# On the generation of ultra-bright and low energy spread electron beams in
laser wakefield acceleration in a uniform plasma

Xinlu Xu<EMAIL_ADDRESS>State Key Laboratory of Nuclear Physics and
Technology, and Key Laboratory of HEDP of the Ministry of Education, CAPT,
Peking University, Beijing, 100871, China  Beijing Laser Acceleration
Innovation Center, Beijing 100871, China  Thamine N. Dalichaouch  Department
of Physics and Astronomy, University of California Los Angeles, Los Angeles,
CA 90095, USA  Jiaxin Liu  State Key Laboratory of Nuclear Physics and
Technology, and Key Laboratory of HEDP of the Ministry of Education, CAPT,
Peking University, Beijing, 100871, China  Qianyi Ma  State Key Laboratory of
Nuclear Physics and Technology, and Key Laboratory of HEDP of the Ministry of
Education, CAPT, Peking University, Beijing, 100871, China  Jacob Pierce
Department of Physics and Astronomy, University of California Los Angeles, Los
Angeles, CA 90095, USA  Kyle Miller  University of Rochester, Laboratory for
Laser Energetics, Rochester, NY 14623, USA  Xueqing Yan  State Key Laboratory
of Nuclear Physics and Technology, and Key Laboratory of HEDP of the Ministry
of Education, CAPT, Peking University, Beijing, 100871, China  Beijing Laser
Acceleration Innovation Center, Beijing 100871, China  Collaborative
Innovation Center of Extreme Optics, Shanxi University, Shanxi 030006, China
Warren B. Mori  Department of Physics and Astronomy, University of California
Los Angeles, Los Angeles, CA 90095, USA  Department of Electrical Engineering,
University of California, Los Angeles, California 90095, USA

###### Abstract

The quality of electron beams produced from plasma-based accelerators, i.e.,
normalized brightness and energy spread, has made transformative progress in
the past several decades in both simulation and experiment. Recently, full-
scale particle-in-cell (PIC) simulations have shown that electron beams with
unprecedented brightness ($10^{20}\sim
10^{21}~{}\mathrm{A}/\mathrm{m}^{2}/\rad^{2}$) and $0.1\sim
1~{}\mathrm{M}\mathrm{eV}$ energy spread can be produced through controlled
injection in a slowly expanding bubble that arises when a particle beam or
laser pulse propagates in density gradient, or when a particle beam self-
focuses in uniform plasma or has a superluminal flying focus. However, in
previous simulations of work on self-injection triggered by an evolving laser
driver in a uniform plasma, the resulting beams did not exhibit comparable
brightnesses and energy spreads. Here, we demonstrate through the use of
large-scale high-fidelity PIC simulations that a slowly expanding bubble
driven by a laser pulse in a uniform plasma can indeed produce self-injected
electron beams with similar brightnesses and energy spreads as for an evolving
bubble driven by an electron beam driver. We consider laser spot sizes roughly
equal to the matched spot sizes in a uniform plasma and find that the
evolution of the bubble occurs naturally through the evolution of the laser.
The effects of the electron beam quality on the choice of physical as well as
numerical parameters, e.g. grid sizes and field solvers used in the PIC
simulations are presented. It is found that this original and simplest
injection scheme can produce electron beams with beam quality exceeding that
of the more recent concepts.

###### pacs:

##  I I. Introduction

Critical metrics for accessing the capability of a particle accelerator are
related to the quality of the beams they can deliver. Several macro or
ensemble averaged quantities of importance to quantify the beams include the
energy $E_{\mathrm{b}}$, energy spread $\sigma_{\mathrm{E_{\mathrm{b}}}}$,
current $I$, normalized emittance $\epsilon_{\mathrm{N}}$, duration
$\tau_{\mathrm{b}}$ and repetition rate $f$. As a promising novel acceleration
method, plasma-based acceleration (PBA) driven by an intense laser pulse or
particle beam can sustain ultrahigh acceleration gradients ($10\sim
100~{}\mathrm{G}\mathrm{V}\mathrm{/}\mathrm{m}$) within the acceleration
medium - a fully ionized plasma Tajima and Dawson (1979); Chen et al. (1985);
Esarey et al. (2009); Joshi et al. (2020). Generation of high-quality electron
beams from PBA is critical for its development as such beams could transform
applications, such as X-ray free-electron lasers (XFELs) Pellegrini et al.
(2016); Kim et al. (2017) and TeV-class colliders Benedetti et al. (2022).
Both of these applications have stringent requirements for the beam quality.
Although high acceleration gradients were demonstrated experimentally in the
early stages of PBA research Clayton et al. (1993); Everett et al. (1994);
Modena et al. (1995), the generated electron beams were characterized by large
divergences and emittances, and Maxwellian energy distributions, i.e., $100\%$
energy spread Modena et al. (1995); Umstadter et al. (1996a); Wagner et al.
(1997); Malka et al. (2002). In 2004, three groups Mangles et al. (2004);
Geddes et al. (2004); Faure et al. (2004) produced monoenergetic $\sim$100 MeV
beams with a few percent energy spread and several mrad divergence by shooting
a ultrashort ($30\sim 60~{}\mathrm{f}\mathrm{s}$) laser pulse with $\sim$joule
energy into a plasma with $\sim 10^{19}~{}\mathrm{c}\mathrm{m}^{-3}$ density.
However, the beam qualities had large shot-to-shot fluctuations due to the
variation of the laser and plasma parameters when operating at the relatively
high plasma densities.

In order to improve the stability and reproducibility of the beams produced
from PBA, a number of controllable injection schemes have been proposed during
the last two decades. These schemes utilize a variety of physical mechanisms,
such as additional lasers Umstadter et al. (1996b); Esarey et al. (1997a);
Hemker et al. (1998); Fubiani et al. (2004); Kotaki et al. (2004); Faure et
al. (2006); Davoine et al. (2009); Lehe et al. (2013); Chen et al. (2022), an
external magnetic field Vieira et al. (2011), a plasma density gradient
Bulanov et al. (1998); Suk et al. (2001); Geddes et al. (2008); Gonsalves et
al. (2011); Buck et al. (2013); Li et al. (2013a), or the vast difference in
the ionization potentials of electrons between different shells of atoms Moore
et al. (1999); Chen et al. (2006); Oz et al. (2007); Rowlands-Rees et al.
(2008); Pak et al. (2010); McGuffey et al. (2010); Clayton et al. (2010); Liu
et al. (2011); Pollock et al. (2011); Maier et al. (2020).

To date, the representative mechanisms which have produced the best
experimental and simulation results are ionization injection and density
downramp injection in the nonlinear blowout regime. In ionization injection,
electrons with high ionization potentials are released inside the wake and
these electrons are more easily trapped (injected) than background electrons.
In density downramp injection, a negative plasma density gradient is used to
reduce the phase velocity of the wake by gradually increasing the wavelength
of the wake and trigger injection of energetic plasma sheath electrons. The
phase space dynamics of the injected electrons in these two schemes have been
thoroughly investigated Xu et al. (2014a, 2017) and many variations have been
proposed to further improve the generated beam quality Xu et al. (2017);
Hidding et al. (2012); Li et al. (2013b); Bourgeois et al. (2013); Martinez de
la Ossa et al. (2013); Yu et al. (2014a); Xu et al. (2014b); Zeng et al.
(2015); Xu et al. (2016a); Tooley et al. (2017); Deng et al. (2019);
Dalichaouch et al. (2020); Xu et al. (2022a); Li et al. (2022); Xu et al.
(2022b); Wang et al. (2022).

Recent full-scale high-fidelity particle-in-cell (PIC) simulations have shown
that downramp injection and other schemes that gradually increase the
wavelength in the nonlinear blowout regime Xu et al. (2017); Dalichaouch et
al. (2020); Li et al. (2022) can produce beams with unprecedented brightness
$B=\frac{2I}{\epsilon_{\mathrm{N}}^{2}}=10^{20}\sim
10^{21}~{}\mathrm{A}/\mathrm{m}^{2}/\rad^{2}$ ($I\sim
20~{}\mathrm{k}\mathrm{A}$ and $\epsilon_{\mathrm{N}}\sim
10~{}\mathrm{n}\mathrm{m}$) and low slice energy spreads of $0.1\sim
1~{}\mathrm{M}\mathrm{eV}$. The large energy chirp formed during the injection
process can be compensated by the chirp of the acceleration gradient of the
nonlinear wake Lu et al. (2006); Tzoufras et al. (2008) during the subsequent
acceleration process Kalmykov et al. (2009, 2011); Zhang et al. (2016); Xu et
al. (2017); Manahan et al. (2017); Kirchen et al. (2021). As a result, the
beams can achieve $\sim 0.1\%$ or even smaller relative projected energy
spreads after they are boosted to GeV-class or higher energies Manahan et al.
(2017); Pousa et al. (2019); Dalichaouch et al. (2020); Ke et al. (2021).

