induced sub-wavelength plasma inside dielectrics: III.
Terahertz radiation emission

Kazem Ardaneh<EMAIL_ADDRESS>FEMTO-ST Institute, Univ. Bourgogne
Franche-Comté, CNRS, 15B avenue des Montboucons, 25030, Besançon Cedex, France
Sorbonne University, Pierre and Marie Curie Campus, 4 place Jussieu, 75252,
Paris Cedex 5, France  Ken-Ichi Nishikawa  Department of Physics, Chemistry
and Mathematics, V. Murry Chambers Bld., Alabama A&M University, Huntsville,
AL 35810, USA  Remo Giust  Benoit Morel  Pierre-Jean Charpin  FEMTO-ST
Institute, Univ. Bourgogne Franche-Comté, CNRS, 15B avenue des Montboucons,
25030, Besançon Cedex, France  Arnaud Couairon  CPHT, CNRS, Ecole
Polytechnique, Institut Polytechnique de Paris, Route de Saclay, F-91128
Palaiseau, France  Guy Bonnaud  CEA, Centre de Paris-Saclay, DRF, Univ. Paris-
Saclay, 91191 Gif-sur-Yvette, France  Francois Courvoisier
<EMAIL_ADDRESS>FEMTO-ST Institute, Univ. Bourgogne Franche-
Comté, CNRS, 15B avenue des Montboucons, 25030, Besançon Cedex, France

###### Abstract

Electromagnetic radiation within the terahertz (THz) frequency range is of
great interest for applications in remote sensing and time-domain
spectroscopy. The laser-induced plasmas are promising mediums for generating
THz radiation. It has been recently reported that focusing femtosecond Bessel
pulses inside dielectrics induces a high aspect ratio over-critical plasmas.
Here we show that the intense resonantly driven electrostatic fields at the
so-called critical surface lead to THz radiation emission. Through three-
dimensional particle-in-cell simulation and analytical derivation, we have
investigated the emission of THz radiation. We show that the THz radiation is
associated with a hot population of electrons trapped in ambipolar electric
fields of the double layers.

##  I Introduction

Terahertz (THz) radiation, typically referred to as the frequency band 100 GHz
$-$ 10 THz, in the infrared and microwaves ranges, has been attracting ongoing
interest because of its broad applications ranging from biomedical imaging,
security or packaged goods inspection, to time-domain spectroscopy. [1, 2, 3,
4] Using femtosecond laser pulses, the techniques for generating THz radiation
are generally classified as either optical rectification,[5, 6, 7, 8, 9, 10,
11] transient current sources,[12, 13, 14, 15, 16, 17, 2, 18, 19, 20, 21, 22,
23] or a combination of these two mechanisms. [24]

Optical rectification can induce THz radiation in non-centrosymmetric
crystals, e.g., ZnTe and GaAs, in which the fundamental frequency of an
infrared femtosecond laser pulse is down-converted to the THz frequency via
the second-order susceptibility where the polarization reads
$\mathbf{P}\left(\omega_{\mathrm{THz}}\right)=\chi^{(2)}\mathbf{E}(\omega+\omega_{\mathrm{THz}})\mathbf{E}^{*}(\omega)$.
[5, 7] The frequency of the rectified pulse envelope is in the range of 3-10
THz.[7] In the centrosymmetric media, e.g., gases, two-color illumination can
be used to mix the fundamental frequency with the second-harmonic in a four-
wave mixing process as
$\mathbf{P}\left(\omega_{\mathrm{THz}}\right)=\chi^{(3)}\mathbf{E}(2\omega\left.-\omega_{\mathrm{THz}}\right)\mathbf{E}^{*}(\omega)\mathbf{E}^{*}(\omega)$.
[8, 11] Correspondingly, the THz component might match with the fundamental
one in an inverse process and leads to second-harmonic generation, which is a
method for THz detection. A THz emission with an estimated field strength
$\sim$ 400 kV/cm has been reported in which the presence of plasma was
essential for the high-efficiency process. [10]

Laser-induced plasmas are attractive for THz radiation generation because of
their ability to sustain extremely intense electromagnetic fields.[13, 14, 25,
26, 27] In this context, femtosecond laser-induced breakdown of gases is
investigated widely,[28, 29] mostly using the two-color approach.[8] The peak
of the THz field however saturates for laser intensities higher than $10^{15}$
W/cm2 because of the strong THz absorption in the long ($\sim$7 mm) air
plasma.[17] Plasma generation in laser-solid interactions offer an
alternative: experiments of ultrashort laser pulses-solid interaction have
shown a monotonous increase in THz radiation with the incident laser intensity
up to $10^{19}$ W/cm2.[18, 19, 21]

Illumination of solid targets by intense ultrashort laser beams results in the
generation of hot electron currents that are the source of the secondary
electromagnetic radiation ranging from x-rays [30, 31, 32, 33] to THz
radiation.[19] Under $p-$polarized laser illumination of a short-scale
inhomogeneous plasma, for moderate laser intensities (bellow $10^{14}\,{\rm
W/cm^{2}}$), resonance absorption is the main mechanism for hot electron
generation.[34, 35, 36]

Figure 1: The energy spectrum of the radiation emitted by a population of hot
electrons: (a) the spatially averaged, (b) angular distribution, and (c)
spatial distribution averaged in the frequency range of 1-30 rad/ps. The hot
electrons are randomly selected.

In previous papers, we have reported that over-critical plasmas, i.e. with
density above the reflection density for the incident laser wavelength, are
generated by focusing Bessel beams with moderate intensities on the order of
$10^{14}\,{\rm W/cm^{2}}$ inside sapphire.[37, 38, 39] A Bessel beam is a
solution of the wave equation in which the wave amplitude is defined by the
Bessel function of the first kind.[40] Importantly, the axial intensity
profile of the Bessel beam is propagation invariant. Therefore, all segments
of the dielectric along the Bessel zone will receive simultaneously the same
amount of energy which results in a high aspect ratio plasma rod.

