# Parametric analysis of electron beam quality in laser wakefield acceleration
based on the truncated ionization injection mechanism

Srimanta Maity∗, Alamgir Mondal, Eugene Vishnyakov, Alexander Molodozhentsev+
ELI Beamlines Facility, The Extreme Light Infrastructure ERIC, Za Radnicí 835,
25241 Dolní Břežany, Czech Republic
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Laser wakefield acceleration (LWFA) in a gas cell target separating injection
and acceleration section has been investigated to produce high-quality
electron beams. A detailed study has been performed on controlling the quality
of accelerated electron beams using a combination of truncated ionization and
density downramp injection mechanisms. For this purpose, extensive two-
dimensional Particle-In-Cell (PIC) simulations have been carried out
considering a gas cell target consisting of a hydrogen and nitrogen mixture in
the first part and pure hydrogen in the second part. Such a configuration can
be realized experimentally using a specially designed capillary setup. Using
the parameters already available in the existing experimental setups, we show
the generation of an electron beam with a peak energy of 500-600 MeV, relative
energy spread less than $5\%$, normalized beam emittance around 1.5 mm-mrad,
and beam charge of 2-5 pC/$\mu$m. Our study reveals that the quality of the
accelerated electron beam can be independently controlled and manipulated
through the beam loading effect by tuning the parameters, e.g., laser focusing
position, nitrogen concentration, and gas target profile. These simulation
results will be useful for future experimental campaigns on LWFA, particularly
at ELI Beamlines.

##  1 Introduction

Laser wakefield acceleration (LWFA)[1, 2] is considered one of the most
promising concepts to revolutionize particle accelerator technology. A typical
laser-plasma accelerator (LPA) [3, 4, 5, 6, 7] can offer acceleration
gradients ($\sim$100 GV/m) more than three orders of magnitude larger than
that achieved in a conventional radio-frequency (rf) accelerator ($\sim$100
MV/m). Thus, LPA can accelerate electrons to high energy (e.g., order of GeV)
over a very short distance (a few centimeters), opening the path for
developing compact electron accelerators.

The concept of particle acceleration using plasma as an accelerating medium
was initially introduced in 1956 in three independent research works carried
out by Budker [8], Veksler [9], and Fainberg [10]. Later in 1979, in a
pioneering theoretical work [1], Tajima and Dawson proposed how an intense
laser pulse can excite large-amplitude plasma wakes and eventually accelerate
electrons to high energy. The first experimental demonstration of electron
acceleration using a laser-plasma system was reported in 1993 in Ref. [11].
Since then, significant progress has been made to experimentally demonstrate a
high acceleration gradient generating high-quality monoenergetic electron beam
using laser-plasma interactions [12, 13, 14, 15, 16, 17, 18]. A breakthrough
was achieved in 2006 [19] experimentally demonstrating GeV electron beams
using a centimeter-scale pre-formed capillary plasma channel. The latest
achievement was reported in 2019 [20] demonstrating a quasi-monoenergetic
electron beam with 7.8 GeV peak energy in a capillary discharge waveguide
using a Petawatt Laser. However, the generation of high-quality electron
beams, compared with the conventional rf accelerator, remains challenging for
a plasma-based accelerator.

A significant effort is being made to improve the quality, e.g., high charge,
low energy spread, small beam emittance, and low divergence of the accelerated
electron beam. The electron beam quality in a laser-plasma accelerator (LPA)
strongly depends on the injection mechanism. Over the past few decades, many
injection schemes, e.g., self-injection [21, 22, 23, 24, 25, 26, 27, 28],
density down-ramp injection [29, 30, 31, 32, 33, 34, 35, 36], colliding pulse
injection [37, 38, 39, 40, 41, 42, 43], and ionization injection [44, 45, 46,
47, 48], have been proposed, investigated, and demonstrated in simulation and
experimental studies. Out of these methods, the ionization-induced injection
mechanism is considered one of the most promising techniques for controlled
injector optimization. This is because it is an easy-to-implement technique to
realize in experiments, and a large set of parameters can be used to
externally control the electron injection and the quality of the accelerated
beam. The ionization injection scheme utilizes the large ionization threshold
of the $K$-shell electrons of a high-$Z$ gas (e.g., nitrogen, oxygen) mixed
with a low-$Z$ gas (e.g., hydrogen, helium) to control the initial phase of
the injected and trapped electrons. The leading edge of the laser pulse is
intense enough to fully ionize the low-$Z$ gas atoms along with the outer-
shells electrons of the high-$Z$ gas atoms. The laser ponderomotive force
pushes these pre-ionized electrons away and excites wakes. However, the
$K$-shell electrons of high-$Z$ gas atoms are only ionized near the peak of
the laser profile and, thus, slip backward relative to the phase velocity of
the wake. Under certain conditions [46], these electrons are trapped and
accelerated further as they move forward with respect to the wake. Thus, using
high-$Z$ gas atoms provides external control over the electron injection,
allowing electron trapping even at lower plasma densities and with lower laser
intensities compared to the self-injection scheme. However, the energy spread
of the accelerated electron beam produced using this scheme is typically very
large because of the continuous injection of electrons until the end of the
mixed gas or until some physical mechanism, e.g., beam loading, pump
depletion, stops injection [49]. On the other hand, the density downramp
injection scheme relies on the decrease in the wake phase velocity and wake
wave breaking that occurs due to the inhomogeneity of the Langmuir frequency
[29]. Thus, this technique can be used for localized injection of electrons in
the wake structure, reducing the energy spread of the accelerated electron
beam. In this study, we have demonstrated that ionization and density downramp
injection mechanisms can be compatible with each other, and their combined
effect can produce high-quality, high-energetic electron beams.