Beside these controllable injection schemes, self-injection induced by the
evolution of an intense laser driver Pukhov and Meyer-ter Vehn (2002); Tsung
et al. (2004); Xu et al. (2005); Lu et al. (2007); Kalmykov et al. (2009);
Kostyukov et al. (2009); Kalmykov et al. (2011); Martins et al. (2010) in a
uniform plasma with density around $10^{17}\sim
10^{18}~{}\mathrm{c}\mathrm{m}^{-3}$ has been commonly used in experiments to
generate GeV-class electrons due to its simplicity Leemans et al. (2006);
Kneip et al. (2009); Froula et al. (2009); Hafz et al. (2008); Wang et al.
(2013); Kim et al. (2013); Leemans et al. (2014); Gonsalves et al. (2019);
Miao et al. (2022); Aniculaesei et al. (2022). Notably, this self-injection
mechanism was used to generate beams with energies above the GeV barrier
Leemans et al. (2006) and, more recently, up to 8 GeV Gonsalves et al. (2019).
However, these beams were characterized by relatively poor quality, i.e.,
large energy spreads ($\frac{\sigma_{\mathrm{E_{b}}}}{E_{\mathrm{b}}}\gg 1\%$)
and large emittance ($\sim\mathrm{\SIUnitSymbolMicro}\mathrm{m}$), in both PIC
simulations and experiments Tsung et al. (2004); Leemans et al. (2006); Kneip
et al. (2009); Froula et al. (2009); Hafz et al. (2008); Kalmykov et al.
(2009); Kostyukov et al. (2009); Martins et al. (2010); Kalmykov et al.
(2011); Wang et al. (2013); Kim et al. (2013); Leemans et al. (2014);
Gonsalves et al. (2019); Corde et al. (2013); Davidson et al. (2018).

In this paper, we demonstrate using large-scale high-fidelity PIC simulations
and theoretical analysis that the dynamics of self-injection induced by the
evolution of a short pulse laser are fundamentally the same as those observed
in downramp injection. Thus, the self-injected electrons can achieve similar
beam quality as from downramp injection, i.e., $10^{20}\sim
10^{21}~{}\mathrm{A}/\mathrm{m}^{2}/\rad^{2}$ brightness, sub-MeV slice energy
spread, and $\lesssim 0.1\%$ relative projected energy spread after the beam
is boosted to GeV-class energies. In Sec. II, we compare simulation results
for both a uniform plasma and a plasma with a density downramp to show the
similarity between these two regimes. A GeV-class beam with $0.3\times
10^{20}~{}\mathrm{A}/\mathrm{m}^{2}/\rad^{2}$ and $0.2\%$ relative projected
energy spread is produced in the example. In Sec. III, we explore the
potential of self-injection in a uniform plasma. We find that for diffraction-
limited laser beams and perfectly uniform plasmas this simple scheme can
generate beams with ultrahigh brightness ($\sim
10^{21}~{}\mathrm{A}/\mathrm{m}^{2}/\rad^{2}$) and ultrahigh current ($\sim
100~{}\mathrm{k}\mathrm{A}$) by operating at $\sim
10^{19}~{}\mathrm{c}\mathrm{m}^{-3}$ plasma densities. Possible physical
factors which prevent the generation of such high-quality beams in experiments
are briefly discussed in Sec. IV and a summary of our findings is given in
Sec. V. The effects of numerical resolution and the choice of the
electromagnetic field solver on the injected beams’ quality are studied in the
Appendix A.

##  II II. Common dynamics of self injection in uniform plasma and density
downramps

We model the self-injection of plasma electrons in a laser wakefield
accelerator using the quasi-three-dimensional (Q3D) version Davidson et al.
(2015) of the PIC code OSIRIS Fonseca et al. (2002) with recently developed
high-fidelity Maxwell solvers Li et al. (2017); Xu et al. (2020). A plasma
column with density $n_{\mathrm{p0}}=2\times
10^{18}~{}\mathrm{c}\mathrm{m}^{-3}$ and isotropic electron temperature
$T=[0.1~{}\mathrm{eV},0.1~{}\mathrm{eV},0.1~{}\mathrm{eV}]$ is initialized at
the beginning of the simulation. An 800 nm, 4.3 J laser pulse with a peak
power $P=155~{}\mathrm{T}\mathrm{W}$ and a full-width half maximum (FWHM)
pulse duration
$\tau_{\mathrm{FWHM}}=38.7~{}\mathrm{f}\mathrm{s}~{}(3.1~{}\omega_{\mathrm{p0}}^{-1})$
is incident on the fully ionized plasma, where
$\omega_{\mathrm{p0}}=\sqrt{\frac{n_{\mathrm{p0}}e^{2}}{m\epsilon_{0}}}$ is
the plasma frequency, $m$ and $e$ are the electron mass and charge, and
$\epsilon_{0}$ is the vacuum permittivity. The laser has a diffraction-limited
Gaussian transverse profile and is focused at the start of the plasma ($z=0$)
with a spot size
$w_{0}=16.9~{}\mathrm{\SIUnitSymbolMicro}\mathrm{m}~{}(4.5k_{\mathrm{p0}}^{-1})$
(the field profile at focus is $e^{-r^{2}/w_{0}^{2}}$), where
$k_{\mathrm{p0}}=\frac{\omega_{\mathrm{p0}}}{c}$ is the plasma wavenumber and
$c$ is the speed of light in vacuum. The normalized vector potential of the
laser is
$a_{0}\equiv\frac{eA_{\mathrm{L}}}{mc^{2}}=\frac{eE_{\mathrm{L}}}{mc\omega_{0}}=4$
at its focus, where $\omega_{0}$ is the laser frequency, $A_{\mathrm{L}}$ is
the peak vector potential, and $E_{\mathrm{L}}$ is its peak electric field.
The spot size used in the simulations is slightly larger than the matched spot
size
($k_{\mathrm{p0}}w_{\mathrm{0,match}}=k_{\mathrm{p0}}r_{\mathrm{b}}=2\sqrt{a_{0}}=4$)
suggested by the nonlinear plasma wave wake theory Lu et al. (2006, 2007),
where $r_{\mathrm{b}}$ is the radius of the blowout wake. The laser is
linearly polarized along the $x$-direction. The grid sizes are chosen as
$\mathrm{d}z=\mathrm{d}r=\frac{1}{512}k_{\mathrm{p0}}^{-1}$ to resolve the
subtle physics involved in the injection process and the acceleration of the
strongly focused injected electrons. Details of the simulation parameters can
be found in Appendix B.

Figure 1:  (a) The charge density distribution of a nonlinear plasma wake
(white-blue-black) driven by a laser pulse (orange) at
$\omega_{\mathrm{p0}}t=150$ in a uniform plasma. The trajectory of an injected
electron (red line) is superimposed. (b) The plasma density profile. The
downramp starts from $k_{\mathrm{p0}}z=50$ with density 1.1 $n_{\mathrm{p0}}$
and ends at $k_{\mathrm{p0}}z=150$ with density $n_{\mathrm{p0}}$. A super-
gaussian upramp profile with order 4 is used between $z=0$ and
$20~{}k_{\mathrm{p0}}^{-1}$. (c) The evolution of the peak $a_{0}$ (blue
lines) and the projected spot size (black lines) of the laser pulse, and the
position of the plasma wake where $E_{\mathrm{z}}=0$ (red lines).

