# Enhanced Terahertz Emission from the Wakefield of CO2 Laser-Created Plasma

Srimanta Maity<EMAIL_ADDRESS>ELI Beamlines Facility, The Extreme
Light Infrastructure ERIC, Za Radnicí 835, 25241 Dolní Břežany, Czech Republic
Garima Arora<EMAIL_ADDRESS>Institute of Plasma Physics of the Czech
Academy of Sciences, 18200 Prague, Czech Republic

###### Abstract

High-field terahertz (THz) pulse generation is investigated through the
interaction of an intense single-color CO2 laser pulse with helium (He) gas
targets. Employing multi-dimensional Particle-In-Cell (PIC) simulations, this
study reveals a substantial enhancement in THz generation efficiency, even
with a single-color laser pulse interacting with gas targets in the self-
modulated-laser-wakefield (SMLWF) regime. Our study demonstrates that in the
presence of photoionization, a synergistic interplay of laser self-modulation,
self-focusing, and local pump depletion leads to the generation of robust THz
pulses polarized parallel to the laser electric field. The dependence of THz
generation efficiency on target density and laser pulse duration has been
investigated. Our study identifies a favourable parametric regime for
producing THz fields with amplitudes reaching hundreds of GV/m, surpassing
those reported in previous studies.

##  I Introduction

Terahertz (THz) radiation, especially in the frequency gap of 1-10 THz and
high field strength, i.e., Gigavolt-per-Meter ( GV/m), has gained wide
interest due to many promising applications, e.g., THz spectroscopyvTonouchi
(2007); Kampfrath  _et al._ (2013), medical imaging Chan  _et al._ (2007),
security screening Kemp  _et al._ (2003), THz-triggered chemistry LaRue  _et
al._ (2015), and biology Alexandrov  _et al._ (2013). The THz spectrum was
earlier produced from photonics, such as black body radiation, using
electronics and oscillators Ferguson and Zhang (2002). However, these sources
only gave nanowatt (nW) power, and their power significantly decreased in this
particular gap regime, which even poses more difficulty in getting high power.
The high-power THz sources are produced using ultrafast laser-pumped crystals
Shalaby and Hauri (2015); Vicario  _et al._ (2014). However, they are limited
to lower power and bandwidth due to the damaging threshold of the crystal at
high power intensity. Quantum cascade lasers Köhler  _et al._ (2002), photonic
crystal lasers Chassagneux  _et al._ (2009), and optically pumped THz gas
lasers have also been developed, but they produce tens of mW power. The chase
of high peak power has made the technology shift to large-scale accelerators
Wu  _et al._ (2013); the THz free electron lasers (FEL) Ramian (1992) produce
Megawatt (MW) peak power, and the THz from large-scale accelerators can
provide GigaWatt (GW) peak powers with GV/m field strength. Nowadays, a new
mechanism of THz generation with high power and field strength in the context
of intense laser-plasma interaction using a table-top setup has been getting a
lot of attention Cook and Hochstrasser (2000); Clerici  _et al._ (2013);
Andreeva  _et al._ (2016); Kim  _et al._ (2008).

A table-top laser-matter interaction is routinely used for electron
acceleration Tajima and Dawson (1979); Joshi (2007); Esarey  _et al._ (2009);
Wen  _et al._ (2020); Shen  _et al._ (2022); Zhu  _et al._ (2021); Maity  _et
al._ (2024), ion acceleration Macchi  _et al._ (2013); Daido  _et al._ (2012),
attosecond pulse generation Zhang  _et al._ (2021), higher harmonic generation
Ganeev (2007); Ganeev  _et al._ (2009), inertial confinement fusion Hu  _et
al._ (2010); Zylstra  _et al._ (2021); Igumenshchev  _et al._ (2023), and so
on. The THz emission can also be produced in table-top laser-matter
interactions when an ultrafast laser pulse interacts with a gas target, even
at moderate intensities. Hamster et al. Hamster  _et al._ (1993) observed the
emission at plasma frequency, which is in the THz regime when the fast laser
pulse interacts with He gas. The first demonstration of efficient generation
of THz pulses was carried out by Cookes et al. Cook and Hochstrasser (2000) by
sending two laser pulses, combining its fundamental frequency with its second
harmonic (called two color scheme) in the air. They explained the mechanism of
THz generation by four-wave rectification. After that, a few groups reported
the THz generation by laser-air interaction using this two-color scheme. The
mechanism of THz generation in this scheme is based on the photoionized
induced radiation (PIR) proposed by Kim et al. Kim  _et al._ (2007) in which
the symmetry of positive and negative half-cycles of the fundamental pump wave
is broken, generating a net transverse drift current in each ionization
instants. Another THz generation approach utilizes laser wakefield excitation
in an inhomogeneous plasma. This mechanism is based on the mode conversion
from the laser wakefields excited in plasma into electromagnetic waves (THz
radiation) Sheng  _et al._ (2005). Recently, this scheme has been verified
experimentally producing millijoule THz radiation Wang  _et al._ (2024). Chen
et. al. Chen and Pukhov (2015) proposed a new method to generate GV/cm THz
field using a single-color ultra-intense laser interaction (i.e., relativistic
effects) with near-critical density plasmas. They have shown that due to local
pump depletion Decker  _et al._ (1996), the front of an initially Gaussian
laser pulse is gradually etched and loses its symmetricity. However, this high
field THz suffers very low transmitted efficiency due to absorption from
plasma to vacuum boundary. Various studies proposed that an external
transverse or longitudinal magnetic field can enhance the transmission
efficiency of generated THz emission produced by the intense laser Wang  _et
al._ (2015); Tailliez  _et al._ (2022).