In the first article of this series (Ardaneh et al.,[38] Paper I hereafter),
we confirmed that the resonance of the plasma waves can explain the
experimental diagnostics of total absorption, and far-field intensity pattern.
We reported electron acceleration up to several keV while surfing the plasma
waves. In the outward propagation of hot electrons, electrostatic ambipolar
fields form at the plasma surface due to the different inertia of the electron
and ion. Moreover, in the second article of this series (Ardaneh et al.,[39]
Paper II hereafter), we reported the second-harmonic generation by a second-
order current of hot electrons near the critical surface. The electron
currents form by the resonance absorption and radiation force of the incident
laser wave.

In the current work as the third in this series, we establish a link between
resonance absorption-driven currents and THz radiation. This is based on
calculating the coherent radiation spectrum of the hot electrons for the
performed Particle-In-Cell (PIC) simulation. The simulation consists of
electron-ion plasma initially induced by multi-photon and collisional
ionizations. The dipole moments are induced due to the radiation force of the
resonance fields. For a laser field with frequency $\omega_{0}$, this force
induces a second-harmonic component at $2\omega_{0}$ and a low-frequency
component by separating the light electrons from the heavy ions. We have
developed an analytical model for THz generation in laser-plasma interactions
to explain the underlying physics, in particular, how the dipole moment is
created in the plasma and the characteristics of the radiation spectrum.

We organized the paper as follows. In Sec. II, we recall the setup of the PIC
simulation as discussed in Paper I[38], we detail the radiation diagnostic,
and the results of the simulation. Then, in Sec. III, we derive an analytical
solution for the current source of THz radiation, and the radiated
electromagnetic fields with their frequency spectrums.

Table 1:  Simulation setup.

Parameter |  Value
---|---
Simulation volume |  $15\times 15\times 30$ $\rm\mu m^{3}$
Grid resolution |  $\Delta_{\rm x:y}k^{0}_{\rm r}=0.04$a, $\Delta_{\rm z}k^{0}_{\rm z}=0.1$b
FDTD order |  Second-order
BC for fieldsc |  Perfectly matched layers
BC for particles |  Outflow
Pulse energy ($E_{\rm p}$) |  1.2 $\rm\mu J$
Pulse frequency ($\omega_{0}$) |  2.35 rad/fs
Pulse cone angle ($\theta$) |  $25^{\circ}$
Pulse temporal profile |  $\exp[-(t-t_{\rm c})^{2}/T^{2}]$
Central time ($t_{\rm c}$) |  $130\,{\rm fs}$
Pulse ${\rm FWHM}=\sqrt{2\ln 2}T$ |  $100\,{\rm fs}$
Pulse spatial profile |  $\exp(-r^{2}/w_{0}^{2})$
Pulse spatial waist ($w_{0}$) |  10 $\rm\mu m$
Maximum density ($n_{{\max}}$) |  $5\,n_{\rm c}$
Density profile (axial) |  $\tanh(z^{\rm{\mu m}})$
Mass ratio ($m_{\rm i}/m_{\rm e}$) |  $102\times 1836$d
Plasma distribution [$f(\mathbf{v}_{\rm e:i})$] |  Maxwellian
Plasma temperature ($T_{\rm e:i}$) |  $1$ eV
Particles per cell per species |  32
Particle shape function |  triangle
Time step |  $\Delta t\omega_{0}=0.07$
Simulation time |  $320\,{\rm fs}$

  * a

$k^{0}_{\rm r}=k_{0}\sin\theta$.

  * b

$k^{0}_{\rm z}=k_{0}\cos\theta$.

  * c

BC: Boundary condition.

  * d

102 is the sapphire molar mass.

##  II PIC Simulation

We performed self-consistent PIC simulation using the three-dimensional
massively parallel electromagnetic code EPOCH [41]. In our simulation, we used
the plasma parameters that could reproduce our experimental measurements (far-
field, near-field, absorption) as reported in Paper I,[38] and II.[39] The
simulation setup is summarized in Table 1. The plasma is fully ionized and
composed of electrons and ions with equal densities (to preserve electric
neutrality) given by $n=n_{\max}\exp(-x^{2}/w_{\rm x}^{2})\exp(-y^{2}/w_{\rm
y}^{2})\tanh(z^{\rm{\mu m}})$ with FWHM${}_{\rm x}=\sqrt{2\ln 2}w_{\rm x}=250$
nm, and FWHM${}_{\rm y}=\sqrt{2\ln 2}w_{\rm y}=600$ nm. There are initially 32
particles per cell per species leading to the total number of particles in the
simulation $\sim 10^{9}$. The collisions are modeled through a binary model as
presented in Refs.[42, 41].

We injected from the $z_{\min}$ boundary a linearly $x-$polarized Gaussian
pulse propagating along the positive $z-$direction. We applied a phase to the
Gaussian beam to create a Bessel-Gauss beam.[43] The peak intensity in the
Bessel zone is $6\times 10^{14}\,{\rm W/cm^{2}}$ in absence of plasma. The
time step is limited by the Courant condition. The minimum frequency in the
simulation is 1.5 rad/ps which is well below the peak frequency of the THz
spectrum at 30 rad/ps.

Figure 2: THz radiation from an electron trapped in the ambipolar electric
fields of double layers. Shown are: (a) $x-$component of the electric field
over-plotted by the trajectory of a representative electron, electron emission
for a time window between: (b) 20-150 fs [shown by blue $\bigotimes$ symbols
in panel (a), and blue line in panel (b)], and 182-312 fs [shown by red
$\bigotimes$ symbols in panel (a) and red line in panel (b)], (c-f)
$cE^{2}_{\rm x}/8\pi$, $cE^{2}_{\rm y}/8\pi$, $cE^{2}_{\rm z}/8\pi$ and the
total intensity of the THz radiation emitted by the electron for the time
window between 182-224 fs. The dashed lines in panel (a) show the expansion of
the plasma at the sound velocity. The color in the electron trajectory
reflects its energy based on the color bar of the panel (b).