Recent works [50, 51, 52, 53] have demonstrated that ionization-induced
injection length can also be controlled, producing a quasi-monoenergetic
electric beam. For example, using an initially unmatched laser pulse (i.e.,
$k_{p}w_{0}\neq 2\sqrt{a_{0}}$, where $k_{p}$, $w_{0}$, and $a_{0}$ represent
wave number of a plasma wave, laser spot size, and amplitude of the normalized
vector potential of the laser pulse) automatically truncates the injection
process due to the violation of the ionization injection conditions. This
technique has been termed as ’self-truncated ionization injection (STII)’ [54,
55, 56], which utilizes the relativistic self-focusing effect [57, 58, 59] of
the laser pulse. In several simulations [51, 52, 53] and experimental studies
[60, 61, 62], the use of a gas target consisting of a few mm-long mixed gas
(doped with high-$Z$ gas) region followed by a large volume of a pure gas
(low-$Z$) has also been proposed and demonstrated to reduce the injection
length without compromising the acceleration length. The influence of a
chirped laser pulse on the electron trapping and the quality of the electron
beam in an LWFA operating in the bubble regime has been investigated in the
past [63, 64, 65]. An analytical study suggesting the scaling and design
parameters for generating a multi-pico Coulomb and multi-GeV electron beam
from LWFA with a cm-scale gas cell has also been reported recently [66].
However, a systematic study showing a direct dependence of various system
parameters on the accelerated electron beam quality remains unexplored.

Our current work investigates the capabilities to produce high-quality
electron beams in a gas cell target in the framework of laser wakefield
acceleration. For this purpose, a systematic and comprehensive study using
two-dimensional PIC simulations has been carried out. The gas cell target
considered in our study consists of two sections. In the first section, a
truncated ionization-induced injection occurs where a short region of
nitrogen-doped hydrogen is considered. In the second section, the injected
electrons are accelerated in a long plasma plateau formed from pure hydrogen.
The dependence of laser parameters and density profile on the quality of
accelerated electron beams has been explored. In particular, we have
explicitly shown the effect of laser focusing position, nitrogen
concentration, and the initial density profile of the gas target on the
injection process as well as on the properties of the accelerated electron
beam. Our study affirms how these parameters can be used to independently
control and manipulate several beam properties, e.g., beam charge, peak
energy, emittance, and beam divergence.

The remaining part of this paper has been organized as follows. In section 2,
PIC simulation details and simulation setup highlighting the parameters used
in this study have been discussed. A detailed discussion of the observations
obtained from simulations is provided in section 3. The subsections therein
contain the simulation results for various system parameters, e.g., laser
focusing position, concentration of nitrogen, and target density profile. A
summary of our study, including the conclusive remarks, has been provided in
section 4.

##  2 Simulation Setup

A set of two-dimensional (2D) Particle-In-Cell (PIC) simulations has been
performed to investigate the dependence of electron beam properties on various
system parameters in a laser wakefield accelerator (LWFA). For this purpose, a
fully relativistic, massively parallel, open source PIC code, EPOCH 4.18.0
[67, 68] has been used. A 2D simulation box with size $80$-$\mu$m $\times$
$160$-$\mu$m in the x-y plane is considered. The cell size is considered to be
$0.02$-$\mu$m $\times$ $0.08$-$\mu$m, which corresponds to $4000$ and $2000$
number of grid cells in the $x$ and $y$ directions, respectively. We used
eight simulation particles in each cell initially. In EPOCH, the standard
Boris algorithm along with a modified leapfrog method has been used for
particle pusher. The details about this code can be found at the Ref. [67].
The moving window scheme has been utilized in our study so that a relatively
very small simulation box can be used to simulate a large acceleration
distance. The open boundary condition has been used for both electromagnetic
(EM) waves and particles, where EM waves and particles are transmitted through
the boundary and removed from the system. The time interval between the two
steps is chosen following the CFL (Courant-Friedrichs-Lewy) condition
($C=c\Delta t/\Delta x+c\Delta t/\Delta y\leq 1$). Here $c$ is the speed of
light in the vacuum, and the dimensionless number $C$ is called the Courant
number. In our study, we have chosen the value of $C$ to be $C=0.99$ and,
thus, simulation time step $\Delta t=0.052$ fs. To resolve all the physical
processes associated with the wakefield acceleration, the Courant number $C$,
following the CFL condition, should be as close as 1.0, and the grid size
($\Delta x$) along the laser propagation direction should be less than
$\lambda_{L}/30$, where $\lambda_{L}$ represents laser wavelength. In our
simulations, we have taken care of these conditions. A Gaussian laser pulse
with wavelength $\lambda_{L}=800$ nm and a Full-Width-Half-Maximum (FWHM)
pulse duration $\tau_{fwhm}=30$ fs is used in this study. The laser also has a
Gaussian transverse profile (along $y$) and is focused at a $f$ distance
(which will be varied in our study) along $\hat{x}$ from the left boundary of
the simulation box. The Full-Width-Half-Maximum (FWHM) of laser spot size
$w_{fwhm}$ at focus is considered to be 30 $\mu$m, which corresponds to the
width (radius) of the laser spot at focus, $w_{0}\simeq 15$ $\mu$m. The
initial laser energy (at focus) is considered to be $1.5$ J, which corresponds
to the peak power ($P_{0}$) of the laser of $50$ TW and a normalized vector
potential $a_{0}$ around 2.0 at focus. Table 1 also provides all the relevant
simulation parameters.

In our simulation study, we have used a combination of nitrogen (N2) and
hydrogen (H2) gas as a target. The schematic of the simulation configuration
and on-axis density profile of the gas mixture has been depicted in Fig. 1. A
detailed discussion related to the gas target configuration is provided in
Sec. 3.

Table 1: Simulation parameters used in this study.