The abundance of Ti-Saphhire lasers emitting central wavelength 800 nm have
been utilized so far for generating THz radiation, and very few studies have
been performed for mid and far-infrared wavelengths. Nevertheless, experiments
have been conducted that show an increase in THz conversion efficiency using
longer wavelengths Koulouklidis  _et al._ (2020); Clerici  _et al._ (2013);
Nguyen  _et al._ (2018). Wang et. al. Wang  _et al._ (2011) have theoretically
demonstrated that lasers of longer wavelengths can produce stronger THz
radiation. Dechard et al. Déchard  _et al._ (2019) studied the wavelength
dependence on THz pulse generation by ultraintense two-color laser fields
starting from near to far infrared. They have shown that long wavelength (10.6
$\mu$m) significantly enhanced the plasma wakefield, generating tens of GV/m
THz radiation in the laser-polarized direction through the photon deceleration
mechanism, which fundamentally differs from the well-known PIR mechanism in
the two-color scheme for short-wavelength pulses.

Our present study investigates THz generation through laser-matter
interactions, employing a system configuration distinct from prior reports.
Here, we have shown the THz generation using a single-color CO2 laser pulse
interacting with helium gas jets and pre-formed plasmas in the self-modulated
laser wakefield (SMLWF) regime Andreev  _et al._ (1992); Krall  _et al._
(1993); Esarey  _et al._ (1994); Bulanov  _et al._ (1995); Le Blanc  _et al._
(1997). The CO2 laser Polyanskiy  _et al._ (2015, 2020); Panagiotopoulos  _et
al._ (2020), with a central wavelength in the range 9.6-10.6 $\mu$m, has
demonstrated its utility in driving larger wakefields while requiring less
power compared to lasers with central wavelengths 0.8-1.0 $\mu$m Kumar  _et
al._ (2019); Brunetti  _et al._ (2022); Pogorelsky  _et al._ (2016a, b).
Additionally, a CO2 laser pulse can also be used to explore the magnetized
regime of laser-matter interaction Maity  _et al._ (2021, 2022); Juneja  _et
al._ (2023); Dhalia  _et al._ (2023); Vashistha  _et al._ (2023), as the
requirement of an external magnetic field in this case is reduced by at least
one order of magnitude. This study employs a CO2 laser pulse to bolster THz
generation efficiency within a table-top laser-matter interaction setup. Our
investigation revealed that a combined effect of photoionization, self-
modulation, and local pump depletion drives THz radiation generation. Notably,
we have observed THz fields with amplitudes in the hundreds of GV/m range
along the laser polarization direction, significantly higher than those in the
previously reported studies. In our study, multi-dimensional PIC simulations
have been carried out considering a single-color CO2 laser pulse with a
central wavelength around 10.6 $\mu$m. We have observed a strong modulation of
the laser pulse while propagating through the medium via forward Raman
scattering instability Estabrook  _et al._ (1980); Joshi  _et al._ (1981);
Mori  _et al._ (1994); Fisher and Tajima (1996); Najmudin  _et al._ (2000),
generating Stokes and anti-Stokes lines in the optical spectrum. Additionally,
electromagnetic radiations of frequencies between 1-10 THz with polarization,
the same as the incident laser pulse, have been generated. These THz signals
are observed to be transmitted in the vacuum through the right plasma-vacuum
boundary along with the transmitted laser pulse. The dependence of THz
generation efficiency on the target density and laser pulse duration for long
pump wavelength cases, which has not been explored in previous studies, has
been investigated.

This paper is structured as follows. Section II discusses the details of the
simulation performed in this study. Section III presents the simulation
results and relevant discussion, focusing on generating THz radiation and its
underlying mechanisms. Finally, Section IV summarizes our research findings,
highlighting some conclusive remarks.