One of the primary advantages of PIC codes is the possibility to access full
information about the particles. We have developed a radiation diagnostic that
utilizes the position and momentum of particles over time and calculates the
radiated fields and energy. For this purpose, let us consider a particle at
position ${\bf r}\left(t\right)$ at time $t$. At the same time, we observe the
radiated electromagnetic fields from the particle at position ${\bf x}$. Due
to the finite velocity of light, we observe the particle at an earlier
position ${\bf r}\left(t^{\prime}\right)$ where it was at the retarded time
$t^{\prime}=t-\mathbf{R}\left(t^{\prime}\right)/c$, where
$\mathbf{R}\left(t^{\prime}\right)=\left|\mathbf{x}-\mathbf{r}\left(t^{\prime}\right)\right|$
is the distance from the charged particle (at the retarded time $t^{\prime}$)
to the observer. The magnetic and electric fields produced from a moving point
charge can be calculated directly from their scalar and vector potentials
known as the Liénard–Wiechert potentials. The electric field reads:[44]

$\begin{split}\mathbf{E}(\mathbf{x},t)=&\overbrace{e\left[\frac{\mathbf{n}-\bm{\beta}}{\gamma^{2}(1-\bm{\beta}\cdot\mathbf{n})^{3}R^{2}}\right]_{\mathrm{ret}}}^{\text{Velocity
field}}\\\
+&\overbrace{\frac{e}{c}\left[\frac{\mathbf{n}\times\\{(\mathbf{n}-\bm{\beta})\times\dot{\bm{\beta}}\\}}{(1-\bm{\beta}\cdot\mathbf{n})^{3}R}\right]_{\mathrm{ret}}}^{\text{Acceleration
field}}\end{split}$ (1)

where
$\mathbf{n}=\mathbf{R}\left(t^{\prime}\right)/\left|\mathbf{R}\left(t^{\prime}\right)\right|$
is a unit vector pointing from the particle retarded position to the observer,
$\bm{\beta}=\mathbf{v}/c$ the particle instantaneous velocity,
$\dot{\bm{\beta}}={\rm d}\bm{\beta}/{\rm d}t$ is the acceleration divided by
$c$, $\gamma$ is the Lorentz factor. The spatial spectra are obtained by the
choice of $\mathbf{n}\left(n_{\rm x}^{2}+n_{\rm y}^{2}+n_{\rm
z}^{2}=1\right)$. The field in Eq. (1) divides itself into ”velocity fields,”
which are independent of acceleration, and ”acceleration fields,” which depend
linearly on $\dot{\bm{\beta}}$. The velocity field is a static field
decreasing as $R^{-2}$ while the acceleration field is a radiation field,
being transverse to the radius vector and falling off as $R^{-1}$. The total
energy $W$ radiated per unit solid angle ${\rm d}\Omega$ per unit frequency
${\rm d}\omega$ from the accelerated charged particle reads:[44]

$\frac{{\rm d}^{2}W}{{\rm d}\omega{\rm
d}\Omega}=\frac{e^{2}\omega^{2}}{4\pi^{2}c}\left|\int_{-\infty}^{\infty}{\rm
d}t^{\prime}{\hat{\mathbf{n}}\times(\hat{\mathbf{n}}\times\bm{\beta})}e^{j\omega(t^{\prime}+R(t^{\prime})/c)}\right|^{2}$
(2)

In our simulations, we collected the THz radiation emissions from the hot
electrons as follows. We have tracked $10^{5}$ electrons in the simulations
and recorded the information of these electrons. We calculated the energy
spectrum of the radiation emitted by 100 randomly selected electrons according
to Eq. (2). The result is shown in Fig. 1(a). We see two peaks around
$\omega=0$, with a width of typically 50 rad/ps. We note that, because of
computing memory limitations, the time resolution of the particle positions is
insufficient to capture the second harmonic emission. The angular and spatial
distributions of the THz emission are obtained by averaging the energy
spectrum in the frequency range of 1-30 rad/ps [Figs. 1(b) and 1(c)].

As one expects, the energy spectrum has sharp maxima at the laser frequency
$\omega_{0}$ due to strong electron acceleration in resonantly driven plasma
waves at the critical surfaces. The spectrum also has maxima at $\omega\approx
30$ rad/ps. The angular distribution of this THz radiation in Fig. 1(b) shows
maxima around $(\theta,\phi)\approx(0,\pi/2)$ and $(0,3\pi/2)$, perpendicular
to the electron acceleration which is mainly in the $x-$direction. The small
tilt in Fig. 1(c) is due to the asymmetric distribution of the randomly
selected electrons in $xy-$space over the integrated time (a similar deviation
occurred for another set of 100 electrons).

We select some representative electrons to calculate the radiated fields in a
spatial window $|x|\leqslant 45$ $\rm{\mu m}$ and $|y|\leqslant 45$ $\rm{\mu
m}$ at $z=0$. Using a time window, we also examined in which part of the
trajectory, the electron emits electromagnetic radiation in the THz frequency
range [Figs. 2(a)]. For each time window, we calculated the intensity
distribution $I(\omega_{\rm THz},x,y)$ by performing a discrete Fourier
transform on each component of the electric field, $E_{\rm x:y:z}(t,x,y)$ and
averaging in the frequency range of 1-30 rad/ps [Figs. 2(b)-2(f)].

Figure 2(a) shows the time evolution of the $E_{\rm x}$ component, parallel to
the incident laser polarization over-plotted with the trajectory of a
representative electron. One can see the resonance plasma waves induced at the
critical surfaces ($x=\pm 0.2\,{\rm\mu m}$), in the time between 70-200 fs
(See Fig. 4 in Paper I[38] for more details). Near the peak of the laser
field, the intense ambipolar fields propagating with the sound speed are
generated at the surface of the plasma (dashed lines). The ambipolar field
sign is positive for $x>0$ and negative for $x<0$. The radiation force (See
Appendix A) due to the intense, localized resonance field ejects electrons
from the resonance region. The electrons are ejected from the critical
surfaces in the positive $x-$direction where $x>0$ and negative $x-$direction
where $x<0$ as shown in Fig. 6 Paper I. Therefore, the electrons ejected with
energy less than the potential barrier of the ambipolar field will be
reflected by $-e\bf{E}$ force. Consequently, these electrons will be trapped
between the ambipolar electric fields on either side of the plasma. An ejected
electron oscillates between the ambipolar fields with a period that increases
with time due to the energy exchange between the electrons and ions.