As shown in Fig 1, we consider the self-injection for two cases: a linear
plasma density downramp with normalized density gradient $g\equiv\frac{\Delta
n/n_{\mathrm{p0}}}{k_{\mathrm{p0}}L_{\mathrm{ramp}}}=0.001$ and a uniform
plasma with $g=0$, where $\Delta n$ is the density drop across the ramp and
$L_{\mathrm{ramp}}$ is the ramp length. When an intense laser pulse propagates
inside a plasma (from left to right), the electrons are pushed outwards and
forward from the pulse center. The ions then pull these electrons back causing
them to form a narrow sheath that surrounds an ion column [Fig. 1(a)]. In
general, when the laser’s peak power exceeds the self-focusing critical power
Esarey et al. (1997b), the plasma’s refractive index is self-consistently
modified to focus and guide the laser Mori (1997). In the simulations
conducted here, the peak power of the laser (155 $\mathrm{T}\mathrm{W}$) is
much higher than the critical power for self-focusing in the plasma
$P_{\mathrm{c}}\approx
17\left(\frac{\omega_{0}}{\omega_{\mathrm{p0}}}\right)^{2}~{}\mathrm{G}\mathrm{W}=14.8~{}\mathrm{T}\mathrm{W}$
for the density in the plateau. Thus, the laser is self-focused and its peak
field (projected spot size) increases (decreases) as shown in Fig 1(c).
Despite the differences in the plasma density profiles, the evolution of the
lasers are similar in these two cases.

Figure 2:  The initial $z_{\mathrm{i}}$ and the axial position after injection
$\xi$ of the injected electrons for $g=0.001$ (a) and $g=0$ (b). The density
distribution and the current (red lines) of the injected beam at
$\omega_{\mathrm{p0}}t=250$ for $g=0.001$ (c) and $g=0$ (d). (e) The
longitudinal phase space and the slice energy spread for both cases at
$\omega_{\mathrm{p0}}t=250$. (f) The slice emittance along $x$-direction (blue
lines) and the brightness (red lines) at $\omega_{\mathrm{p0}}t=250$. The
duration of each slice is $\frac{1}{128}k_{\mathrm{p0}}^{-1}$ when calculating
the slice properties.

The wavelength of a laser-driven 3D nonlinear plasma wake depends on the spot
size and the peak intensity of the laser driver in a complicated way,
particularly when the beam is not well matched. In general, the wavelength
tends to increase with both spot size and laser intensity. However, for the
parameters studied here, the projected spot size decreases while the intensity
increases through self-focusing in the ion channel. In this case, the increase
in the intensity dominates, leading to a slow expansion of the wake size
(wavelength). Previous simulations have also shown the wake expands as the
laser diffracts for an initially tightly focused beam Xu et al. (2005). Since
it is difficult to locate the precise end of the wake which is occupied by the
self-injected electrons, we measure the axial position in the first wave
bucket where $E_{\mathrm{z}}=0$ to quantify the expansion of the wake [red
lines in Fig 1(c)]. Note the position where $E_{\mathrm{z}}=0$ is roughly the
center of the wake and the rear of the wake expands with approximately twice
this velocity. In the uniform plasma case, the self-focusing of the laser
driver causes an expansion of the wake with a velocity, $v_{\phi,E_{z}=0}$, of
$\sim 0.003c$ between $\sim 50k_{\mathrm{p0}}^{-1}$ and $\sim
150k_{\mathrm{p0}}^{-1}$. The wake expands with a faster velocity of $\sim
0.005c$ in the case with a density downramp which indicates the downramp
speeds up the wake expansion.

In a nonlinear wake, plasma electrons originating from
$r_{\mathrm{i}}\sim\kappa r_{\mathrm{b}}$ form the high-density sheath
surrounding the ion channel Xu et al. (2017) and gain large forward velocities
when they move to the back of the wake, where $r_{\mathrm{i}}$ is the initial
radial position of the electron and $\kappa$ is a value between $0.5$ and 1
which depends on the driver. If their forward velocities are faster than the
phase velocity of the end of the wake, these sheath electrons would remain
just inside at the end of the wake where they would then be accelerated
continuously. The trajectory of a sample injected electron is superimposed on
the wake shape as shown in Fig. 1(a).

Since the electrons are always injected at the rear of the continuously
expanding wake, there is a mapping between the initial positions
($z_{\mathrm{i}}$) of the electrons and their axial positions inside the wake
after injection ($\xi$) Kalmykov et al. (2009); Kostyukov et al. (2009);
Kalmykov et al. (2011); Xu et al. (2017) which can be seen in Figs. 2(a) and
(b). The duration of the beam is roughly equal to the difference of the wake
wavelength at the start and the end of the injection. The injected beam from a
plasma downramp has a longer duration since the wake is smaller at the start
of the injection where the plasma density is $1.1n_{\mathrm{p0}}$. The
compression factor thus scales as $\sim\gamma_{\phi,E_{z}=0}^{2}$ Xu et al.
(2017); Tooley et al. (2017), where
$\gamma_{\phi,E_{z}=0}=1/\sqrt{1-v_{\phi,E_{z}=0}^{2}/c^{2}}$. Thus, there is
a significant compression of the beams’ duration during the injection, i.e.,
for these simulations the electrons initially distributed with a length of
$\sim 100~{}k_{\mathrm{p0}}^{-1}$ and are compressed and form a beam with a
duration of $\sim k_{\mathrm{p0}}^{-1}$. This enables the generation of beam
currents of 10s of kA. As shown in Figs. 1(c) and (d), the current of the core
of the beams are $\sim 30~{}\mathrm{k}\mathrm{A}$. Here the core of the beams
is defined as part with brightness $\geq
10^{19}~{}\mathrm{A}/\mathrm{m}^{2}/\rad^{2}$, which is
$11.3<k_{\mathrm{p0}}\xi<12.1$ for the $g=0.001$ case and
$11.47<k_{\mathrm{p0}}\xi<11.7$ for the $g=0$ case. The charge of the beam
core is 180 $\mathrm{p}\mathrm{C}$ for $g=0.001$ and 76 $\mathrm{p}\mathrm{C}$
for $g=0$. We note that in addition to the continuous injection during the
expansion of the wake, there is an isolated injection near the end of the
upramp ($z=20k_{\mathrm{p0}}^{-1}$) Li et al. (2013a). This injection forms an
attosecond peak with $\sim 70~{}\mathrm{k}\mathrm{A}$ current at the head of
the beams in both cases Li et al. (2013a).

Based on analyzing numerous PIC simulations of downramp injection and its
variants by the authors and others Grebenyuk et al. (2014); Xu et al. (2017);
Dalichaouch et al. (2020); Li et al. (2022), we have inferred that the current
of the core of the beam scales as
$I\sim\frac{\Lambda}{2}I_{\mathrm{A}}=I_{\mathrm{d}}$ for a beam-driven
nonlinear wake and $I\sim\frac{a_{0}}{2}I_{\mathrm{A}}$ for a laser-driven
case (with a nearly matched spot size), where
$\Lambda\equiv\frac{2I_{\mathrm{d}}}{I_{\mathrm{A}}}$, $I_{\mathrm{d}}$ is the
peak current of the beam driver and $I_{\mathrm{A}}\approx
17~{}\mathrm{k}\mathrm{A}$ is the non-relativistic Alfven current. While we
currently do not have simple arguments for obtaining these scaling laws, they
may arise due to a fine balance between the initial injection and a subsequent
quenching via self-beam loading, i.e., the injected electrons can modify the
shape of the wake which reduces the forward velocity of the sheath electrons
and hinders injection Dalichaouch et al. (2021). This observed scaling
suggests that beams with hundreds of kA current and nanocoulombs of charge can
be injected into a wake driven by an intense laser driver with $a_{0}\gtrsim
10$.

Figure 3:  (a) The evolution of the average energy (blue solid line) and the
projected energy spread (red dashed line) of the core of the beam ($g=0$) and
the acceleration gradient (black dotted line). (b) Longitudinal phase space of
the injected electrons at three acceleration distances. The shadowed region
represents the beam core.