##  II Simulation setup

Figure 1: A schematic of the simulation setup has been shown. A CO2 laser
pulse with 10.6 $\mu$m central wavelength is incident from the simulation
box’s left-boundary ($x=0$) on a helium (He) gas target. The He gas target has
a trapezoidal density profile along the laser propagation direction, i.e.,
$\hat{x}$ with a 400-$\mu$m-long plateau (in between the dashed vertical lines
at $x=200$ $\mu$m and 600 $\mu$m) and 25 $\mu$m ramps. After the interaction,
a terahertz (THz) signal is observed to be transmitted through the right
plasma-vacuum boundary, propagating in the forward direction along with the
transmitted laser pulse. Table 1: Simulation parameters

Laser wavelength ($\lambda_{0}$) |  Laser spot size (FWHM) ($w_{fwhm}$) |  Laser pulse duration (FWHM) ($\tau_{fwhm}$) |  Peak value of normalized vector potential ($a_{0}$)
---|---|---|---
10.6 $\mu$m | 50 $\mu$m | 100-500 fs | 5.0
Laser frequency ($\nu_{0}$) |  Laser peak power ($P_{0}$) | Laser peak intensity ($I_{0}$) |  Laser peak energy ($E_{0}$)
$28.3$ THz |  $4.3$ TW |  $3.0\times 10^{17}$ W/cm2 |  $\sim$ 0.4-2.1 J
Helium density in the plateau ($n_{He}$) |  Critical density of plasma electrons ($n_{c}$) |  Plasma frequency ($\nu_{pe}$) corresponding to $n_{He}$ | $\nu_{0}/\nu_{pe}$
$(0.05-0.5)\times 10^{18}$ cm-3 |  $\sim$ $1.0\times 10^{19}$ cm-3 |  $\sim$ 3-9 THz |  $\sim$ 10-3

A set of 2D3V (2D in space and 3D in velocity) PIC simulations has been
performed using a fully relativistic, massively parallelized, open source PIC
code, EPOCH Arber  _et al._ (2015); Bennett  _et al._ (2017). Second-order Yee
scheme Yee (1966) and Boris rotation algorithm Boris  _et al._ (1970) with a
modified leapfrog method are used for field solver and particle pusher,
respectively. A 2D rectangular Cartesian geometry (x-y) has been considered as
a simulation domain. The simulation box is extended from $0$ to
$100\lambda_{0}$ along $\hat{x}$ and from $-50\lambda_{0}$ to $50\lambda_{0}$
along $\hat{y}$. Here, $\lambda_{0}$ represents the laser wavelength. The grid
size is considered to be $\lambda_{0}/30$ $\times$ $\lambda_{0}/30$,
corresponding to the $3000$ number of grid cells in each direction. The
simulation time step is considered $\Delta t=0.58$ fs, which is low enough to
resolve the time scales of all the physical processes of our interest in this
study. To reduce numerical dispersion, Courant number ($C$), following the CFL
(Courant-Friedrichs-Lewy) condition ($C=c\Delta t/\Delta x+c\Delta t/\Delta
y\leq 1$) was chosen to be $C=0.99$, i.e., close to 1.0. Here $c$ defines the
speed of light in vacuum, and $\Delta x$, $\Delta y$ represent the grid size
along $\hat{x}$, $\hat{y}$ direction, respectively. In our simulations,
sixteen simulation particles are initially considered in each cell. Open
boundary conditions are used in all four directions for both the
electromagnetic fields and particles.

The simulation configuration considered in this study has been illustrated by
the schematic in Fig. 1. Helium gas (He), or a preformed plasma channel, is
considered as a target. The target has a trapezoidal density profile along
$\hat{x}$ with a 400-$\mu$m-long plateau and $25$ $\mu$m ramps. The plateau
starts from $x=200$ $\mu$m and extends up to $x=600$ $\mu$m. In EPOCH Arber
_et al._ (2015), different ionization models are included to account for
various modes of the ionization process, e.g., collisional ionization and
field ionization. We have not considered collisions between the simulation
particles, as their effect can be ignored for higher irradiances
($I_{0}\lambda_{0}^{2}\geq 10^{15}$ Wcm-2$\mu$m2) of the laser pulse Gibbon
(2005). However, in our simulations, we have incorporated field ionization
modules, i.e., multi-photon ionization Delone and Krainov (2000) and tunneling
ionization based on the Ammosov-Delone-Krainov (ADK) formula Ammosov  _et al._
(1986) as well as barrier-suppression ionization (BSI) process Krainov (1995).

A single-color Gaussian laser pulse with central wavelength $\lambda_{0}=10.6$
$\mu$m is considered. The laser starts from the simulation box’s left boundary
($x=0$) and propagates along $+\hat{x}$ direction. The laser is polarized
along $\hat{z}$ and initially focused at $x=200$ $\mu$m, i.e., at the
beginning of the target plateau. We have considered a Gaussian profile of the
laser pulse in the transverse plane (along $\hat{y}$) with a Full-Width-Half-
Maximum (FWHM) spot size $w_{fwhm}=50$ $\mu$m at focus. In our simulations,
the laser pulse duration has been varied from $\tau_{fwhm}=100$ to 500 fs. The
initial peak value of normalized vector potential is considered to be
$a_{0}=5.0$ at focus, which corresponds to the laser peak intensity
$I_{0}\approx 3.0\times 10^{17}$ W/cm2 and peak power $P_{0}\approx 4.3$ TW.
The central frequency of the laser pulse is $\nu_{0}=\omega_{0}/2\pi\approx
28$ THz. Thus, the value of critical density of plasma electrons, above which
the laser pulse can no longer penetrate (beyond the skin depth distance) the
target, becomes $n_{c}\approx 1.0\times 10^{19}$ cm-3. Table 1 also provides
all the relevant simulation parameters.