We monitored the electron radiation using a time window of 130 fs. The
electron emission between 20-150 fs [shown by blue $\bigotimes$ symbols in
panel (a)], is sharply peaked at the laser frequency $\omega_{0}$ [blue line
in panel (b)]. This emission is due to the electron acceleration while surfing
the resonantly driven plasma waves (See Paper I[38] for more details). During
the time interval 182-312 fs [shown by red $\bigotimes$ symbols in panel (a)],
the electron is trapped between two ambipolar fields and emits the THz
radiation with a peak frequency at $\omega=24$ rad/ps [red line in panel (b)].
Figures 2(c-f) show respectively the intensity of the electric field
components computed using Eq. (1), $c(E^{2}_{\rm x},E^{2}_{\rm y},E^{2}_{\rm
z})/8\pi$, and the total intensity radiated by the electron during the time
between 182-312 fs. The THz radiation is mainly polarized in the $x-$direction
because $E_{\rm x}$ component is dominant in the radiated field. This
polarization is the same as the incident pulse and second-harmonic detailed in
Paper II[39]. In Sec. III, we will show that the emission pattern corresponds
with an electron current in the $x-$direction.

##  III Electron THz emission in ambipolar electric fields

The starting point in understanding the mechanism responsible for THz
radiation is the identification of its current sources. We have seen earlier,
in Figs. 2, that the electrons emit THz radiation while they are trapped in
the ambipolar electric fields of plasma double layers. Taking the strength of
the resonance electric field of about 50 GV/m integrated over its width of 70
nm (See Paper I[38]), one arrives at a potential of about a few keV which
corresponds to the temperature of the hottest electrons in the simulation. The
hot electrons propagate outside of the plasma and consequently, the separation
of charges forms an electric double layer where an ambipolar electric field is
present. An analytical solution for this field is possible by using the two-
fluid plasma equations for continuity and momentum (See Appendix B).

Here for simplicity, we considered a $s-$polarized monochromatic laser wave as
$\mathbf{E}=\mathbf{E}_{\rm s}(\mathbf{r})\cos(\omega_{0}t)$ where
$\mathbf{E}_{\rm s}(\mathbf{r})$ includes the spatial dependence. The
ambipolar electric field $E_{\rm a}$ is described by an inhomogeneous second-
order differential equation for a classical, damped, driven harmonic
oscillator given by [See Eq. (18) in Appendix B]:

${\partial_{\rm t}^{2}E_{\rm a}}+2\Gamma{\partial_{\rm t}E_{\rm
a}}+\Omega_{\rm p}^{2}E_{\rm a}=\Omega_{\rm
p}^{2}\left[\mathscr{E}_{0}+\mathscr{E}_{2}\cos(2\omega_{0}t)\right]$ (3)

where $\Gamma={\nu_{\rm ei}}/{2}\left(1+{Zm_{\rm e}}/{m_{\rm i}}\right)$,
$\nu_{\rm ei}$ is the electron-ion collision frequency, $\Omega_{\rm
p}^{2}=\omega_{\rm{pe}}^{2}\left(1+{Zm_{\rm e}}/{m_{\rm i}}\right)$,
$\omega_{\rm pe}=(4\pi n_{\rm e}e^{2}/m_{\rm e})^{1/2}$ is the electron plasma
frequency (in cgs units), and

$\displaystyle\begin{split}\mathscr{E}_{0}=&\frac{4\pi e}{\Omega_{\rm
p}^{2}}\left[{\partial_{\rm x}}\left(Z\frac{P_{\rm i}}{m_{\rm i}}-\frac{P_{\rm
e}}{m_{\rm e}}+Zn_{\rm i}v_{\rm i}^{2}-n_{\rm e}v_{\rm e}^{2}\right)\right]\\\
&-\frac{4\pi e}{m_{\rm e}{\Omega_{\rm
p}^{2}}}\frac{\omega_{\rm{pe}}^{2}}{\omega_{0}^{2}}\partial_{\rm
x}\left\langle\frac{{E}^{2}}{8\pi}\right\rangle\end{split}$ (4a)
$\displaystyle\mathscr{E}_{2}=$ $\displaystyle-\frac{4\pi e}{m_{\rm
e}{\Omega_{\rm
p}^{2}}}\frac{\omega_{\rm{pe}}^{2}}{\omega_{0}^{2}}\partial_{\rm
x}\left\langle\frac{{E}^{2}}{8\pi}\right\rangle$ (4b)

with the standard notation $(t,x,v,P,m_{\rm s},n_{\rm s},Z)$ for the time,
space, velocity, pressure, mass and density of a particle of species $s$, and
ion charge respectively. The $\langle\rangle$ denotes an average over a laser
cycle. The coupling to the laser was included in the momentum equation via the
radiation force density ${\bf f}_{\rm RF}=(\epsilon-1)/{8\pi}\bm{\nabla}E^{2}$
where $\epsilon$ is the plasma permittivity (See Appendix A). One can find an
equation similar to Eq. (3) in Refs.[16, 2, 20, 24] but with a different
right-hand side (different current sources of the THz radiation).

The solution of Eq. (3) under the initial conditions of $\left(E_{\rm
a},\partial_{\rm t}E_{\rm a}\right)=\left(0,0\right)$ reads (See for example
Ref.[45]):

Figure 3: Temporal profile of the THz component of the ambipolar electric
field from Eq. 5, panel (a), the frequency spectrum of the electromagnetic
radiation, panel (b), angular distributions of the radiated energy from Eq.
7(b), panel (c), and electric field components, panels (d)-(f).