Wavelength ($\lambda_{L}$) |  Spot size (FWHM) ($w_{fwhm}$) |  Pulse duration (FWHM) ($\tau_{fwhm}$) |  Laser peak power ($P_{0}$)
---|---|---|---
0.8 $\mu$m | 30 $\mu$m | 30 fs |  $50$ TW
Laser frequency ($\omega_{L}$) |  Laser energy ($E_{L}$) |  Peak value of normalized vector potential ($a_{0}$) | Peak intensity $I_{0}$ (W/cm2)
$2.35\times 10^{15}$ Hz | 1.5 J | 2.1 | $1.0\times 10^{19}$
Hydrogen density in the plateau ($n_{Hp}$) |  Plasma frequency ($\omega_{pe}$) corresponding to $n_{Hp}$ |  N2 concentration ($C_{N}$) | $\omega_{L}/\omega_{pe}$
$1.6\times 10^{18}$ cm-3 |  $7.13\times 10^{13}$ Hz | $1-17\%$ | 33

##  3 Simulation Results

Figure 1: (a) A schematic of the ‘dual-stage’ configuration of LWFA considered
in our simulation study has been shown. In the first section of the target
(injection stage), a mixture of nitrogen (N2) and hydrogen (H2) gas has been
considered. In the second section (acceleration stage), only pure hydrogen
(H2) is considered. (b) A schematic of the 1D density profiles of H2 and N2
along $x$ is shown. Here, $n_{Hm}$, $n_{Nm}$ represent the hydrogen and
nitrogen neutral (atomic) density in the first section, respectively and
$n_{Hp}$ represents the (atomic) density of the pure hydrogen in the second
section of the gas cell target. (c) Variations of $n_{Hm}$, $n_{Nm}$, $n_{Hp}$
and corresponding net electron density $n_{e}$ along $x$ at $y=0$ are shown
for a particular case. Here, in this particular case, ($90\%+10\%$) N2-H2
mixture has been used and the values of P1, P2, and P3 are chosen to be $1.02$
mm, $1.12$ mm, and $2.12$ mm, respectively. The black dashed line at a
distance $f$ from the left boundary of the simulation box represents the laser
focusing position.

In this study, the ionization-based injection mechanism and its dependence on
various system parameters of an LWFA have been explored in detail. In
particular, a dual-stage configuration of LWFA has been considered, as
illustrated by the schematic in subplot (a) of Fig. 1. In the first section of
this configuration, a gas mixture of nitrogen (N2) and hydrogen (H2) has been
considered. This section ranges over a shorter distance ($\sim 2$ mm) and has
been termed as Injection stage. As the laser propagates through this stage,
the leading edge of the laser pulse ionizes the neutral hydrogen atoms and the
outer five electrons of neutral nitrogen atoms. Thus, these electrons
immediately start to experience laser ponderomotive force and are pushed out
in both longitudinal and transverse directions to excite plasma wake. However,
two K-shell electrons of each nitrogen atom having higher ionization
potentials are ionized near the peak of the laser pulse profile and injected
into the electric field of the fully formed plasma wake. The second section of
the target, termed as Acceleration stage, covers a larger distance ($\sim 6$
mm) and comprises pure hydrogen (H2) gas. In this section, no further
injection of electrons occurs. This section only provides the accelerating
electric field as the laser pulse propagates through it to further accelerate
the trapped electrons [69] entering from the Injection stage. The gas target
configuration considered in our study can be realized in experiments using a
specially designed single capillary setup with an axial length of around 10-15
mm and a radius of 300-500 $\mu$m. A similar concept was recently implemented
[51, 52, 53] to experimentally demonstrate the generation of high-quality
electron beams.

A schematic of the 1D density profile along $x$ considered in this study has
been demonstrated in Fig. 1(b). Here, $n_{Hm}$ and $n_{Nm}$ represent the
atomic density of H2 and N2 in the first stage (Injection stage),
respectively. The parameter $n_{Hp}$ defines the atomic density of H2 in the
second stage, i.e., Acceleration stage. In our study, we have also
investigated the dependence on the initial density profile in the first stage
by changing the values of $L$ (distance between the positions P3 and P2). The
1D density profile along $x$ for $L=1.0$ mm has been shown in Fig. 1(c), where
we have considered the atomic density of H2 (hence the electron density) in
the plateau region of the second stage to be $1.6\times 10^{18}$ $cm^{-3}$. In
our simulations, we have varied the three system parameters, i.e., laser
focusing position ($f$), $N_{2}$ concentration ($C_{N}$), and initial density
profile (i.e., changing values of $L$). We have analyzed various features of
our observation, focusing on the dependence of electron beam properties on
these system parameters, and presented them in the following subsections.

###  3.1 Effect of Laser Focusing Position

Figure 2: Distributions of electron density in the x-y plane at three
different instants of time with changing values of laser focusing position
($f$) have been illustrated. In subplots (a1) $\omega_{pe}t=430$, (a2)
$\omega_{pe}t=500$, and (a3) $\omega_{pe}t=570$ laser focusing position is
considered to be $f=3.0$ mm. Whereas, in subplots (b1)-(b3) and (c1)-(c3), the
laser focusing positions are considered to be $f=3.3$ mm and $3.7$ mm for the
same instants of time, i.e., $\omega_{pe}t=430$, $500$, and $570$,
respectively. Here, red solid lines in each subplots represent the on-axis
longitudinal electric field ($E_{x}$) for the corresponding cases. Here,
$n_{e0}$ is the electron density in the plateau regime of the second stage. In
these simulations, N2 concentration is chosen to be $C_{N}=10\%$. Figure 3:
Longitudinal phase-space ($P_{x}$-$x$) distributions of electrons for a
particular simulation run with $f=3.5$ mm and $C_{N}=10\%$ at time (a)
$\omega_{pe}t=715$, (b) $\omega_{pe}t=1070$, (c) $\omega_{pe}t=1425$, and (d)
$\omega_{pe}t=1780$ have been shown. The electrons originated from H2 are
represented by green markers. Whereas, electrons ($N$) from N2 are depicted by
pseudo-color symbols from yellow to red as shown by the color-bar. Figure 4:
Properties of electrons injected and trapped in the plasma wake structures
behind the laser pulse as a function of laser propagation distance is shown
for different values of laser focusing position ($f$). In subplots (a) and
(b), the absolute values of charge ($Q_{fwhm}$) located within FWHM of energy
histogram of the trapped electron beam and corresponding normalized beam
emittance ($\epsilon_{n}$) are shown, respectively. Energy peak values
($E_{pk}$) and corresponding energy spread ($\delta E_{fwhm}$) of the beam are
depicted in subplots (c) and (d), respectively. Figure 5: The histograms of
electron beam energy and beam divergence at a particular time
$\omega_{pe}t=2212$ (i.e., laser after exiting from the plasma) for different
simulations with changing values of $f$ are shown in subplots (a) and (b)
respectively. The peak value energy ($E_{pk}$), energy spread ($\delta
E_{fwhm}$) of the beam, net charge ($Q_{b}$), and normalized beam emittance
$\epsilon_{n}$ (considering all the trapped electrons) at the same instant of
time with changing values of $f$ are shown in subplots (c) and (d),
respectively.