##  III Results and Discussion

In the present study, we have investigated the interaction of a CO2 laser
pulse with a helium (He) gas target in the self-modulated laser wakefield
(SMLWF) regime. As the laser pulse propagates through the neutral He target,
it ionizes the medium and forms a plasma channel. As soon as electrons are
ionized, they start to experience laser ponderomotive force
($F_{p}\propto\nabla{I}/m$) Kruer (1988) and are eventually expelled from the
region of the laser pulse. The generated ions also experience the laser
ponderomotive force in the same direction as the electrons. However, ions
remain stationary in the electron timescale because of their higher inertia.
As a result, large amplitude plasma waves (wakefields) are excited with a
phase velocity approximately equal to the group velocity of the laser pulse
inside the medium. The parametric regime where the longitudinal length of the
laser pulse $c\tau_{fwhm}$ becomes longer than the plasma wavelength
$\lambda_{p}$ is known as the self-modulated laser wakefield (SMLWF) regime.
In the SMLWF regime, even for the small-amplitude density perturbation
originating from the ponderomotive force, the laser pulse encounters a
periodically fluctuating index of refraction within the medium, rather than a
constant one. As a result, the laser pulse envelope gets modulated at plasma
wavelength $\lambda_{p}$ generating beats. The ponderomotive force associated
with this modulated laser pulse resonantly amplifies the plasma wave, leading
to a higher density perturbation, thereby producing an instability known as
self-modulation instability Andreev  _et al._ (1992); Krall  _et al._ (1993);
Esarey  _et al._ (1994). As a signature of this instability, the transmitted
laser spectrum exhibits forward Raman scattered sidebands at
($\omega_{0}+\omega_{pe}$, $k_{0}+k_{p}$ (anti-Stokes line) and
($\omega_{0}-\omega_{pe}$, $k_{0}-k_{p}$) (Stokes line). Here,
$\omega_{0}=2\pi\nu_{0}$ and $k_{0}=2\pi/\lambda_{0}$ denote the fundamental
frequency and wavenumber of the laser pulse, while $\omega_{pe}=2\pi\nu_{pe}$
and $k_{p}=\omega_{pe}/c$ represent the plasma frequency and its associated
wavenumber, respectively. Our present simulation study shows that in addition
to the Raman scattered sidebands, electromagnetic (EM) signals in the
frequency range $1-10$ THz are also generated in the interaction process.
Below, we have discussed the mechanism and parametric dependence of these
generated THz radiation.

Figure 2:  (a)-(b) Distribution of electric field $E_{z}$ in the x-y plane
produced by a single-color CO2 laser pulse with $\lambda_{0}=10.6$ $\mu$m and
$a_{0}=5.0$ for (a) ionizing He (i.e., with photoionization) and (b) pre-
ionized plasma target at $t=3.5$ ps. (c) Fourier spectra of $E_{z}(t)$
evaluated at $x=700$ $\mu$m (i.e., $\sim$100 $\mu$m after the target) for
ionizing He (red circle-marked line) and pre-ionized plasma (blue star-marked
line). Here, two solid black vertical lines locate the peak positions of anti-
Stokes and Stokes sidebands, and a dashed vertical line identifies the THz
peak. (d) Longitudinal electric field $E_{x}$ inside the target for these two
cases at $t=2.0$ ps. In both cases, the plateau electron density is
$n=0.10n_{c}$.