$\begin{split}E_{\rm
a}(t)=&\overbrace{\mathscr{E}_{0}\left\\{1-\exp{\left(-\Gamma
t\right)}\left[\cos\left(\varpi t\right)+\frac{\Gamma}{\varpi}\sin\left(\varpi
t\right)\right]\right\\}}^{\text{Terahertz oscillation}}\\\
+&\overbrace{\Omega_{\rm p}^{2}\mathscr{E}_{2}\frac{\left(\Omega_{\rm
p}^{2}-4\omega_{0}^{2}\right)\cos(2\omega_{0}t)+4\omega_{0}\Gamma\sin(2\omega_{0}t)}{\left(\Omega_{\rm
p}^{2}-4\omega_{0}^{2}\right)^{2}+16\Gamma^{2}\omega_{0}^{2}}}^{\text{Second-
harmonic oscillation}}\end{split}$ (5)

where $\varpi^{2}=\Omega_{\rm p}^{2}-\Gamma^{2}$. This solution includes two
components. The first component oscillates with a frequency close to the
plasma frequency $\omega_{\rm{pe}}$ when $\omega_{\rm{pe}}\gg\nu_{\rm ei}$.
This oscillation, however, decays exponentially at a rate close to the
collision frequency. This component is established by the spatial gradients of
the pressure difference between the light electrons and the heavy ions as
represented in Eq. (4a). This part induces the dipole moment in the plasma by
separating the electrons from the ions. After a time $t\gg 1/\nu_{\rm ei}$,
neglecting the electron and ion velocities and assuming $T_{\rm e}\gg T_{\rm
i}$ (See Paper I[38]), a nearly constant electric field remains $eE_{\rm
a}\approx-{1}/{n_{\rm e}}{{\rm d}P_{\rm e}}/{{\rm d}x}=-\gamma T_{\rm e}{{\rm
d}\ln n_{\rm e}}/{{\rm d}x}$, considering an adiabatic equation of state with
the adiabatic index $\gamma$. Therefore, the ambipolar field oscillations are
driven by the electron density gradient. The work function of the electrons
that moved from the plasma interior (density $n_{1}$) to the exterior (density
$n_{2}$) then reads $-e\Delta\phi=\gamma T_{\rm e}\ln(n_{1}/n_{2})\approx
4\,{\rm keV}$.

The second part in Eq. 5 arises where gradients of the laser intensity induce
a second harmonic longitudinal field oscillation. This term has a resonance at
$2\omega=\Omega_{\rm p}\approx\omega_{\rm pe}$ (four times the critical
density) for the evanescent part of the wave causing a very steep increase of
the oscillation amplitude. This resonance for $s-$polarized lasers is
different from the Denisov resonance absorption occurring under oblique
incidence of $p-$polarized lasers. [46, 47]

An example of the electric field given by Eq. (5) is shown in Fig. 3(a) for
$\varpi=30$ rad/ps (corresponds to an edge plasma density of $n_{2}/n_{\rm
c}=10^{-4}$), and $\Gamma=3$ rad/ps where we have supposed that the electron
collision frequency is small compared to the plasma frequency. The wave shows
2-3 oscillations and damps out within a time scale of $\sim$ 2 ps. The Fourier
spectrum of the electric current associated with the quasi-static electric
field, $4\pi J_{\rm a}(\omega)=j\omega E_{\rm a}(\omega)/(1+j\nu_{\rm
ei}/\omega)$, is shown in Fig. 3(b). As one can see, it has a maximum at
$\omega\approx\varpi=30$ rad/ps.

To compare with the numerical results of the radiated emission in Fig. 1(b),
we derive the angular distribution of the radiated energy from a current
source. For a number of accelerated charges, the integrand in Eq. (2) involves
the replacement $e\bm{\beta}e^{j\omega R(t^{\prime})/c}\rightarrow\sum_{\rm
m=1}^{N}e_{\rm m}\bm{\beta}_{\rm m}e^{j\omega R_{\rm m}(t^{\prime})/c}$. In
the limit of a continuous distribution of charge, the summation becomes an
integral over the current density as $e\bm{\beta}e^{j\omega
R(t^{\prime})/c}\rightarrow 1/c\int{\rm d}^{3}{\bf r}^{\prime}\
\mathbf{J}({\bf r}^{\prime},t^{\prime})e^{j\omega R(t^{\prime})/c}$. Hence,
the radiation energy per solid angle per frequency of the current source
reads:[44]

$\frac{{\rm d}^{2}W}{{\rm d}\omega{\rm
d}\Omega}=\frac{\omega^{2}}{4\pi^{2}c^{3}}\left|\int{\rm d}t^{\prime}\int{\rm
d}^{3}{\bf r}^{\prime}\
{\hat{\mathbf{n}}}\times\left[{\hat{\mathbf{n}}}\times\mathbf{J}({\bf
r}^{\prime},t^{\prime})\right]e^{j\omega[t^{\prime}+R(t^{\prime})/c]}\right|^{2}$
(6)

We consider an emission length of $L$ for the plasma rod oriented parallel to
the $z-$direction, and a current of electron in the $x-$direction as ${\bf
J}({\bf r}^{\prime},t^{\prime})={\hat{\mathbf{x}}}J_{\rm
a}(t^{\prime})\delta(x^{\prime})\delta(y^{\prime})\exp{\left(jkz^{\prime}\right)}$.
For a coordinate system with the spherical angle
$\theta=\cos^{-1}\left({z}/{r}\right)$ and the azimuth angle
$\phi=\tan^{-1}\left({y}/{x}\right)$ defining the direction of observation
$\mathbf{n}$, Eq. (6) reduces to:

$\displaystyle\frac{\mathrm{d}^{2}W}{\mathrm{d}\omega\mathrm{d}\Omega}=$
$\displaystyle\frac{\left|J_{\rm
a}(\omega)\right|^{2}}{\pi^{2}c}f(\phi,\theta)$ (7a) $\displaystyle
f(\phi,\theta)=$
$\displaystyle\frac{\sin^{2}\theta\sin^{2}\phi+\cos^{2}\theta}{\left(1-\cos\theta\right)^{2}}\sin^{2}\left(Lk\sin^{2}\frac{\theta}{2}\right)$
(7b)

where $k$ is the emission wave-vector.