To study the effect of laser focusing position on the properties of the
accelerated electron beam, a set of PIC simulations with changing values of
$f$ has been performed. The laser focusing position has been varied from
$f=2.5$ mm to $4.0$ mm. In these simulations, ($10\%+90\%$) N2-H2 mixture has
been used in the first stage (Injection stage) with a plateau density of
atomic hydrogen $n_{Hp}=1.6\times 10^{18}$ $cm^{-3}$ in the second stage
(Acceleration stage). The first stage extends up to $x=2.12$ mm, from where
the plateau regime of the second stage starts. In our study, we have also
simulated the cases with $f<2.5$ mm and $f>4.0$ mm. However, we have observed
that for $f<2.5$ mm, although the net charge of the injected electron bunch
becomes very high, the quality of the accelerated beam is very poor. Whereas,
for $f>4.0$ mm, no significant injection of electrons was observed for our
chosen values of system parameters.

Net electron density ($\rho$) distributions in the $x-y$ plane have been
depicted in Fig. 2 with changing values of $f$ at three particular time
instants. In subplots (a1)-(a3) of Fig. 2, we have considered three different
instants of time, i.e., $\omega_{pe}t=430$, $500$, and $570$ with laser
focusing position fixed at $f=3.0$ mm, respectively. Whereas, for subplots
(b1)-(b3) and (c1)-(c3) of Fig. 2, laser focusing positions are considered to
be $f=3.3$ mm and $3.7$ mm for the same instants of time, respectively. It is
seen that, in all cases, electrons get injected and trapped into the plasma
wake. The injection process starts approximately at $x\sim 1.76$ mm, as can be
seen from subplots (a1)-(c1) of Fig. 2. It is clear that in all the cases,
injections and trapping of electrons occur only within the density down-ramp
regime of the first stage. Thus, in our case, both ionization injection and
density down-ramp mechanisms effectively contribute to the injection and
trapping of electrons. However, the timings of injection and trapping of
electrons get shifted with the increase of $f$, as seen from subplots
(a2)-(c2) Fig. 2. The on-axis longitudinal electric field ($E_{x}$) profile
shown by red solid lines for different values of $f$ at different times also
manifests these phenomena. Fig. 2(a2) shows that the profile of $E_{x}$ gets
modified (around $x=2.065$ mm) for $f=3.0$ mm at this instant of time. This
clearly indicates the beam loading effect [70], which occurs when electrons
get trapped within the first quarter of $E_{x}$. However, at the same time,
for $f=3.3$ mm and $3.7$ mm, no effect of beam loading is observed, as can be
seen from subplots (b2) and (c2) Fig. 2, respectively. Finally, at time
$\omega_{pd}=570$ (Fig. 2(b3)), the effect of beam loading starts to reveal
for the case with $f=3.3$. It is to be noticed that by this time, the laser
pulse, as well as trapped electrons, had already entered the plateau regime
($x>2.12$ mm) of the second stage. However, beam loading does not occur for
the case with $f=3.7$ mm, as shown in Fig. 2(c1)-(c3). The effect of beam
loading on the accelerated electron bunch’s quality will be discussed in
section 3.2.

The longitudinal phase-space ($P_{x}$-$x$) distributions of electrons at
different times of the simulation run with $f=3.5$ mm have been depicted in
subplots (a)-(d) of Fig. 3. Here, green markers represent the electrons
originating from hydrogen, whereas yellow to dark red color symbols are for
electrons ionized from nitrogen neutrals. As mentioned before, no nitrogen gas
is present in the second stage ($x>2.12$ mm). Thus, it is clear that only the
electrons ionized from the nitrogen atoms in the first stage get trapped and
are accelerated further. It also reveals that no self-injection of electrons
originating from H2 occurs during the propagation of the laser through the
second stage. Electrons from H2 only form wake structures and provide the
accelerating field, as shown in the inset of Fig. 3(d).

We have characterized several properties of these trapped electrons as they
travel through the second stage for different simulation setups with changing
values of $f$. We have presented this in subplots (a)-(d) of Fig. 4. In our
simulations, we have observed that the net charge of the trapped electron beam
almost remains constant throughout the propagation. This indicates that the
injected electrons do not get lost during their propagation through the second
stage after being trapped in the wake structure in the first stage. However,
the net charge located within the full width at half maximum (FWHM) of the
energy histogram of the trapped electron beam changes with the laser
propagation distance. In Fig. 4(a), we have depicted the absolute value of
FWHM charge ($Q_{fwhm}$) of the trapped electrons for different values of $f$.
For $f\leq 3.5$ mm, $Q_{fwhm}$ first increases with the propagation distance.
However, after a certain distance, the value of $Q_{fwhm}$ decreases again and
finally becomes constant at a certain value. For $f\geq 3.7$ mm, $Q_{fwhm}$
increases first as the laser propagates and finally gets saturated after a
certain distance.