The mapping between the initial positions of the electrons and their axial
positions after injection leads to two consequences of the beams’ energy
distribution. The first is a low slice energy spread since the electrons in
one axial slice originate from similar longitudinal locations and experience
the same acceleration gradient after injection. Fig. 2(e) shows that the beams
have a slice energy spread of $\sim 0.5~{}\mathrm{M}\mathrm{eV}$ except for
near their heads. The second is a roughly linear energy chirp along the beam
since the electrons at the beam head are injected earlier and accelerated over
a longer distance. As shown in Fig. 2(e), the chirps at
$\omega_{\mathrm{p0}}t=250$ are
$174~{}\mathrm{M}\mathrm{eV}/(k_{\mathrm{p0}}\xi)~{}(46~{}\mathrm{M}\mathrm{eV}/\mathrm{\SIUnitSymbolMicro}\mathrm{m})$
for $g=0.001$ and
$423~{}\mathrm{M}\mathrm{eV}/(k_{\mathrm{p0}}\xi)~{}(112~{}\mathrm{M}\mathrm{eV}/\mathrm{\SIUnitSymbolMicro}\mathrm{m})$
for $g=0$.

These positive chirps can be compensated by the chirp of the acceleration
gradient during the subsequent acceleration. The beam loaded wake has a lower
gradient at the head and higher gradient at the rear. Thus there is an
optimized acceleration distance where the beam can achieve a projected energy
spread on the order of MeV Xu et al. (2017); Ke et al. (2021). Due to the
constant evolution of the laser driver, the acceleration gradient and its
chirp change and we rely on simulations to find this optimized distance. We
present the evolution of the average energy, the projected energy spread of
the beam core ($11.47<k_{\mathrm{p0}}\xi<11.7$) for $g=0$ and the acceleration
gradient in Fig. 3(a). The optimized acceleration distance occurs at
$z=810k_{\mathrm{p0}}^{-1}~{}(3.05~{}\mathrm{m}\mathrm{m})$ where the core of
the beam achieves a projected energy spread (red dashed line) as low as 1.6
MeV while its average energy (blue solid line) is 0.76 GeV. The energy of the
beam can still be boosted since there is $\sim 75\%$ energy left in the laser
pulse. We continue the simulation to
$z=1230k_{\mathrm{p0}}^{-1}~{}(4.63~{}\mathrm{m}\mathrm{m})$ where the core of
the beam has 1.04 GeV energy and the laser pulse contains $\sim 60\%$ of its
initial energy. The average acceleration gradient drops from $\sim
0.35~{}\mathrm{T}\mathrm{V}\mathrm{/}\mathrm{m}$ at
$z=250k_{\mathrm{p0}}^{-1}~{}(0.94~{}\mathrm{m}\mathrm{m})$ to $\sim
0.1~{}\mathrm{T}\mathrm{V}\mathrm{/}\mathrm{m}$ at
$z=1230k_{\mathrm{p0}}^{-1}~{}(4.63~{}\mathrm{m}\mathrm{m})$. A simulation
with lower resolution
($\mathrm{d}z=\mathrm{d}r=\frac{1}{128}k_{\mathrm{p0}}^{-1}$) shows the beam
energy reaches its maximum 1.40 GeV at
$z=2440k_{\mathrm{p0}}^{-1}~{}(9.19~{}\mathrm{m}\mathrm{m})$ and then starts
to lose energy. The longitudinal phase space of the injected beam at three
acceleration distances are shown in Fig. 3(b) where the evolution of the
energy chirp can be seen. Secondary and tertiary phases of injection occur
around $z=300k_{\mathrm{p0}}^{-1}~{}(1.13~{}\mathrm{m}\mathrm{m})$ and
$750k_{\mathrm{p0}}^{-1}~{}(2.82~{}\mathrm{m}\mathrm{m})$ which are
characterized by low current, large emittance and large slice energy spreads.
The aforementioned analysis and simulation results show that the injection
mechanism and the longitudinal mapping are similar in a downramp and a uniform
plasma.

Figure 4:  (a) The evolution of the laser pulse driver and the plasma wake
wave under different laser spot sizes: $k_{\mathrm{p0}}w_{0}=3.5,4$ and 4.5.
(b) The charge density distribution of the injected electrons and their
current profiles for these cases.

As shown in Ref. Xu et al. (2017), sheath electrons experience a transverse
defocusing force from the nonlinear wake at the very rear of the channel prior
to injection Dalichaouch et al. (2021). This transverse defocusing force
reduces the transverse momentum of these electrons as they approach the axis
leading to a beam with ultra-low emittance. The same dynamics occur for
electrons injected from an elongating wake in a uniform plasma. The emittance
of the beams along the laser polarization direction ($x$) are shown in Fig.
2(f) where the middle of the beams can achieve an emittance as low as
$<0.01~{}k_{\mathrm{p0}}^{-1}~{}(38~{}\mathrm{n}\mathrm{m})$ while the head
and the tail are characterized by a larger emittance of $0.01\sim
0.04~{}k_{\mathrm{p0}}^{-1}~{}(\sim 100~{}\mathrm{n}\mathrm{m})$. The
relatively large emittance at the beam tail is due to the lack of symmetry of
the injected electrons at the end of the injection process, i.e., the initial
angular distribution of the injected electrons is asymmetric as shown in the
Appendix C. The emittance along the other transverse direction has a similar
profile. The peak brightness of the beams is $0.5\times
10^{20}~{}\mathrm{A}/\mathrm{m}^{2}/\rad^{2}$ for $g=0.001$ and $0.3\times
10^{20}~{}\mathrm{A}/\mathrm{m}^{2}/\rad^{2}$ for $g=0$.

In principle, the injection of ultrahigh quality electrons in uniform plasmas
can be controlled by the initial parameters of the laser pulse driver (the
intensity, the spot size, the vacuum focal plane and the pulse duration) and
the plasma density. However, it is challenging to derive an analytical
expression to describe with good accuracy how the injection depends on these
parameters. In instead we rely on simulations to show how the injection varies
when different laser spot sizes are used. In Fig. 4, we see that the plasma
wake expands and injection occurs for three cases near the matched spot size,
$k_{\mathrm{p0}}w_{0}$=3.5, 4 (matched) and 4.5, for fixed intensity
($a_{0}=4$) and thus different laser powers. When a laser pulse with a smaller
spot size is used, the injection distance shrinks. The isolated injection that
occurs at the end of the upramp around $k_{\mathrm{p0}}z=20$ for
$k_{\mathrm{p0}}w_{0}=4$ and 4.5 is absent when $k_{\mathrm{p0}}w_{0}=3.5$.
Note that the aforementioned injection of ultrahigh quality beams also occurs
for laser pulse drivers with spot sizes far away from the matched spot size.
We choose nearly matched laser spot sizes in this paper for the subsequent
acceleration since a nearly matched laser pulse can be guided over many
Rayleigh lengths of distance in plasma Lu et al. (2007).

The motion of plasma ions Rosenzweig et al. (2005); An et al. (2017) which
becomes important when the parameter
$\Omega_{\mathrm{b}}\frac{\sigma_{\mathrm{zb}}}{c}=\sqrt{\frac{n_{\mathrm{b}}}{n_{\mathrm{p0}}}\frac{m}{m_{\mathrm{i}}}}k_{\mathrm{p0}}\sigma_{\mathrm{zb}}$
exceeds unity would modify the distribution of the acceleration and focusing
fields inside the wake and degrade the beam quality. Here
$\Omega_{\mathrm{b}}=\frac{n_{\mathrm{b}}e^{2}}{m_{\mathrm{i}}\epsilon_{0}}$
is the ion plasma frequency for the beam density, $m_{\mathrm{i}}$ is the ion
mass, $n_{\mathrm{b}}$ and $\sigma_{\mathrm{zb}}$ are the peak density and the
duration of the injected beam. For the $g=0$ case, this value is
$\Omega_{\mathrm{b}}\frac{\sigma_{\mathrm{zb}}}{c}\sim 0.1$ even for the
lightest hydrogen ions, so the ion motion is weak. Simulations performed with
mobile hydrogen ions confirm that ion motion has little effect on the injected
beam quality.