The angular distribution given in Eq. (7b) is shown in Fig. 3(c) for an
emission length of $L=10\,{\rm\mu m}$. It fits well with the distribution
obtained from the PIC simulation in Fig. 1(b) where the emission is beamed in
the positive $z-$direction and has two maxima in the $y-$direction,
perpendicular to the current source. The vector potential for this current
source is in the $x-$direction and at far-field, it reads:[44]

$\begin{split}\mathbf{A}(\mathbf{x},\omega)=&\frac{1}{c}\frac{e^{jkR}}{R}\int{\rm
d}^{3}\mathbf{r}^{\prime}\mathbf{J}\left(\mathbf{r}^{\prime},\omega\right)e^{-jk\mathbf{n}\cdot\mathbf{r}^{\prime}}\\\
=&{\hat{\mathbf{x}}}\frac{2J_{\rm
a}(\omega)}{c}\frac{e^{jkR}}{kR}\frac{\sin\left(Lk\sin^{2}\frac{\theta}{2}\right)}{1-\cos\theta}\end{split}$
(8)

One can derive the scalar potential $\Phi$ using the Lorenz gauge and then the
components of the electric field as follows:

$\displaystyle\bm{\nabla}\cdot\mathbf{A}=$
$\displaystyle-\frac{1}{c}\frac{\partial\Phi}{\partial t}$ (9a)
$\displaystyle\mathbf{E}=$
$\displaystyle-\frac{1}{c}\frac{\partial\mathbf{A}}{\partial
t}-\bm{\nabla}\Phi$ (9b)

The components of the radiated electric field calculated using Eq. (9b) are
shown in Fig. 3(d)-(f). In agreement with the results of the PIC simulation
[Figs. 2(c-d)], the radiated emission is polarized in the $x-$direction.
Moreover, the angular distributions of the electric field components agree
with the results of the PIC simulation [Figs. 2(c)-(e)]. The quadrupole
pattern in Fig. 2(d) is not symmetric like Fig. 3(e). The asymmetry is because
the trajectory of the electron is not symmetric as the electron spends more
time in the $x>0$ region relative to $x<0$ [Fig. 2(a)].

Figure 4:  The frequency spectrum of the THz wave [Eq. (5)] is shown for
different plasma frequencies, panel (a), different electron-ion collision
frequencies, panel (b), different pulse durations, panel (c), and for
different pulse central wavelengths, panel (d).

This analytical model allows us to explain the THz radiation emission in PIC
simulations. (i) We show that the current source of the THz emission
originates from the electrons which are trapped between the double layers.
(ii) The current source is parallel to the incident laser polarization, and
consequently, the THz radiation is polarized like the incident laser
polarization. (iii) The THz radiation shows a much higher signal along the
angles corresponding to the forward direction ($0^{\circ}<\theta\leq
90^{\circ}$) than for the backward direction ($90^{\circ}<\theta\leq
180^{\circ}$). This is due to the coherence of the phases of the dipole
moments induced along the plasma rod [Eq. (7)].

It would be of interest to see what parameters affect the THz radiation. The
THz radiation forms due to the oscillating dipole moments in the plasma.
Hence, the peak frequency of the THz spectrum is determined by plasma
frequency ($\varpi^{2}=\Omega_{\rm p}^{2}-\Gamma^{2}$) as shown in Fig. 4(a).
A higher plasma frequency will cause a greater radiation force and increases
the energy of THz radiation. The electron-ion collision frequency $\Gamma$ is
one of the important factors affecting the THz spectrum. Increasing the
collision time slows down the thermal equilibrium between the electrons and
ions and leads to a longer-lasting ambipolar electric field and a broader THz
spectrum as shown in Fig. 4(b).

The THz wave amplitude is proportional to the radiation force driven
${\mathscr{E}_{0}}$ field as expressed in Eq. (5). To have an estimate of this
amplitude, let us suppose a pulse intensity given by
$I=I_{0}/\sqrt{\pi}\exp\left(-{r^{2}}/{w_{0}^{2}}\right)\exp\left(-{t^{2}}/{T^{2}}\right)$,
where $T$ is the duration of the pulse, and $w_{0}$ the waist of the pulse.
Under the equilibrium between radiation and space charge forces, the THz wave
amplitude reads ${\mathscr{E}_{0}}={4E_{\rm
p}^{2}}/{(ec\omega_{0}^{2}T^{3})}$, where the pulse energy is $E_{\rm
p}=I_{0}{\pi w_{0}^{2}T}$. Hence, reducing the pulse duration while keeping
the pulse energy constant strongly increases the radiated THz energy [Fig.
4(c)]. This is due to the higher peak intensity of the incident field.
Moreover, increasing the laser wavelength enforces the exerted radiation force
on electrons during a laser cycle [Eq. (11)]. It leads to a stronger net
electron current and amplification of the THz radiation [Fig. 4(d)].

##  IV Discussion

In this work, we have extended our study of femtosecond Bessel beam-induced
plasmas inside the dielectrics. A single-shot Bessel beam can generate a high
aspect ratio over-critical plasma inside the dielectric.[37] The generated
plasma offers a promising medium for the THz radiation due to the current hot
electrons driven by the resonance absorption.

Based on an analytical approach, we derived the current source, the electric
field components, and the angular distribution of THz radiation. The
analytical derivation reproduces the main characteristics of the THz radiation
calculated using the radiation diagnostic of PIC simulation. Under the linear
mode conversion, the radiation force of the resonantly driven plasma waves
kicks the electrons from the critical surfaces. Due to the different mobility
of the plasma species, charge separations known as double layers, and
consequently, ambipolar electric fields form at the plasma surfaces. Most of
the ejected electrons from the critical surfaces trap in the potentials of the
ambipolar electric fields at plasma edges. The trapped electrons oscillate
with a period of around 130 fs and radiate in the THz frequency domain.

Although in this work we have examined the over-critical plasma, the presented
study is valid for the sub-critical plasma, because the radiation force of an
intense laser ($\gtrsim 10^{18}$ W/cm2) can also induce the quasi-static
fields and the associated THz radiation. The second-harmonic part of the
ambipolar field offers an experimental diagnostic for the detection of THz
radiation. Its pattern at the far-field (a central spot) differs from the one
generated at the critical surfaces (two lobes parallel with the incident laser
polarization discussed in Paper II[39]).