First, we investigate the impact of photoionization in the interaction process
in the SMLWF regime by comparing results from two scenarios. In the first
scenario, a CO2 laser propagates through a neutral helium gas target. In the
second case, we consider the laser interaction with a pre-ionized plasma
having an electron density profile ($n=0.10n_{c}$) identical to that generated
from the photoionizing helium. The simulation results in these two cases are
shown in Fig. 2. The distribution of laser electric field $E_{z}$ transmitted
to the vacuum after propagating through the target is demonstrated in Fig.
2(a), (b) for pre-ionized plasma and neutral He targets, respectively. The 2D
profiles of transmitted $E_{z}$ exhibit distinct differences between these two
cases, implying a modification in the interaction process due to
photoionization. To dig into more details, we have performed Fourier
transformation of the on-axis transmitted laser field ($E_{z}$) and presented
in Fig. 2(c). In both the cases, i.e., pre-ionized plasma and neutral He
targets, in addition to the fundamental frequency at $\nu_{0}\sim 28$ THz, a
strong Stokes sideband at $\nu=\nu_{0}-\nu_{pe}$ ($\sim 20$ THz) appears in
the corresponding Fourier spectrum. However, the anti-Stokes sideband
($\nu=\nu_{0}+\nu_{pe}$) is more apparent in the case of photoionizing He. A
similar observation was also reported by Kumar et al. Kumar  _et al._ (2019).
Moreover, our investigation unveils a notable feature in the Fourier spectrum,
showcasing a prominent peak at a frequency $\nu<10$ THz (indicated by a dashed
vertical line). Importantly, this peak exhibits higher intensity in the
presence of photoionizing He than the pre-ionized target. Notably, this
observation was not addressed in the previous study by Kumar et al. Kumar  _et
al._ (2019), which considered a hydrogen target. The detailed mechanism and
analysis of the THz emission will be explored in subsequent discussions.
However, an understanding of the effect of photoionization in the interaction
process can be elucidated as follows. In the presence of an ionizing front,
the dielectric function of a medium undergoes temporal variation rather than
remaining constant. Mori et al. Mori and Katsouleas (1992) conducted a
theoretical investigation into the laser-driven ponderomotive force within
such a medium. Their findings suggest that when there is a co-moving
ionization front, the ponderomotive force amplifies significantly compared to
a pre-ionized medium. As a result, the wakefield ($E_{x}$) amplitude becomes
higher in the case of photoionizing He compared to the pre-ionized He plasma
target, as illustrated in Fig. 2(d). The stronger wakefield amplifies the
initiation of the forward Raman scattering instability, enhancing the
intensity of the Stokes and anti-Stokes sidebands.

Thus, it has been established that photoionization plays a distinct and
significant role in the interaction process. Moreover, we will demonstrate the
pivotal contribution of photoionization to generating THz radiation. From now
on, our analysis will focus primarily on THz radiation and its parametric
dependencies.

Figure 3:  (a) Terahertz signal ($E_{z}^{THz}$) extracted from the on-axis
($y=0$) $E_{z}(t)$ transmitted to vacuum at $x=700$ $\mu$m after filtering in
the frequency range $\nu<\nu_{0}/2$ for simulations considering ionizing He
with different values of electron density ($n$) at the plateau. (b)
Corresponding Fourier spectra of $E_{z}^{THz}(t)$ for these cases. Insets in
(a) and (b) show the extracted (at $x=700$ $\mu$m) on-axis THz signal and its
Fourier spectrum for pre-ionized plasma target with plateau electron density
$n=0.1n_{c}$, respectively. We have considered the laser pulse duration
$\tau_{fwhm}=300$ fs for all the cases.

The THz signals transmitted into the vacuum ($x=700$ $\mu$m) have been shown
in Fig. 3(a) for different simulations with changing values of plateau
(electron) density $n$. Fig. 3(b) illustrates the corresponding Fourier
spectra. In the insets of Fig. 3(a) and (b), THz signals and corresponding
Fourier spectrum have been shown for the case with pre-ionized plasma with
electron density at the plateau $n=0.10n_{c}$. The THz signals have been
extracted from the transmitted on-axis $E_{z}$ (t) at the location $x=700$
$\mu$m performing inverse Fourier Transform within the frequency window
$\nu\equiv\omega/2\pi<\nu_{0}/2$. It is seen that THz fields with 200 GV/m
maximum amplitude transmit to the vacuum along with the remaining laser field
after the interaction. The amplitude of these THz fields increases with an
increase in target density. For $n\lesssim 0.01n_{c}$, the THz field strength
becomes negligible. It is also seen (from the inset of Fig. 3) that for the
same plateau electron density, THz field strength further enhances when a
neutral medium (e.g., He) is considered instead of a pre-ionized plasma. The
Fourier spectra of these THz radiations illustrated in Fig. 3(b) show peaks
around the corresponding relativistic plasma frequency
$\tilde{\nu}_{pe}\equiv\nu_{pe}/\sqrt{\gamma}<10$ THz, where $\gamma$ defines
the Lorentz factor.

Figure 4:  (a1), (b1), and (c1) Longitudinal momentum $p_{X}$ (color patches)
with on-axis longitudinal electric field $E_{x}$ (black solid curves) at
$t=2.0$ ps. (a2), (b2), and (c2) Transverse momentum $p_{z}$ (color patches)
along with on-axis electric field $E_{z}$ (black solid curves) at $t=2.0$ ps.
For (a1)-(a2) and (b1)-(b2), ionizing He was considered with $n=0.1n_{c}$ and
$0.01n_{c}$, respectively. Whereas, for (c1)-(c2), pre-formed plasma channel
with $n=0.1n_{c}$ was considered. Here, the label $N_{e}$ in the colorbar
represents the number of ionized electrons. We have considered
$\tau_{fwhm}=300$ fs for all the cases.