We emphasize that fine grid sizes and advanced field solvers are necessary to
model the ultrahigh quality electron beams generation in the highly nonlinear
plasma wakes. Due to their ultra-low emittance and the ultra-strong focusing
fields inside the ion channel, the injected beams are tightly focused down to
spot sizes of $\sim 0.1~{}k_{\mathrm{p0}}^{-1}$ with peak densities as high as
$10^{3}\sim 10^{4}~{}n_{\mathrm{p0}}$. Thus a fine grid size is needed to
resolve them. When these high-density relativistic electrons propagate on the
numerical grids, they can excite unphysical numerical fields since the grids
can be viewed as a medium with a complicated dielectric tensor. Two important
kinds of the unphysical effects are numerical Cherenkov radiation Godfrey
(1974); Godfrey and Vay (2013); Xu et al. (2013) and numerical space-charge
fields Xu et al. (2020). These unphysical fields can modify the beams’
evolution and degrade their qualities. Maxwell field solvers based on spectral
solvers Yu et al. (2014b, 2015a, 2015b, 2016); Godfrey and Vay (2015) or with
finite difference solvers with extended stencils Xu et al. (2020); Li et al.
(2021) have recently been developed to suppress some or combinations of these
numerical fields. In the aforementioned simulations, a combination of fine
grid sizes ($\mathrm{d}z=\mathrm{d}r=\frac{1}{512}k_{\mathrm{p0}}^{-1}$) and
the recently developed Xu solver Xu et al. (2020) are used to model the
injected electrons with high-fidelity. For comparison, results with the Yee
solver and/or coarse resolutions are presented in the Appendix A.

##  III III. Generation of ultrahigh brightness and ultrahigh current beams
with hundreds of MeV

The properties of self-injected beams are determined by the plasma density and
the evolution of the laser driver in the plasma in terms of both the $\xi$ and
propagation distance variables. Thus, there is a large parameter space to
explore and electron beams with different properties can be injected. In the
previous section, a GeV-class high-quality beam with $10^{19}\sim
10^{20}~{}\mathrm{A}/\mathrm{m}^{2}/\rad^{2}$ brightness is produced by
focusing a 4.3 J laser pulse into a plasma with $n_{\mathrm{p0}}=2\times
10^{18}~{}\mathrm{c}\mathrm{m}$. We show two more representative cases in this
section: one is the generation of $\sim 100$ MeV beams with $\sim
10^{21}~{}\mathrm{A}/\mathrm{m}^{2}/\rad^{2}$ and the other is generation of
beams with $\sim 100~{}\mathrm{k}\mathrm{A}$ current.

Each simulation corresponds to a family of physical instances where the
normalized parameters remain fixed. Thus, the emittance of the injected beams
in downramp injection and self-injection in a uniform plasma scales with the
background plasma density as $\epsilon_{\mathrm{N}}\propto
k_{\mathrm{p0}}^{-1}\propto n_{\mathrm{p0}}^{-1/2}$ in beam driven plasma
wakefield accelerators if the normalized dimensions of the beam drivers
($k_{\mathrm{p0}}\sigma_{x,y,z}$) and their normalized peak density
($\frac{n_{\mathrm{b}}}{n_{\mathrm{p0}}}$) are assumed fixed Xu et al. (2017).
This scaling for $\epsilon_{\mathrm{N}}$ also holds for laser-driven wakes if
laser parameters are also scaled, including the laser frequency. However, as
high power lasers are presently available in a limited range of wavelengths,
then as the density changes the frequency ratio
($\frac{\omega_{\mathrm{0}}}{\omega_{\mathrm{p0}}}$) will not be scaled
appropriately. At lower frequency ratios the laser evolves more rapidly and
thus the scaling with $\epsilon_{\mathrm{N}}$ with density will be
approximate. This scaling thus indicates that ultra-bright beams can be
produced using a high density plasma for lasers as well as particle beam
drivers. To confirm this, results from a simulation with
$n_{\mathrm{p0}}=10^{19}~{}\mathrm{c}\mathrm{m}^{-3}$ are shown in Fig. 5. An
800 nm laser pulse with duration
$\tau_{\mathrm{FWHM}}=25.9~{}\mathrm{f}\mathrm{s}~{}(4.6\omega_{\mathrm{p0}}^{-1})$
and linear polarization in the $x$ direction is focused inside the plasma at
[$z_{\mathrm{f}}=67.3~{}\mathrm{\SIUnitSymbolMicro}\mathrm{m}~{}(40k_{\mathrm{p0}}^{-1})$]
with a focal spot size
$w_{0}=5.9~{}\mathrm{\SIUnitSymbolMicro}\mathrm{m}~{}(3.5k_{\mathrm{p0}}^{-1})$.
The laser contains 0.35 J energy and its peak power is
$18.7~{}\mathrm{T}\mathrm{W}$. Its normalized vector potential at the vacuum
focal plane is $a_{0}=4$. The spot size and the focal position Götzfried et
al. (2020); Dickson et al. (2022) have a significant effect on the laser
evolution and thus where the injection happens. These parameters are first
scanned using simulations with low resolution to find optimal operating
parameters.

As shown in Fig. 5(a), the laser is focused down even after its vacuum focal
plane due to the self-focusing and starts to expand around $z\approx
135~{}\mathrm{\SIUnitSymbolMicro}\mathrm{m}~{}(80k_{\mathrm{p0}}^{-1})$.
Injection starts at $z\approx 135~{}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$
where the $a_{0}$ of the laser is focused to $a_{0}\approx 6$. The injection
ceases at $z\approx
235~{}\mathrm{\SIUnitSymbolMicro}\mathrm{m}~{}(140k_{\mathrm{p0}}^{-1})$ due
to a combination of beam loading from the injected electrons and the decrease
of the laser intensity. Compared with the $g=0$ case in Fig. 1, the wake
expands with a faster velocity of $0.01\sim 0.02c$ which leads to a smaller
energy chirp at the end of the injection [31.4
$\mathrm{M}\mathrm{eV}/(k_{\mathrm{p0}}\xi)$]. This small chirp is compensated
quickly by the slope of the acceleration gradient thus the energy at the
optimized acceleration distance is much lower than for the $g=0$ case in Fig.
1.

Figure 5:  Generation of electron beams with $\sim
10^{21}~{}\mathrm{A}/\mathrm{m}^{2}/\rad^{2}$ brightness. (a) The evolution of
the normalized vector potential (blue), the projected spot size (black) of the
laser diver, and the expansion of the wake (red). The purple dots show the
initial axial and radial positions of the injected electrons. (b) The
longitudinal phase space and the slice parameters of the injected beam at
$k_{\mathrm{p0}}z=194$: the normalized emittance, the brightness, the current
and the slice energy spread. The shadowed region represents the beam core.

Simulations show the core of the injected beam ($12.2<k_{\mathrm{p0}}\xi<13.3$
with $B\geq 0.5\times 10^{21}~{}\mathrm{A}/\mathrm{m}^{2}/\rad^{2}$) achieves
its minimum projected energy spread (1.6 MeV) at
$z=326~{}\mathrm{\SIUnitSymbolMicro}\mathrm{m}~{}(194k_{\mathrm{p0}}^{-1})$
with an average energy 74.4 MeV. The slice energy spread is $\sim 0.3$ MeV and
the current is $\sim$13 kA. The charge contained in the beam core is 81 pC. As
shown in Fig. 5(b), the emittance of the beam core is $\sim$3 nm ($x$),
$\sim$9 nm ($y$) and its peak brightness reaches $1.5\times
10^{21}~{}\mathrm{A}/\mathrm{m}^{2}/\rad^{2}$. The unequal emittance (and spot
sizes) along the two transverse directions of the injected beam can be traced
back to the fact that only electrons with initial azimuthal angles around
$\theta_{i}\equiv\mathrm{atan2}(y_{i},x_{i})=\pm\frac{\pi}{2}$ are injected
and the details are presented in Appendix C. The dephasing between the
injected electrons and the laser driver sets a limit on the maximum energy
gain of the injected beam as $\sim 100$ MeV in a plasma with $\sim
10^{19}~{}\mathrm{c}\mathrm{m}^{-3}$ density Lu et al. (2007). Injection and
acceleration in such a high-density plasma can serve as an injector with 100
MeV-class ultra-bright electron beams.