To estimate power radiated within the THz range, equating the radiation force
with the force due to the space charge field generated from electron-ion
separation gives an acceleration $a={e{\mathscr{E}_{0}}}/{m}={4E_{\rm
p}^{2}}/{(mc\omega_{0}^{2}T^{3})}$, and the power using the Larmor formula[44]
$P=2e^{2}a^{2}/{3c^{3}}$, is $P={32e^{2}E_{\rm
p}^{2}}/{(3m^{2}c^{5}\omega_{0}^{4}T^{6})}$, or $P_{\rm W}\approx
10^{8}\left[E_{\rm p}^{\rm{\mu
J}}(\lambda_{0}^{\mu\rm{m}})^{2}/(T^{\rm{fs}})^{3}\right]^{2}$.

Assuming a microjoule laser pulse with a duration of 100 fs at 800 nm
wavelength, the laser to THz efficiency is predicted to be about $\sim
10^{-8}$. This value appears to be very small compared with the THz efficiency
of $\sim 10^{-3}-10^{-6}$ for femtosecond pulses with the energy of $\sim
10^{-3}-1$ Joules.[48, 19, 23] Unlike our works, the absorption process for
interactions of $10^{-3}-1$ Joules class lasers with solids relies on the
Brunel mechanism[49, 50] and THz radiation generation is due to the surface
currents[48, 18] or highly relativistic particles passing through the
different dielectrics, the so-called transition radiation.[23] The THz energy
might be improved using the Bessel beams. The long plasma (recently, we
reached cm-scale over-critical plasmas inside dielectrics)[51] created by
Bessel beams yields a longer double layer at the plasma surface. A longer
double layer traps the ejected hot electrons from the critical surface on a
longer distance, for a longer time which results in a longer THz pulse.

Assuming that individual electrons radiate incoherently, we might estimate the
THz intensity and conversion efficiency in the presented PIC simulation. As
reported in Paper I[38], the high-energy electrons represent around 4% of the
electrons in the simulation ($\sim 10^{9}$). Considering the THz intensity for
a single electron of $~{}0.04$ W/cm2 [Fig. 2(f)], the radiated THz intensity
amounts to about $~{}10^{6}$ W/cm2, corresponding to a conversion efficiency
of $10^{-8}$, in agreement with the above predicted efficiency.

We require a picosecond timescale to observe the complete process of the THz
wave generation (for example, a THz wave at 0.5 THz corresponds to a timescale
of 2 ps). However, the numerical heating appearing for high-density plasmas in
several picoseconds imposes a limitation on the maximum duration of the PIC
simulations.[41] For this reason, we did not run our simulations beyond 320
fs. The THz radiation corresponds to a frequency around 30 rad/s. Another
challenge is calculating the radiation integral using the whole set of
electrons in the PIC simulation. This requires the implementation of the
radiation integral in the MPI-based parallel PIC codes, as done in Refs.[52,
53]

###### Acknowledgements.

We thank the EPOCH support team for help https://cfsa-pmw.warwick.ac.uk. The
authors acknowledge the financial supports of: European Research Council (ERC)
682032-PULSAR, Region Bourgogne-Franche-Comte and Agence Nationale de la
Recherche (EQUIPEX+ SMARTLIGHT platform ANR-21-ESRE-0040), Labex ACTION
ANR-11-LABX-0001-01, I-SITE BFC project (contract ANR-15-IDEX-0003), and the
EIPHI Graduate School ANR-17-EURE-0002. This work was granted access to the
PRACE HPC resources MARCONI-KNL, MARCONI-M100, and GALILEO at CINECA,
Casalecchio di Reno, Italy, under the Project ”PULSARPIC” (PRA19_4980), PRACE
HPC resource Joliot-Curie Rome at TGCC, CEA, France under the Project
”PULSARPIC” (RA5614), HPC resource Joliot-Curie Rome/SKL/KNL at TGCC, CEA,
France under the projects A0070511001 and A0090511001, and Mésocentre de
Calcul de Franche-Comté.

##  Appendix A Radiation force density

The radiation force per unit volume, force density $\mathbf{f}_{\rm RF}$, on
the free electrons can be written:[54]

${\mathbf{f}_{\rm
RF}}=\frac{\epsilon-1}{8\pi}\bm{\nabla}E^{2}+\frac{\epsilon-1}{4\pi
c}{\partial_{\rm t}}({\mathbf{E}}\times{\mathbf{B}})$ (10)

Usually, the average values of the force density during one period of the
laser wave are considered. This is because the time envelope of the laser wave
is much slower in comparison with the frequency of the laser wave. Hence, one
can neglect the time average of the Poynting term, the last term in Eq. (10).
Let us consider the monochromatic solutions of the wave equation
${\mathbf{E}}={\mathbf{E}}_{\rm s}(\mathbf{r})\cos(\omega_{0}t)$ where
${\mathbf{E}}_{\rm s}(\mathbf{r})$ includes the field’s spatial dependence.
The radiation force density then reads

$\begin{split}\mathbf{f}_{\rm{RF}}=&-\frac{\omega_{\rm{pe}}^{2}}{8\pi\omega_{0}^{2}}\cos^{2}(\omega_{0}t)\bm{\nabla}{E}_{\rm
s}^{2}\\\
=&-\frac{\omega_{\rm{pe}}^{2}}{16\pi\omega_{0}^{2}}\bm{\nabla}{E}_{\rm
s}^{2}-\frac{\omega_{\rm{pe}}^{2}}{16\pi\omega_{0}^{2}}\cos(2\omega_{0}t)\bm{\nabla}{E}_{\rm
s}^{2}\\\
=&-\frac{\omega_{\rm{pe}}^{2}}{\omega_{0}^{2}}\bm{\nabla}\left\langle\frac{{E}^{2}}{8\pi}\right\rangle-\frac{\omega_{\rm{pe}}^{2}}{\omega_{0}^{2}}\cos(2\omega_{0}t)\bm{\nabla}\left\langle\frac{{E}^{2}}{8\pi}\right\rangle\end{split}$
(11)

We have used $\left\langle E^{2}\right\rangle={E_{\rm s}^{2}}/{2}$ in Eq.
(11).