To comprehensively understand our observations, we delve into the target
during its interaction with the laser pulse. Specifically, we have examined
the phase-space distribution of electrons as well as longitudinal and
transverse electric fields. Figures 4(a1)-4(b1) depict a comparison between
the longitudinal phase-space ($P_{x}$-$x$) distributions of electrons and the
on-axis longitudinal electric field $E_{x}$ at a particular time ($t=2.0$ ps)
for two different plateau densities $n=0.1n_{c}$ and $0.01n_{c}$,
respectively, considering photoionization of helium in both scenarios. In both
cases, the laser pulse duration is considered to be $\tau_{fwhm}=300$ fs. It
is seen that wakefield amplitude and, consequently, the longitudinal momentum
gained by electrons are much higher for higher target density since
$E_{x}\propto\sqrt{n}$. Electrons are also observed being injected into the
accelerating phase of the wakefield for $n=0.1n_{c}$, which will be
accelerated further as the laser propagates through the medium. These
accelerated electrons will radiate x-ray through betatron motion inside the
wakefield Albert  _et al._ (2017). Upon crossing the plasma-vacuum interface,
they will also emit radially polarized coherent transition radiation (CTR) in
the THz band range Leemans  _et al._ (2003); Déchard  _et al._ (2019).
However, our present study mainly focuses on the THz radiation in the laser
polarization direction (i.e., $\hat{z}$).

Transverse momentum ($P_{z}$) of electrons (color patches) and on-axis laser
field $E_{z}$ (black solid curves) have been depicted as a function of x
($t=2.0$ ps) in Fig. 4(a2) and Fig. 4(b2) for these two cases, i.e.,
$n=0.1n_{c}$ and $0.01n_{c}$, respectively. Although the laser intensity
($a_{0}$) was the same for both cases, the peak values of $E_{z}$ are
significantly larger for higher plateau density ($n=0.10n_{c}$). This outcome
is attributed to the combined effect of relativistic self-focusing Sprangle
_et al._ (1987, 1990) and ponderomotive self-channeling Sun  _et al._ (1987);
Kurki-Suonio  _et al._ (1989) of laser pulse inside the target. The
relativistic self-focusing occurs when laser power exceeds a critical power
($P_{c}$), i.e., $P_{0}\gtrsim P_{c}=17\omega_{0}^{2}/\omega_{pe}^{2}$ GW. It
is based on the change of plasma refractive index through the relativistic
mass correction of electrons. However, this mechanism is mainly effective for
a long laser pulse, i.e., $\tau_{fwhm}>\omega_{pe}^{-1}$, as the modification
of the index of refraction by the laser pulse requires a time
$\sim\omega_{pe}^{-1}$. Additionally, for a long laser pulse
($L=c\tau_{fwhm}>\lambda_{p}$), ponderomotive self-channeling enhances the
relativistic self-focusing effect. This is because, as the laser propagates
through the plasma, it pushes away electrons from its path, creating a density
channel. Consequently, the radial gradient (from the laser propagation axis)
of the refractive index becomes negative, leading to the self-focusing of the
laser spot size. Even though both scenarios with $n=0.1n_{c}$ and
$n=0.01n_{c}$ satisfy the condition $P_{0}>P_{c}$, it is noteworthy that only
in the former case does $L>\lambda_{p}$. Consequently, the on-axis laser
electric field $E_{z}$ and thus $P_{z}$, demonstrate significantly higher
values for $n=0.1n_{c}$, as evident in Fig. 4(a2)-(b2). Additionally, for the
case with $n=0.1n_{c}$, as the longitudinal length of the laser pulse
($c\tau_{fwhm}$) is longer compared to the plasma wavelength ($\lambda_{p}$),
the laser pulse gets modulated via self-modulation instability Andreev  _et
al._ (1992), as can be seen from Fig. 4(a2). In the self-modulated regime, the
constant feedback of wake on the pulse, coupled with the self-consistent
evolution of both, results in the self-steepening and self-shortening of the
laser pulse. These combined effects enhance the wakefield amplitude for higher
plateau density. It is to be noticed that all these phenomena occur even in a
preformed plasma medium with higher density (e.g., $n=0.1n_{c}$), as depicted
in Fig. 4(c1)-(c2). An additional contribution from photoionization is that
the wakefield amplitude and, thus, longitudinal momentum ($P_{x}$) gained by
the electrons increases Déchard  _et al._ (2019), as discussed previously and
also can be seen from Fig. 4(a1) $\&$ (c1). Furthermore, at higher densities,
with and without photoionization, the leading edge of the laser envelope
etches back and erodes due to the effect of local pump depletion. Moreover, in
higher density targets where $L>\lambda_{p}$, the laser pulse undergoes self-
modulation instability, resulting in the breakdown of symmetry between the
positive and negative cycles of the pulse. These phenomena are illustrated in
Fig. 4(a2) and (c2). Conversely, these effects are absent in lower-density
targets, allowing the laser envelope to maintain its Gaussian symmetry, as
depicted in Fig. 4(b2).