The normalized beam emittance ($\epsilon_{n}$) as a function of time presented
in Fig. 4(b) shows a strong dependence on the laser focusing position. Here,
we have calculated $\epsilon_{n}$ using the transverse phase-space coordinates
($P_{y}$-$y$) of the particles located within the FWHM of the energy
histogram. For lower values of $f$ ($f<3.5$ mm), beam emittance initially
increases with the propagation distance, attains a maximum value, and
decreases again as the beam propagates further. However, for all the cases,
the peak energy ($E_{pk}$) of the trapped electron beam increases almost
linearly with the laser propagation distance. Finally, it gets saturated at
certain values, as seen in Fig. 4(c). This is consistent with the previously
reported theoretical scaling [71] where the energy gain of an electron beam in
an LWFA is linearly proportional to the accelerating distance within the
dephasing length. The energy gain gets saturated after a certain distance for
all the cases. This corresponds to the time when the laser pulse and trapped
electron beams exit the plateau region of the second stage. It is also seen
that the final saturated peak energy of the beam is higher for the lower value
of the laser focusing position. In Fig. 4(d), the energy spread ($\delta
E_{fwhm}$) measured from the FWHM of energy histogram of the trapped electron
beam is shown for different simulation runs with changing values of $f$. We
have observed that the energy spread does not remain constant, at least up to
a certain time of evolution, and has a strong dependence on the laser focusing
position. For lower values of $f$ ($<3.5$ mm), energy spread increases with
time, attains a maximum value, and decreases again before it finally saturates
at a constant value. However, the final saturated values of $\delta E_{fwhm}$
strongly depend on the laser focusing position. It is interesting to notice
that the evolution of $Q_{fwhm}$, $\epsilon_{n}$, and $\delta E_{fwhm}$ follow
a similar trend revealing a strong inter-correlation. The dependence of
electron beam properties as a function of propagation distance is attributed
to the combined effect of beam loading and evolution of the bubble size due to
the self-focusing of the laser pulse. A detailed discussion has been provided
in section 3.2.

The final beam properties of the trapped electrons after exiting from the
plasma ($x>9.0$ mm) at time $\omega_{pe}t=2212$ have been characterized for
different simulation runs with changing values of $f$ and illustrated in Fig.
5. The energy histogram of the electron beam presented in Fig. 5(a) shows that
for lower values of $f$, electron energy is spread over a larger range with
lower values of maximum energy. Whereas, for higher values of $f$, energy
distribution becomes narrower with higher values of maximum gained energy. The
beam divergence representing the angular spread of the electron beam in the
transverse plane also strongly depends on the laser focusing position, as can
be observed from Fig. 5(b). It is seen that the angular spreading of the beam
in the transverse plane becomes narrower with the increase of $f$. The peak
value of gained energy of the electron beam and energy spreading as a function
of $f$ has been depicted in Fig. 5(c). The peak energy ($E_{pk}$) increases
monotonically with the increase of $f$, as also presented in Fig. 4. Whereas,
the relative energy spread ($\delta E_{fwhm}/E_{pk}$) decreases with an
increase of $f$ and finally it gets saturated beyond a certain value $f$
($f\sim 3.7$ mm). The net absolute charge ($Q_{b}$) and normalized beam
emittance ($\epsilon_{n}$) (calculated by considering all the trapped
electrons) as a function of $f$ is shown in Fig. 5(d), which reveals that both
$Q_{b}$ and $\epsilon_{n}$ decrease with an increase of $f$. It is to be
noticed that the net charge of the beam can also be obtained from the energy
histogram shown in Fig. 5(a). The total number of trapped particles can be
obtained from the energy histogram by calculating the sum of the number of
particles in each bin multiplied by the bin width (4 MeV, in our case). From
this, one can calculate the charge (per unit length as for the 2D slab
geometry) of the accelerated electron beam, which would be the same as
presented in Fig. 5(d).

###  3.2 Effect of Nitrogen Concentration

Figure 6: Electron density ($\rho$) distributions in the x-y plane at time
$\omega_{pe}t=715$ for simulation runs with N2 concentration (a) $C_{N}=7\%$,
(b) $C_{N}=10\%$, (c) $C_{N}=12.5\%$, and (d) $C_{N}=15\%$ are shown. Here,
the red lines in each subplots represent the longitudinal electric field
($E_{x}$) for the corresponding cases. In these simulations, laser focusing
position is kept fixed at $f=3.5$ mm. Figure 7:  Time history of the
properties of electrons injected and trapped in the plasma wake structures
behind the laser pulse is shown for different values of N2 concentration
($C_{N}$). In subplots (a) and (b), the absolute values of charge ($Q_{fwhm}$)
located within the FWHM of the energy distribution function of the trapped
electron beam and corresponding normalized beam emittance ($\epsilon_{n}$) are
shown as a function of time, respectively. Whereas, the time evolution of
energy peak values ($E_{pk}$) and corresponding energy spread ($\delta
E_{fwhm}$) are depicted in subplots (c) and (d), respectively. Figure 8: On-
axis longitudinal electric field $E_{x}$ (red lines) and electron densities
$n_{eH}$ (black lines), $n_{eN}$ (blue lines) originated from hydrogen,
nitrogen neutrals, respectively, are shown at different times of evolution.
Subplots (a1)-(a3) represent the case with nitrogen concentration $C_{N}=5\%$.
Whereas, for subplots (b1)-(b3), $C_{N}=15\%$ was considered. Green dashed
vertical lines define the locations where $E_{x}=0$ inside the first bubble
for each case. Figure 9: Energy histogram and beam divergence at a particular
time $\omega_{pe}t=2212$ for different simulations with changing values of N2
concentration ($C_{N}$) are shown in subplots (a) and (b) respectively. Peak
energy ($E_{pk}$) of electron beam, energy spread ($\delta E_{fwhm}$), beam
charge ($Q_{b}$), and normalized beam emittance ($\epsilon_{n}$), with
changing values of $C_{N}$ are depicted in subplots (c) and (d) at the same
instant of time. Here, $Q_{b}$ and $\epsilon_{n}$ are calculated from the
information of all the trapped electrons.