##  Appendix B Ambipolar electric field of double layer

Following Refs.,[55, 46, 47] we use the two-fluid plasma equations for
continuity and momentum to derive an analytical solution for the ambipolar
electric field of the double layer. The continuity equations read

$\displaystyle\partial_{\rm t}\left(n_{\rm{e}}m_{\rm{e}}\right)+\partial_{\rm
x}\left(n_{\rm{e}}m_{\rm{e}}v_{\rm{e}}\right)=$ $\displaystyle 0$ (12a)
$\displaystyle\partial_{\rm t}\left(n_{\rm{i}}m_{\rm{i}}\right)+\partial_{\rm
x}\left(n_{\rm{i}}m_{\rm{i}}v_{\rm{i}}\right)=$ $\displaystyle 0$ (12b)

where indexes $e$ and $i$ refer to electrons and ions, respectively. The
equations for conservation of momentum read:

$\displaystyle\begin{split}\partial_{\rm
t}\left(n_{\rm{e}}m_{\rm{e}}v_{\rm{e}}\right)=&-\partial_{\rm
x}\left(n_{\rm{e}}m_{\rm{e}}v_{\rm{e}}^{2}\right)-{\partial_{\rm
x}P_{\rm{e}}}-en_{\rm{e}}E_{\rm a}\\\ &-n_{\rm{e}}m_{\rm{e}}\nu_{\rm
ei}\left(v_{\rm{e}}-v_{\rm{i}}\right)+f_{\rm{RF}}\end{split}$ (13a)
$\displaystyle\begin{split}\partial_{\rm
t}\left(n_{\rm{i}}m_{\rm{i}}v_{\rm{i}}\right)=&-\partial_{\rm
x}\left(n_{\rm{i}}m_{\rm{i}}v_{\rm{i}}^{2}\right)-{\partial_{\rm
x}P_{\rm{i}}}+en_{\rm{i}}ZE_{\rm a}\\\ &+n_{\rm{e}}m_{\rm{e}}\nu_{\rm
ei}\left(v_{\rm{e}}-v_{\rm{i}}\right)\end{split}$ (13b)

In Eq. (13a), the radiation force density is given by Eq. (11). We have
neglected the radiation force on the ions ${Zm_{\rm e}}/{m_{\rm
i}}f_{\rm{RF}}$ in Eq. (13b).

The Gauss law for the electric field $E_{\rm a}$ reads:

${\partial_{\rm x}}E_{\rm a}=-4\pi e\left(n_{\rm e}-Zn_{\rm i}\right)$ (14)

Taking the time derivative of the Gauss law, using the equations of continuity
in Eqs. (12), and spatial integration gives:

$\partial_{\rm t}E_{\rm a}=4\pi e\left(n_{\rm{e}}v_{\rm{e}}-Zn_{\rm{i}}v_{\rm
i}\right)$ (15)

The second derivative in time results in:

${\partial_{\rm t}^{2}}E_{\rm a}=4\pi e\left[{\partial_{\rm t}}\left(n_{\rm
e}v_{\rm e}\right)-Z{\partial_{\rm t}}\left(n_{\rm i}v_{\rm i}\right)\right]$
(16)

Substituting from the equations of momentum in Eqs. (13) results in

$\begin{split}\frac{1}{4\pi e}{\partial_{\rm t}^{2}}E_{\rm a}=-&{\partial_{\rm
x}\left(n_{\rm e}v_{\rm e}^{2}\right)}-\frac{1}{m_{\rm e}}{\partial_{\rm
x}P_{x}}-\frac{en_{\rm e}E_{\rm a}}{m_{\rm e}}\\\ +&\nu_{\rm ei}n_{\rm
e}\left(v_{\rm i}-v_{\rm e}\right)+\frac{f_{\rm{RF}}}{m_{\rm e}}\\\
+Z&{\partial_{\rm x}\left(n_{\rm i}v_{\rm i}^{2}\right)}+\frac{Z}{m_{\rm
i}}{\partial_{\rm x}P_{\rm i}}-\frac{Z^{2}en_{\rm i}E_{\rm a}}{m_{\rm i}}\\\
+Z&\nu_{\rm ei}n_{\rm e}\left(v_{i}-v_{e}\right)\frac{m_{\rm e}}{m_{\rm
i}}\end{split}$ (17)

The rearrangements of the terms result in the following differential equation
that described a damped oscillator subjected to an external force
(inhomogeneous second-order differential equation).

${\partial_{\rm t}^{2}E_{\rm a}}+2\Gamma{\partial_{\rm t}E_{\rm
a}}+\Omega_{\rm p}^{2}E_{\rm a}=\Omega_{\rm
p}^{2}\left[\mathscr{E}_{0}+\mathscr{E}_{2}\cos(2\omega_{0}t)\right]$ (18)

where

$\displaystyle\Gamma=$ $\displaystyle\frac{\nu_{\rm
ei}}{2}\left(1+\frac{Zm_{\rm e}}{m_{\rm i}}\right)$ (19a)
$\displaystyle\Omega_{\rm p}^{2}=$
$\displaystyle\omega_{\rm{pe}}^{2}\left(1+\frac{Zm_{\rm e}}{m_{\rm i}}\right)$
(19b) $\displaystyle\begin{split}\mathscr{E}_{0}=&\frac{4\pi e}{\Omega_{\rm
p}^{2}}\left[{\partial_{\rm x}}\left(Z\frac{P_{\rm i}}{m_{\rm i}}-\frac{P_{\rm
e}}{m_{\rm e}}+Zn_{\rm i}v_{\rm i}^{2}-n_{\rm e}v_{\rm e}^{2}\right)\right]\\\
&-\frac{4\pi e}{m_{\rm e}{\Omega_{\rm
p}^{2}}}\frac{\omega_{\rm{pe}}^{2}}{\omega_{0}^{2}}\partial_{\rm
x}\left\langle\frac{{E}^{2}}{8\pi}\right\rangle\end{split}$ (19c)
$\displaystyle\mathscr{E}_{2}=$ $\displaystyle-\frac{4\pi e}{m_{\rm
e}{\Omega_{\rm
p}^{2}}}\frac{\omega_{\rm{pe}}^{2}}{\omega_{0}^{2}}\partial_{\rm
x}\left\langle\frac{{E}^{2}}{8\pi}\right\rangle$ (19d)

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