Based on the empirical relation between the current of the injected beams and
the $a_{0}$ of the laser pulse drivers, beams with hundreds of kA are expected
to be produced when using laser pulses with $a_{0}\gtrsim 10$. We present
simulation results in Fig. 6 for a case where a PW laser pulse and plasma
density $n_{\mathrm{p0}}=10^{19}~{}\mathrm{c}\mathrm{m}^{-3}$ are used. An 800
nm, linearly polarized ($x$) laser pulse with duration
$\tau_{\mathrm{FWHM}}=38.8~{}\mathrm{f}\mathrm{s}~{}(6.9\omega_{\mathrm{p0}}^{-1})$
is focused outside of the plasma
[$z_{\mathrm{f}}=-640~{}\mathrm{\SIUnitSymbolMicro}\mathrm{m}~{}(-380k_{\mathrm{p0}}^{-1})$]
with a focused spot size
$w_{0}=11.8~{}\mathrm{\SIUnitSymbolMicro}\mathrm{m}~{}(7k_{\mathrm{p0}}^{-1})$.
The laser contains 33.4 J energy and its peak power is
$1.2~{}\mathrm{P}\mathrm{W}$. Its normalized vector potential at its focal
plane is $a_{0}=16$.

Self-injection starts at $\sim 100k_{\mathrm{p0}}^{-1}$ and ends at $\sim
300k_{\mathrm{p0}}^{-1}$. A beam with $\sim 60~{}\mathrm{f}\mathrm{s}$
duration and $\sim$100 kA current is formed at $\omega_{\mathrm{p0}}t=400$.
The total injected charge is $\sim 6~{}\mathrm{n}\mathrm{C}$ which can be
further improved if a plasma downramp is introduced to elongate the injection
distance. Beams with 10s of MA current may be obtained by controlling the
acceleration distance to form an energy chirp along this beam and compress it
in a small chicane Emma et al. (2021). A high-fidelity simulation of the
generation of this high current beam requires a resolution finer than
$\frac{1}{512}k_{\mathrm{p0}}^{-1}$. However, a resolution with
$\frac{1}{256}k_{\mathrm{p0}}^{-1}$ is used in Fig. 6 due to the significant
computational cost to model the large plasma wave wake, and this leads to a
highly suspect emittance ($\sim\mathrm{\SIUnitSymbolMicro}\mathrm{m}$) and
artificial ripples in the longitudinal phase space of the beam (see Fig. 6).
Based on our experience for other cases where we are able to carry fully
resolved simulations, we believe that these effects are numerical and not
physical.

Figure 6:  Generation of electron beams with $\sim 100~{}\mathrm{k}\mathrm{A}$
current by sending a $1.2~{}\mathrm{P}\mathrm{W}$ laser pulse into a plasma
with $10^{19}~{}\mathrm{c}\mathrm{m}^{-3}$ density: the longitudinal phase
space and the current profile at $\omega_{\mathrm{p0}}t=400$.

##  IV IV. Discussion on physical effects which would degrade the beam quality

The aforementioned results have demonstrated the ability to inject ultrahigh
quality electron beams in the simplest configuration of laser wakefield
accelerators. However, the beam quality reported in experiments (either based
on a density downramp or the evolution of the laser pulse driver) are
significantly worse than the predictions of PIC simulations. In this section
we discuss the possible reasons behind this large gap.

The transverse deceleration process which results in injection of low-
emittance beams depends on having a well defined sheath and the axial symmetry
of the nonlinear wake. Thus, any factors that affect the sheath properties and
the symmetry of the wake degrade the emittance and brightness of the injected
beams. For instance: non diffraction-limited lasers, a finite temperature of
the plasma electrons Xu et al. (2017); Zhang et al. (2019), an asymmetric
driver (intensity and/or phase distortions) Glinec et al. (2008); Mangles et
al. (2009); Popp et al. (2010); Beaurepaire et al. (2015), a finite transverse
bulk velocity of a gas produced from a gas jet, transversely non-uniform
plasma density distribution at spatial scales smaller or larger than plasma
skin depth Stupakov (2017); Ma et al. (2018). These imperfections not only
affect the injection process but also degrade the beam quality during the
subsequent acceleration by modifying the linear focusing field and the
transversely uniform acceleration field in an axisymmetrical nonlinear wake.
However, it is not straightforward to study these effects in the Q3D geometry
since significant effort is needed to understand how many modes are necessary
to model these asymmetry wakes with high-fidelity. Meanwhile, the
computational costs of full 3D simulations are currently unaffordable even
using GPU based hardware. Therefore, we will focus on the effects of the
plasma temperature in this section.

Figure 7:  Comparison of the emittance of the self-injected electron beams
under different plasma electron temperature. The parameters of the laser pulse
and the plasma are the same as that of Fig. 1.

In practical experiments, the plasma is created in many ways, e.g., high-
voltage discharge or optical ionization by a low energy laser pulse. Depending
on the ionization process, the plasma electrons can will be distributed with
different temperatures which may affect the quality of the injected electron
beams. In the self-injection results shown in Sec. II, we assumed an initial
plasma electron temperature of $T=0.1~{}\mathrm{eV}$. In Fig. 7, we compare
the emittance of the self-injected beams from plasmas with higher temperatures
($1~{}\mathrm{eV}$, $10~{}\mathrm{eV}$ and $100~{}\mathrm{eV}$). While the
emittance within the beam cores are similar for $T=1~{}\mathrm{eV}$ and
$T=0.1~{}\mathrm{eV}$, it grows by a factor of $\sim$2 for
$T=10~{}\mathrm{eV}$. The emittance increases dramatically and approaches
$\sim\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ when the plasma electron
temperature is $T=100~{}\mathrm{eV}$. This indicates that eV-level plasma
temperatures are necessary for generation of high-quality beams with
nanometer-scale emittances.

Even when these ultra-bright beams are produced in the plasma, the subsequent
transport may also degrade the emittance due to the mismatch of the beams’
transverse phase space Mehrling et al. (2012); Antici et al. (2012);
Migliorati et al. (2013); Floettmann (2014); Dornmair et al. (2015); Xu et al.
(2016b); Zhao et al. (2020); Ariniello et al. (2019), especially when other
macro parameters of the beams (e.g., energy, pointing angles and transverse
positions) fluctuate from shot-to-shot Labat et al. (2018). Thus, carefully
designed transport stages are additionally necessary for utilization of high-
quality self-injected beams from PBA.

##  V V. Conclusions

We have shown that the dynamics of the self-injection in uniform plasmas is
the same as that of density downramp injection Xu et al. (2017) and its
variants Dalichaouch et al. (2020); Li et al. (2022), thus beams with
$10^{20}\sim 10^{21}~{}\mathrm{A}/\mathrm{m}^{2}/\mathrm{rad}^{2}$ and
$0.1\sim 1$ MeV slice energy spreads can be produced. Fine grid sizes and
advanced field solvers are necessary to model this ultrahigh quality beams
generation in PIC codes.

Due to its low emittance, the GeV beam described in Sec. II is focused tightly
to a spot size as small as $240~{}\mathrm{n}\mathrm{m}$ and reaches a peak
density as high as $2\times 10^{21}~{}\mathrm{c}\mathrm{m}^{-3}$. The
generated beams with higher brightness or current in Sec. III may be focused
to an even higher beam density if their energy is boosted to GeV-levels. These
extremely dense beams can enable many novel applications, such as driving all-
optical XFELs Xu et al. (2023), generating ultrabright $\gamma$-rays through a
beam-plasma instability Benedetti et al. (2018) or beam-multifoil collisions
Sampath et al. (2021), driving plasma wakefield accelerators with $\sim 10$
TeV/m acceleration gradient in a solid density plasma Sahai et al. (2020) and
studying strong-field QED with beam-beam collisions Yakimenko et al. (2019) or
beam-plasma collisions Matheron et al. (2022).

Using high-fidelity large-scale numerical simulations, we have shown that
beams with extreme parameters can be produced when a laser pulse propagates
and evolves in a uniform plasma. This approach can dramatically simplify the
complexity of plasma-based accelerators. Our findings may also stimulate
future research to study why current experiments can not deliver these
ultrahigh quality beams and on what is needed to finally produce these beams
in experiments to enable many novel plasma acceleration driven applications.

##  VI acknowledgments

This work was supported by the Fundamental Research Funds for the Central
Universities, Peking University, the National Natural Science Foundation of
China (NSFC) (No. 11921006) and the National Grand Instrument Project (No.
2019YFF01014400), the U.S. Department of Energy under Contracts No. DE-
SC0010064, the U.S. National Science Foundation under Grants No. 2108970, and
the DOE Scientific Discovery through Advanced Computing (SciDAC) program
through a Fermi National Accelerator Laboratory (FNAL) subcontract No. 644405.
The simulations were performed on the resources of the National Energy
Research Scientific Computing Center (NERSC), a U.S. Department of Energy
Office of Science User Facility located at Lawrence Berkeley National
Laboratory.