A set of simulation runs has also been carried out with changing values of N2
concentration ($C_{N}$) to study its effect on the accelerated electron beam
properties. In these simulations, the laser focusing position is fixed at
$f=3.5$ mm for a particular density configuration with $L=1.0$ mm (as shown in
Fig. 1). In these cases also, electrons ionized from N2 are observed to be
injected and trapped inside the wake structures as the laser propagates
through the medium. The net charge density distributions in the x-y plane at a
particular instant of time $\omega_{pe}t=715$ have been shown in subplots
(a)-(d) of Fig. 6 for different values of $C_{N}$. It is seen that an almost
circular (spherical in 3D) bubble regime is created behind the laser pulse for
all the cases. The bubble is a nonlinear stage of a wake structure that forms
when a driver laser pulse is intense enough to expel almost all the electrons
from near the laser pulse. The optimal condition [72] to form a bubble is
$k_{p}R\approx 2\sqrt{a_{0}}$, where $k_{p}=c/\omega_{p}$ is the plasma wave
vector, $c$ is the speed of light in vacuum, $\omega_{p}$ represents the
plasma frequency, $R$ defines the radius of the laser spot size, and $a_{0}$
is the normalized vector potential associated with the driver laser pulse. For
our chosen values of simulation parameters, the value of $a_{0}$ should be
approximately 3.1 to match this condition. Although, in our simulations, we
have initially considered $a_{0}$ to be around 2.0 at the focus, due to the
self-focusing effect inside the plasma, the value of $a_{0}$ reaches close to
3.0, satisfying the condition for the bubble regime to occur. The electrons
are pushed forward and radially by almost the same strength of ponderomotive
force as in our case $c\tau_{fwhm}\approx w_{0}$ incorporating the self-
focusing effect. Consequently, the shape of the bubble becomes almost
circular. The amount of charge trapped inside this wake structure increases
with the increase of $C_{N}$. This is also apparent from the profile of the
on-axis longitudinal electric field ($E_{x}$) shown by the red solid line in
the subplots of Fig. 6. A significant modification in the profile of $E_{x}$
is observed at the locations of trapped particles. This is a clear signature
of the beam loading effect where an electric field is originated from the
space charge of the trapped electrons and modifies the wake electric field
profile. Fig. 6 demonstrates that the beam loading effect, hence the number of
trapped electrons, increases with an increase of $C_{N}$.

The temporal evolution of trapped electron dynamics and its characteristic
properties are presented in Fig. 7 for different simulation runs with changing
values of $C_{N}$. We have observed that the in these cases also, the net
charge ($Q_{b}$) of the trapped electrons remains constant throughout the
evolution. However, the charge ($Q_{fwhm}$) within the FWHM of the energy
distribution function of the trapped electrons is strongly dependent on the
propagation distance, as can be seen from Fig. 7(a). For $C_{N}>7\%$,
$Q_{fwhm}$ first increases and attains a maximum value at a particular
propagation distance. Later, after a certain distance of propagation,
$Q_{fwhm}$ becomes constant. For $C_{N}\leq 7\%$, we have observed that
$Q_{fwhm}$ increases first and then gets saturated after a certain distance of
proportion. The normalized emittance ($\epsilon_{n}$) of the electron beam
also strongly depends on the time of evolution for different values $C_{N}$,
as seen from Fig. 7(b). For $C_{N}>7\%$, the value of $\epsilon_{n}$ first
increases with time, and then after a certain time, it decreases again before
being saturated at constant values. However, for $C_{N}\leq 7\%$, beam
emittance ($\epsilon_{n}$) almost remains constant throughout the evolution.
The peak energy ($E_{pk}$) of the beam defining the energy acquired by the
maximum number of trapped electrons is shown as a function of time in Fig.
7(c) for different values of $C_{N}$. It is seen that $E_{pk}$ increases
monotonically (and almost linearly) with time for all the cases and finally
gets saturated as the electron beams leave the plateau region of the second
stage at $\omega_{pe}t\approx 2100$. However, it is observed that the final
energy an electron beam gains is lower for the higher values of $C_{N}$. This
is a consequence of the beam-loading effect. The amount of electrons injected
and trapped inside the wakefield increases for higher nitrogen concentrations.
Consequently, the accelerating field inside the bubble gets flattened due to
the effect of the space charge electric field of the trapped electrons, as
shown in Fig. 6(d). As a result, the local electric field gradient,
responsible for the acceleration of electrons, decreases for higher values of
$C_{N}$, causing the peak value of final gained energy to be lower. In Fig.
7(d), the energy spread has been shown as a function of time for different
simulation runs with changing values of $C_{N}$. The energy spread ($\delta
E_{fwhm}$) is calculated from the full width at half maxima of the
corresponding energy histogram of the trapped electron beam at different
evolution times. It is interesting to observe that $\delta E_{fwhm}$ strongly
depends on the time of evolution (i.e., propagation distance), and the value
of $C_{N}$ significantly influences the nature of dependence. For example,
with $C_{N}>7\%$, $\delta E_{fwhm}$ initially increases with time, reaches a
maximum value, and after a certain propagation distance, its value gradually
decreases. Whereas, for the simulations with $C_{N}\leq 7\%$, the energy
spread ($\delta E_{fwhm}$) of the trapped electron beams increases up to a
certain time of evolution and finally gets saturated, as can be seen from of
Fig. 7(d). In these simulations also, the evolution of $Q_{fwhm}$,
$\epsilon_{n}$, and $\delta E_{fwhm}$ follow a similar trend.