Figure 5:  (a) On-axis transverse current ($J_{z}$) profile along the laser
propagation direction ($\hat{x}$) at $t=2.0$ ps for different values of target
density with and without photoionization. (b) Distribution of THz field in t-x
plane extracted from $E_{z}(t)$ in the frequency range $\nu<\nu_{0}/2$ for
$n=0.10n_{c}$.

The THz radiation is generated through the transverse drift current modulated
at relativistic plasma frequency. In our present study, a combined effect of
different physical processes produces this transverse drift current. The
transverse current ($J_{z}$) as a function of $x$ has been shown in Fig. 5(a)
at a particular time $t=2.0$ ps. In the case of low-density targets
(illustrated by the green star-marked curve), the generated transverse current
is negligible, confined mainly in the ionization zone, approximately around
$x\sim 500$ $\mu$m. Conversely, with higher density, whether with or without
photoionization, the transverse current $J_{z}$ not only appears within the
ionization zone but also extends to the rear pulse ($x<400$ $\mu$m). Notably,
photoionization (red circle-marked curve) enhances the $J_{z}$. In the case of
preformed plasma where photoionization is absent, the net transverse drift
current is produced due to the breaking of the symmetry of the Gaussian laser
envelope through the self-modulation and local pump depletion effect Chen and
Pukhov (2015); Decker  _et al._ (1996), as demonstrated in Fig. 4(c2). In the
case of neutral helium, besides the effects mentioned above, photoionization
significantly contributes to the generation of transverse drift currents. For
a low-intense, short-wavelength laser pulse, ionization occurs over many laser
cycles around the peak of the envelope. Consequently, the net transverse
momentum generated by ionization is effectively averaged out. However, when
the intensity of the laser is increased, and its wavelength is longer,
photoionization can occur at the front edge within a single cycle or even half
a cycle of the laser, resulting in a net transverse momentum of electrons
associated with each ionization event Wang  _et al._ (2011). At higher target
density, the dynamic interplay between laser self-focusing, self-modulation
instability, and local pump depletion synergistically amplifies the generation
mechanism of transverse drift current associated with photoionization. The
combined effect results in heightened laser intensity, resonant amplification
of density accumulation, and erosion of the laser front edge. Consequently, it
substantially reinforces the underlying process.

The temporal and spatial evolution of THz radiation generation for a
particular case considering photoionizing He target with $n=0.1n_{c}$ has been
traced and illustrated in Fig. 5(b). The laser pulse duration is fixed at
$\tau_{fwhm}=300$ fs. As mentioned before, here also, THz fields are extracted
by employing a low-pass filter on the on-axis $E_{z}(x,t)$ within the
frequency range $\nu<\nu_{0}/2$. It is observed that THz radiation generation
initiates after the laser has propagated through a certain distance ($x\approx
400$ $\mu$m) within the target. The intensity of the resulting THz fields
demonstrates a steady growth as the laser pulse advances through the plateau
region of the target (up to $x=600$ $\mu$m). These observations are consistent
with our previously outlined mechanism for THz generation. The effect of self-
focusing, self-modulation instability, and local pump depletion manifests
notably only after a certain distance of propagation. It becomes more and more
prominent as the laser continues its propagation within the target. Despite
being considerably higher than previous reports, only a fraction of the
generated THz fields transmit through the plasma-vacuum interface. The
transmission efficiency can be controlled by applying an external magnetic
field and adjusting the exit ramp Tailliez  _et al._ (2022). Additionally, as
depicted in Fig. 5(b), the generated THz radiation predominantly propagates
forward alongside the laser pulse. This observation is consistent with the
previously reported studies.

Figure 6:  (a), (b) Extracted THz signal $(E_{z}^{THz})$ from on-axis electric
field $E_{z}$ transmitted to the vacuum (at $x=700$ $\mu$m) and corresponding
Fourier spectra for three different values of laser pulse duration
($\tau_{fwhm}$) with a fixed target (photoionized He electron) plateau density
$n=0.05n_{c}$, respectively. (c), (d) Same information for $n=0.1n_{c}$.