##  Appendix A Appendix A: Comparison of the injected beams from simulations
with different resolutions and solvers

Figure 8:  Comparison of the injected beams at $\omega_{\mathrm{p0}}t=250$ in
a uniform plasma from simulations with different resolutions and different
solvers. Simulation Xu/fine:
$\mathrm{d}z=\mathrm{d}r=\frac{1}{512}k_{\mathrm{p0}}^{-1}$ and the Xu solver;
Simulation Yee/fine:
$\mathrm{d}z=\mathrm{d}r=\frac{1}{512}k_{\mathrm{p0}}^{-1}$ and the Yee
solver; Simulation Yee/coarse:
$\mathrm{d}z=\frac{1}{150}k_{\mathrm{p0}}^{-1},\mathrm{d}r=\frac{1}{10}k_{\mathrm{p0}}^{-1}$
and the Yee solver.

In Sec. II, we showed that the beam injected in a uniform plasma shares the
same dynamics as in a plasma downramp and is characterized by ultrahigh
brightness and low slice energy spread. However, the beam quality from
simulations is closely related to the simulation setup, such as the grid sizes
and the Maxwell field solver. In Sec. II, a combination of fine grid sizes
($\mathrm{d}z=\mathrm{d}r=\frac{1}{512}k_{\mathrm{p0}}^{-1}$) and the recently
developed Xu solver Xu et al. (2020) are used to model the injected electrons
with high-fidelity, which will be referred as ‘Simulation Xu/fine’ in the
following discussions. In Fig. 8, we show that the qualities of the injected
beams in simulations with coarse resolutions or the Yee solver are much worse
than the results from Simulation Xu/fine.

Figure 9:  (a) Comparison of the evolution of the laser, the average energy of
the injected electrons from simulations with different settings. (b) The
energy of the electrons at different slices
($\frac{1}{16}k_{\mathrm{p0}}^{-1}$ width) from these simulations at
$\omega_{\mathrm{p0}}t=250$.

Simulation Yee/fine uses the same setup of Simulation Xu/fine except the Yee
field solver Yee (1966) is used instead of the Xu solver. Simulation
Yee/coarse uses a much coarser grid resolution
($\mathrm{d}z=\frac{1}{150}k_{\mathrm{p0}}^{-1},\mathrm{d}r=\frac{1}{10}k_{\mathrm{p0}}^{-1}$)
and the Yee solver with 128 macro-particles per cell to improve the
statistics. The time step of $\mathrm{d}t=0.00625\omega_{\mathrm{p0}}^{-1}$ is
close the Courant limit. The longitudinal phase space and the real space
distribution of the injected beams in these three simulations are compared in
Fig. 8. In Simulation Yee/fine, the beam is focused to a similar real space
distribution as in Simulation Xu/fine. However, these high-density electrons
excite unphysical high-frequency electromagnetic fields with large amplitude
Xu et al. (2020) when the Yee solver is used. These unphysical fields then
modulate the longitudinal phase space of the beam and lead a much larger slice
energy spread (several $\sim 100~{}\mathrm{M}\mathrm{eV}$). The beam in
Simulation Yee/coarse has a large emittance
($>1~{}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$) and thus a large spot size.
Since the electrons are not tightly focused, their density is much lower than
in Simulation Xu/fine and Yee/fine and they excite numerical fields with lower
amplitude than in Simulation Yee/fine. Thus the energy modulation is not as
severe as in Simulation Yee/fine and the slice energy spread is several $\sim
10~{}\mathrm{M}\mathrm{eV}$. The total charge of the injected beams in all
three simulation is very similar but the current profiles in Simulations
Yee/fine and Yee/coarse are modulated as a result of the slippage between the
electrons with unphysical energy modulation.

As shown in Fig. 8, the simulation resolution and field solver can
significantly impact the quality of the injected beams due to the subtle self-
interactions between the high density particles. However, we want to stress
that quantities that depend on physics on larger spatial scales are not
sensitive to the choices of the grid sizes and the solver, such as the
evolution of the laser driver and the average energy of the injected beam as
shown in Fig. 9(a). We also compare the slice mean energy of the injected
beams in these simulations in Fig. 9(b). When the duration of the slice is
chosen as $\frac{1}{16}k_{\mathrm{p0}}^{-1}$ which is longer than the
wavelength ($\sim 0.03k_{\mathrm{p0}}^{-1}$) of the unphysical energy
modulation, the slice mean energy has similar profiles in the three
simulations. This indicates simulations with low resolutions and the Yee
solver can be used to predict the coarse energy space of the injected beam,
and they may be recommended in some situations due to the much lower
computational cost. Machine learning algorithms may be combined with these low
cost simulations to predict these properties for injected electrons.

##  Appendix B Appendix B: Simulation setup

For the simulations shown in Sec. II, we use a moving window propagating at
speed of light in vacuum $c$ with a box size of $14k_{\mathrm{p0}}^{-1}\times
12k_{\mathrm{p0}}^{-1}$ and $7168\times 6144$ cells along the $z$ and $r$
directions, respectively. The grid sizes are chosen as
$\mathrm{d}z=\mathrm{d}r=\frac{1}{512}k_{\mathrm{p0}}^{-1}\approx
7.35~{}\mathrm{n}\mathrm{m}$ to resolve the tightly focused injected electrons
while the time step is
$\mathrm{d}t=\frac{1}{1024}\omega_{\mathrm{p0}}^{-1}\approx
12.2~{}\mathrm{a}\mathrm{s}$ to satisfy the CFL condition. To represent the
plasma electrons, we use 8 macro-particles per cell (ppc) where they are
distributed at one $r-z$ location and 8 different values of the azimuthal
angle $\theta$. All physical quantities in Q3D code Lifschitz et al. (2009);
Davidson et al. (2015) are decomposed into azimuthal modes
$\mathrm{exp}(im\theta)$, where $\theta$ is the azimuthal angle. Two modes
with $m=0$ and $1$ are included in the simulations shown here to describe the
linearly polarized laser pulses. The Xu solver Xu et al. (2020) is used to
eliminate the numerical Cherenkov radiation Godfrey (1974); Godfrey and Vay
(2013); Xu et al. (2013) and the numerical space charge field Xu et al. (2020)
from the high-density relativistic injected electrons and model the injection
and acceleration with high-fidelity. The laser’s electric field has a
symmetric temporal profile of $10\tau^{3}-15\tau^{4}+6\tau^{5}$, where
$\tau=\frac{\sqrt{2}(t-t_{0})}{\tau_{\mathrm{FWHM}}}$ while the radius of the
plasma column is
$41.4~{}\mathrm{\SIUnitSymbolMicro}\mathrm{m}~{}(11k_{\mathrm{p0}}^{-1})$.

The simulation in Fig. 5 uses a moving window with a box size of $16\times
16~{}(c/\omega_{\mathrm{p0}})^{2}$ whose grid sizes are
$\mathrm{d}z=\mathrm{d}r=\frac{1}{512}k_{\mathrm{p0}}^{-1}\approx
3.29~{}\mathrm{n}\mathrm{m}$. The time step is
$\mathrm{d}t=\frac{1}{1024}\omega_{\mathrm{p0}}^{-1}\approx
5.48~{}\mathrm{a}\mathrm{s}$ and 8 macro-particles per cell (distributed at 8
different values of $\theta$ at one $r-z$ location) are used to represent the
plasma electrons. The simulation in Fig. 6 uses a larger moving window with a
box size of $32k_{\mathrm{p0}}^{-1}\times 32k_{\mathrm{p0}}^{-1}$ to model the
larger plasma wave wake excited by the more intense laser pulse. Limited by
computational cost, a resolution with
$\mathrm{d}z=\mathrm{d}r=\frac{1}{256}k_{\mathrm{p0}}^{-1}\approx
6.58~{}\mathrm{n}\mathrm{m}$ and a time step
$\mathrm{d}t=\frac{1}{512}\omega_{\mathrm{p0}}^{-1}\approx
10.96~{}\mathrm{a}\mathrm{s}$ is used. We still use two modes with $m=0$ and 1
in the simulations and the Xu solver. We use 8 macro-particles per cell
(distributed at 8 different values of $\theta$ at one $r-z$ location) for the
plasma electrons. The plasma electrons in both simulations have an initial
$0.1~{}\mathrm{eV}$ temperature. Note $n_{\mathrm{p0}}=2\times
10^{18}~{}\mathrm{c}\mathrm{m}^{-3}$ in Sec. II and
$n_{\mathrm{p0}}=10^{19}~{}\mathrm{c}\mathrm{m}^{-3}$ in Sec. III.