As mentioned in section 3.1, the beam loading effect determines the nature of
variation of electron beam properties as a function of propagation distance.
The beam loading effect occurs when the space charge force of the accelerated
electron bunch becomes significant enough to modify the structure of a wake
and, consequently, the profile of the accelerating electric field. To
demonstrate this phenomenon, we have evaluated 1D on-axis electron density
($n_{eH}$, $n_{eN}$) as well as the longitudinal electric field ($E_{x}$) at
three different stages of evolution and depicted in Fig. 8. Here, we have
considered two cases with $C_{N}=5\%$ and $15\%$ and illustrated in Fig.
8(a1)-(a3) and Fig. 8(b1)-(b3), respectively. Fig. 8(a1)-(a3) shows that for
$C_{N}=5\%$, the charge of the trapped electron bunch (blue solid line) is not
enough to make any significant modification of the wake structure. In this
case, the size of the wake (i.e., in the first bubble) evolves slightly only
due to the evolution of the laser spot size. Thus, for this case, the electron
beam properties remain unchanged after a certain propagation distance, as
shown in Fig. 7. However, for $C_{N}=15\%$, the charge of the injected
electron bunch becomes high and significantly modifies the bubble size as all
as the profile of accelerating field $E_{x}$, as shown in Fig. 8(b1)-(b3). The
positions of the density ($n_{eH}$) peak of the electrons originating from
hydrogen (shown by black solid lines) indicate that the size of the bubble
increases with the propagation distance. It is interesting to notice that
there is an intermediate regime of propagation (Fig. 8(b2)) where a
significant number of trapped electrons becomes out of phase with respect to
the accelerating electric field (i.e., electrons lie in the regime where
$E_{x}>0$). Thus, in this intermediate regime, energy spread $\delta
E_{fwhm}$, and consequently $Q_{fwhm}$, $\epsilon_{n}$ increases, as shown in
Fig. 7. However, as the laser propagates further, the wake structure and,
consequently, the longitudinal electric field $E_{x}$ gets modified in such a
way that all the trapped electrons get confined again within the accelerating
phase of the field, as illustrated in Fig. 8(b3). Thus, electron beam
properties, i.e., $\delta E_{fwhm}$, $Q_{fwhm}$, and $\epsilon_{n}$ decreases
and become constant after a certain distance of propagation.

In these cases also, we have analyzed the properties of the electron beam at
the final simulation time, i.e., after exiting from the plasma ($x>9.0$ mm)
and shown in subplots (a)-(d) of Fig. 9. The energy histogram of the
accelerated electron beam shown in Fig. 9(a) reveals that a quasi-
monoenergetic beam is produced for all the values of $C_{N}$. It is also seen
that the maximum energy gained by the electrons increases with the decreasing
value of $C_{N}$. The energy distribution of trapped electrons also becomes
narrower with the decrease of $C_{N}$. Fig. 9(b) shows that the divergence of
the electron beam increases with an increase of $C_{N}$. However, beam
divergence remains within $\sim 5$ mrad in all the cases. The peak energy
($E_{pk}$) of the beam decreases, whereas energy spreading ($\delta E_{fwhm}$)
increases with an increase of $C_{N}$, as can be seen from Fig. 9(c). At the
same time, it is observed that the net charge of the beam ($Q_{b}$) as well as
normalized beam emittance ($\epsilon_{n}$) increases with the increase of
$C_{N}$, as shown in Fig. 9(d). For the chosen values of parameters, we have
observed that the $E_{pk}$ varies from 450-650 MeV with $\delta E_{fwhm}$
remains in between 1-10$\%$. The charge of the electron beam varies from
0.5-8.5 pC/$\mu$m in 2D slab geometry (approximately 5-85 pC in 3D if we
assume the transverse width of the beam is 10 $\mu$m, which is a typical beam
width in LWFA). Whereas the values of beam emittance $\epsilon_{n}$ remain
within the range 1.5-5 mm-mrad.

###  3.3 Effect of Density Profile

Figure 10:  The variations initial electron density along $x$ contributed from
H2 and N2 (considering only five electrons from each nitrogen atoms) for four
different simulation configurations with changing values of $L$ are shown.
Here, the value of $L$ has been changed only by changing the position P2,
keeping P1 and P3 fixed. Figure 11:  The properties of electrons injected and
trapped in the plasma wake structures behind the laser pulse as a function of
laser propagation distance have been shown for different initial density
profiles with changing values of $L$. In subplots (a) and (b), the FWHM charge
$Q_{fwhm}$ and corresponding normalized emittance $\epsilon_{n}$ are shown as
a function of propagation distance, respectively. Whereas, the evolution of
$E_{pk}$ and $\delta E_{fwhm}$ are depicted in subplots (c) and (d),
respectively.

Our study also investigated the effect of the initial longitudinal density
profile along the gas cell on the quality of accelerated electron beams. For
this purpose, we have carried out simulations with changing values of $L$ with
a fixed nitrogen concentration $C_{N}=10\%$ and laser focusing position
$f=3.7$ mm. In particular, we have varied the position P2, keeping P1 and P3
at fixed positions, as illustrated in Fig. 1(b). The net electron (associated
with the two and five electrons from each hydrogen and nitrogen atom,
respectively) density ($n_{e}$) profile along $x$ for different values of $L$
has been illustrated in Fig. 10. As expected, the down-ramp gradient of
$n_{e}(x)$ in the first stage ($x<2.12$ mm) increases with the decrease of
$L$. In these cases also, we have analyzed several time-scanned properties of
the trapped and accelerated electron beam and presented them in Fig.
11(a)-(d). Fig. 11(a) reveals that in all the cases, the absolute value of
charge ($Q_{fwhm}$), located within the FWHM of the energy distribution
function of the trapped electrons, increases up to a certain propagation
distance and then gets saturate at particular values. We have observed a
significant increase in electron beam charge with the decrease of $L$, as seen
from Fig. 11(a). These results are consistent with the density down-ramp
injection mechanism, where it has been demonstrated that injection and
trapping of electrons become more efficient with the increase of density
gradient [29, 30, 31]. Moreover, in our case, the increase of beam charge is
also attributed to the fact that the effective amount of N2 becomes higher for
lower values of $L$. The normalized beam emittance ($\epsilon_{n}$) calculated
from the electrons within the FWHM of energy histogram almost remains constant
throughout the propagation, as presented in Fig. 11(b). It is also seen that
$\epsilon_{n}$ has a higher value for the lower values of $L$. This is because
$Q_{fwhm}$ is higher for lower values of $L$, and there is a direct
correlation between $Q_{fwhm}$ and $\epsilon_{n}$.