Our study also investigates the impact of laser pulse duration ($\tau_{fwhm}$)
on the THz generation process and its efficiency. Figures 6(a) and (b) depict
the extracted THz field ($E_{z}^{THz}$) and corresponding Fourier spectra at
$x=700$ $\mu$m i.e., right vacuum of the target, for three different values of
$\tau_{fwhm}$ considering photoionizing He target with a fixed electron
density plateau $n=0.05n_{c}$. Similarly, Fig. 6(c) and (d) illustrate the
same for $n=0.1n_{c}$. For lower density target ($n=0.05n_{c}$), THz
generation efficiency and, thus, the amplitude of the transmitted THz field
increases with a decrease in $\tau_{fwhm}$. This effect is a consequence of
the fact that for lower values of $\tau_{fwhm}$, laser ponderomotive force
becomes more robust, which amplifies wakefield $E_{x}$, as shown in Fig. 7(a).
As a result, there is an increase in density accumulation at the leading edge
of the laser pulse for lower $\tau_{fwhm}$. This effect sharpens the pulse’s
front edge through local pump depletion, enhancing the transverse drift
current. Moreover, the sharper envelope associated with lower $\tau_{fwhm}$
intensifies the asymmetry between the positive and negative half cycles for
the photoionization events, consequently leading to a significant increase in
the photoionizing current. It is worth noting that when $n=0.05n_{c}$ and
$\tau_{fwhm}=100$ fs, the longitudinal length of the laser ($L=c\tau_{fwhm}$)
becomes shorter than the plasma wavelength ($\lambda_{p}$). Consequently, the
laser pulse does not undergo self-modulation instability. In this scenario,
the dominant processes for THz generation are solely the photoionizing current
and the etching of the pulse’s leading edge via local pump depletion.
Interestingly, for higher density targets ($n=0.1n_{c}$), the relationship
between the amplitude of the transmitted THz fields and $\tau_{fwhm}$ does not
follow a simple monotonic trend, as demonstrated in Figures 6(c) and (d). In
the case of a higher density target, as the duration of the laser pulse
decreases, the pump depletion length ($L_{pd}\propto\tau_{fwhm}/\omega_{pe}$)
Lu  _et al._ (2007) also diminishes. Consequently, after traveling a certain
distance within the target, the laser pulse loses most of its energy to the
wakefield and accelerated electrons. At a density of $n=0.1n_{c}$ and with a
pulse duration of $\tau_{fwhm}=100$ fs, the pump depletion length is
approximately $L_{pd}\approx 300$ µm, which is less than the plateau length of
the target considered in this study. Thus, for smaller values of
$\tau_{fwhm}$, beyond a certain propagation distance ($\sim L_{pd}$), the
residual energy of the laser pulse becomes insufficient to effectively drive
the wakefield and transverse current, as depicted in Figure 7(b).

Figure 7:  (a), (b) On-axis longitudinal electric field $E_{x}$ at $t=2.0$ ps
with two different values of pulse duration ($\tau_{fwhm}$) for (a)
$n=0.05n_{c}$ and (b) $n=0.1n_{c}$. (c), (d) Kinetic energy distribution of
electrons at $t=2.0$ ps for these cases, respectively.

Our study further demonstrates that despite using the modest laser power
($\sim 4$ TW), electrons undergo self-injection into the wake structure,
leading to significant acceleration to higher energies. The kinetic energy
distributions of electrons at $t=2$ ps with two different values of
$\tau_{fwhm}$ have been shown in 7(c) and (d) for $n=0.05n_{c}$ and
$n=0.1n_{c}$, respectively. For a low-density target ($n=0.05n_{c}$),
electrons gained much higher energy for $\tau_{fwhm}=100$ fs compared to
$\tau_{fwhm}=300$ fs. This outcome results from higher ponderomotive force
associated with the sharp laser profile. Conversely, for a higher density
target ($n=0.1n_{c}$), the maximum energy gained by the electrons is higher
for $\tau_{fwhm}=300$ fs. This effect is attributed to the reduced pump
depletion length associated with lower values of $\tau_{fwhm}$ at high-density
targets, as discussed earlier. These highly energetic electrons will also emit
strong radially polarized THz fields via a coherent transition radiation (CTR)
mechanism Leemans  _et al._ (2003); Déchard  _et al._ (2019) while passing
through the plasma-vacuum boundary.

##  IV Summary and conclusions

This study investigates high-field terahertz (THz) generation through the
interaction of single-color CO2 laser pulses with helium gas targets.
Utilizing multi-dimensional Particle-In-Cell (PIC) simulations, we explore the
influence of photoionization on this process by comparing results with the
preformed plasma condition. Our findings highlight the significant role of
photoionization in the nonlinear laser-matter interaction, particularly in
enhancing THz generation efficiency when employing the CO2 laser pulse. We
demonstrate a strong dependence of THz radiation generation on target density
and laser pulse duration. Through detailed analysis, we disclose the
underlying mechanism of THz radiation generation in our study. Specifically,
the synergistic coupling of photoionization current, self-modulation
instability, and local pump depletion induces a transverse drift current. This
current, modulated at the relativistic plasma frequency, emits terahertz (THz)
signals polarized parallel to the laser electric field within the frequency
range of 3 to 9 THz. Our investigation reveals a favorable parametric regime
in which THz fields with peak amplitudes of 150-200 GV/m have been generated
and transmitted into the vacuum, which is higher than the previous reports by
at least one order of magnitude.

## ACKNOWLEDGMENTS

This work was supported by the Ministry of Education, Youth and Sports of the
Czech Republic through the e-INFRA CZ (ID:90254). The authors also acknowledge
the ELI Beamlines HPC facility for computational resources.

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