From Fig. 11(c), it is seen that in these cases also, peak energy ($E_{pk}$)
of the trapped electron beam increases monotonically (and almost linearly)
with time, i.e., with the propagation distance. However, the final saturated
value of $E_{pk}$ (i.e., after exiting from the density plateau in the second
stage) is higher for higher values of $L$. This is also because of the fact
that $Q_{fwhm}$ decreases with the increase of $L$. Thus, the modification of
the accelerating field of the wake through the beam loading effect originating
from the space charge of the trapped electrons becomes less significant for
higher values of $L$. The energy spread ($\delta E_{fwhm}$) of the beam as a
function of propagation distance has been shown in Fig. 11(d) for changing
values of $L$. It has been observed that in all the cases, $\delta E_{fwhm}$
first increases, and after a certain distance of propagation, its value gets
saturated. This behavior also is consistent with the nature of the evolution
of $Q_{fwhm}$ and can be easily understood from the point of view of the space
charge effect of the trapped electrons. It is to be seen that in the case
corresponding to profile4, $Q_{fwhm}$ increases again after 8 mm. This is
consistent with the fact that for this case, $\delta E_{fwhm}$ increases again
after an 8 mm propagation distance, as can be seen from Fig. 11(d). We have
not observed any second injection at the exiting downramp, as the net charge
of the beam remains constant throughout the evolution. However, the exact
cause that triggers the increase of $\delta E_{fwhm}$ at the exit is still
unclear. This may be attributed to the fact that for the case of profile4, the
self-focusing of the laser pulse is higher as it propagates comparatively a
longer distance in the higher density regime. Thus, even after propagating an
8 mm distance, the electric field inside the expanding bubble at the exiting
downramp may affect the beam quality. In this set of simulation runs, for our
chosen values of parameters, we have observed the generation of the electron
beam with beam charge (2d) $Q_{b}$ varies in between 1-5 pC/$\mu$m (which is
approximately 10-50 pC in 3D assuming a typical 10 $\mu$m transverse beam
width) and normalized beam emittance ($\epsilon_{n}$) remains within 1.5-1.7
mm-mrad. The beam’s peak energy $E_{pk}$ varies between 450-600 MeV with the
corresponding relative energy spread 3-5$\%$.

##  4 Summary and Conclusions

This study explores the parametric dependence of the properties of accelerated
electron beams in a laser wakefield acceleration within a gas cell target. In
particular, a systematic and comprehensive study to obtain high-quality
electron beams in an LWFA based on a combination of ionization-induced
injection and density down ramp injection mechanism has been carried out using
Particle-In-Cell (PIC) simulations. The gas target considered in this study
consists of two sections. In the first section, nitrogen-doped hydrogen gas
was used. The first section is responsible for injecting and trapping the
electrons in the wake structure. In the second section, pure hydrogen gas was
considered. The second section provides the accelerating electric field to
accelerate the trapped electrons further. Our study demonstrates the
dependence of accelerated electron beam properties on laser focusing position,
nitrogen concentration, and gas density profile. In particular, it has been
explicitly shown that several important properties of an electron beam, e.g.,
beam charge, peak energy, relative energy spread, and beam emittance, are
strongly influenced by the system parameters mentioned above. It has been
observed that the energy peak value ($E_{pk}$) of the accelerated electron
beam increases with the increase of laser focusing position ($f$). Whereas,
beam charge ($Q_{b}$), normalized beam emittance ($\epsilon_{n}$), and
relative energy spread ($\delta E_{fwhm}/E_{pk}$) decrease with an increase of
$f$. Our simulation study also reveals that $E_{pk}$ decreases with an
increase in nitrogen concentration ($C_{N}$). Whereas, beam charge ($Q_{b}$),
normalized beam emittance ($\epsilon_{n}$), and relative energy spread
($\delta E_{fwhm}/E_{pk}$) increase with the increase of $C_{N}$. The optimal
zones and suitable parametric regimes for beam delivery have been identified
from the time analysis of the trapped electron beam properties. The effect of
beam loading on the properties of accelerated electron beams at various stages
of evolution has been analyzed and discussed. In our study, using the laser
with 50 TW peak power and 30 $\mu$m (FWHM) spot size, an optimum parametric
regime ($f=3.3-3.7$ mm; $C_{N}=5-12\%$) has been identified for which electron
beams with peak energies of 500-600 MeV, relative energy spread less than
$5\%$, normalized beam emittance around 1.5 mm-mrad, and 2D beam charge of 2-5
pC/$\mu$m (approximately 20-50 pC in 3D if we assume the transverse width of
the beam is 10 $\mu$m) have been generated. These results suit nicely for the
requirements of incoherent undulator physics and will be useful for the
related experimental campaigns [73] at the ELI Beamlines facility in the
future.

## Data availability statement

The data cannot be made publicly available upon publication because they are
not available in a format that is sufficiently accessible or reusable by other
researchers. The data that support the findings of this study are available
upon reasonable request from the authors.

## Acknowledgments

This work was supported by the Ministry of Education, Youth and Sports of the
Czech Republic through the e-INFRA CZ (ID:90254). This work was also supported
by the project “Advanced Research using High Intensity Laser produced Photons
and Particles” (ADONIS) (CZ.02.1.01/0.0/0.0/16019/0000789) from European
Regional Development Fund (ERDF). The authors gratefully acknowledge Prof.
Sergei V. Bulanov for his valuable suggestions, which have significantly
improved the quality of the manuscript. The authors thank Dr. Petr Valenta, a
colleague from ELI Beamlines, for the helpful discussion and acknowledge the
ELI Beamlines HPC facility for computational resources.

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