##  Appendix C Appendix C: Initial azimuthal distribution of the injected
electrons

Figure 10:  The charge density distribution of the injected electrons in the
$\theta_{i}-z_{i}$ plane. (a) Q3D simulation for $n_{\mathrm{p0}}=2\times
10^{18}~{}\mathrm{c}\mathrm{m}^{-3}$ and $g=0.001$. (b) Q3D simulation for
$n_{\mathrm{p0}}=2\times 10^{18}~{}\mathrm{c}\mathrm{m}^{-3}$ and $g=0$. (c)
Q3D simulation for $n_{\mathrm{p0}}=10^{19}~{}\mathrm{c}\mathrm{m}^{-3}$ and
$g=0$. (d) Q3D simulation for
$n_{\mathrm{p0}}=10^{19}~{}\mathrm{c}\mathrm{m}^{-3}$, $g=0$ and the laser
pulse driver is linearly polarized along $y$-direction. (e) Q3D simulation for
$n_{\mathrm{p0}}=10^{19}~{}\mathrm{c}\mathrm{m}^{-3}$, $g=0$ and the laser
pulse driver is circularly polarized. (f) Q3D simulation with low resolution
and more macro-particles per cell for
$n_{\mathrm{p0}}=10^{19}~{}\mathrm{c}\mathrm{m}^{-3}$ and $g=0$. (g) 3D
simulation for $n_{\mathrm{p0}}=10^{19}~{}\mathrm{c}\mathrm{m}^{-3}$ and
$g=0$. Note the peak intensity of the circularly polarized laser pulse is
reduced by a factor of $\sqrt{2}$ to ensure it has the same ponderomotive
force as the linearly polarized laser.

When the plasma density is much lower than the critical density of the laser
driver and the laser pulse consists of many cycles, the ponderomotive
approximation is valid and the plasma wake is nearly axisymmetric. As a
result, the initial positions of the injected electrons are distributed
uniformly along the azimuthal direction. As the plasma density increases, a
shorter laser pulse driver is needed to excite the wake effectively, thus the
oscillatory laser electric field starts to play a role and the injection
becomes non-uniform azimuthally which leads to a transversely asymmetric beam,
i.e., the beam has different spot sizes and emittance along the two transverse
directions.

The initial azimuthal angle $\theta_{i}$ of the injected electrons for
different cases are shown in Fig. 10. When $n_{\mathrm{p0}}=2\times
10^{18}~{}\mathrm{c}\mathrm{m}^{-3}$, there are electrons injected from all
angles for both the $g=0$ and $g=0.001$ cases. Although their distribution
with the initial angle is not uniform, i.e., more electrons originate from
$\theta_{i}=\pm\frac{\pi}{2}$ [Fig. 10(a)-(b)], the distribution is
approximately symmetric between $k_{\mathrm{p0}}z_{i}\approx 40$ and $\sim
140$. At the end of the injection ($k_{\mathrm{p0}}z_{\mathrm{i}}\gtrsim
140$), the angular distribution becomes asymmetric which leads to off-axis
beam tails as shown in Fig. 2. The reason behind this asymmetric angular
distribution has not been understood thoroughly. Note that 8 macro-particles
are initialized azimuthally with angles
$\left(-\pi,-\frac{3}{4}\pi,-\frac{1}{2}\pi,-\frac{1}{4}\pi,0,\frac{1}{4}\pi,\frac{1}{2}\pi,\frac{3}{4}\pi\right)$
in these Q3D simulations.

When $n_{\mathrm{p0}}=10^{19}~{}\mathrm{c}\mathrm{m}^{-3}$ and a laser pulse
driver with fewer cycles is used, only electrons originating from two angles,
$\theta_{i}=\pm\frac{\pi}{2}$, are injected as shown 10(c). Since the angular
distribution is still symmetric about the $x$ and $y$ axes, the injected beam
is characterized by a low emittance. The values of the injected angles depends
on the polarization direction of the laser driver as shown in Fig. 10(c)-(d).
When a circularly polarized laser is used, the augular distribution becomes
much more uniform. This concentration on initial angles when a linearly
polarized laser pulse is used is confirmed in a Q3D simulation with lower
resolution ($\mathrm{d}z=\mathrm{d}r=\frac{1}{256}k_{\mathrm{p0}}^{-1}$) but
more macro-particles per cell (f) and a full-3D simulation (g) with lower
resolution
($\mathrm{d}z=\mathrm{d}x=\mathrm{d}y=\frac{1}{80}k_{\mathrm{p0}}^{-1}$).

## References

  * Tajima and Dawson (1979) T. Tajima and J. M. Dawson, Phys. Rev. Lett. 43, 267 (1979).
  * Chen et al. (1985) P. Chen, J. Dawson, R. W. Huff, and T. Katsouleas, Physical review letters 54, 693 (1985).
  * Esarey et al. (2009) E. Esarey, C. Schroeder, and W. Leemans, Reviews of modern physics 81, 1229 (2009).
  * Joshi et al. (2020) C. Joshi, S. Corde, and W. Mori, Physics of Plasmas 27, 070602 (2020).
  * Pellegrini et al. (2016) C. Pellegrini, A. Marinelli, and S. Reiche, Reviews of Modern Physics 88, 015006 (2016).
  * Kim et al. (2017) K.-J. Kim, Z. Huang, and R. Lindberg, _Synchrotron radiation and free-electron lasers_ (Cambridge university press, 2017).
  * Benedetti et al. (2022) C. Benedetti, S. Bulanov, E. Esarey, C. Geddes, A. Gonsalves, A. Huebl, R. Lehe, K. Nakamura, C. Schroeder, D. Terzani, et al., arXiv preprint arXiv:2203.08366 (2022).
  * Clayton et al. (1993) C. Clayton, K. Marsh, A. Dyson, M. Everett, A. Lal, W. Leemans, R. Williams, and C. Joshi, Physical review letters 70, 37 (1993).
  * Everett et al. (1994) M. Everett, A. Lal, D. Gordon, C. Clayton, K. Marsh, and C. Joshi, Nature 368, 527 (1994).
  * Modena et al. (1995) A. Modena, Z. Najmudin, A. Dangor, C. Clayton, K. Marsh, C. Joshi, V. Malka, C. Darrow, C. Danson, D. Neely, et al., nature 377, 606 (1995).
  * Umstadter et al. (1996a) D. Umstadter, S.-Y. Chen, A. Maksimchuk, G. Mourou, and R. Wagner, Science 273, 472 (1996a).
  * Wagner et al. (1997) R. Wagner, S.-Y. Chen, A. Maksimchuk, and D. Umstadter, Physical Review Letters 78, 3125 (1997).
  * Malka et al. (2002) V. Malka, S. Fritzler, E. Lefebvre, M.-M. Aleonard, F. Burgy, J.-P. Chambaret, J.-F. Chemin, K. Krushelnick, G. Malka, S. Mangles, et al., Science 298, 1596 (2002).
  * Mangles et al. (2004) S. P. Mangles, C. Murphy, Z. Najmudin, A. G. R. Thomas, J. Collier, A. E. Dangor, E. Divall, P. Foster, J. Gallacher, C. Hooker, et al., Nature 431, 535 (2004).
  * Geddes et al. (2004) C. Geddes, C. Toth, J. Van Tilborg, E. Esarey, C. Schroeder, D. Bruhwiler, C. Nieter, J. Cary, and W. Leemans, Nature 431, 538 (2004).
  * Faure et al. (2004) J. Faure, Y. Glinec, A. Pukhov, S. Kiselev, S. Gordienko, E. Lefebvre, J.-P. Rousseau, F. Burgy, and V. Malka, Nature 431, 541 (2004).
  * Umstadter et al. (1996b) D. Umstadter, J. Kim, and E. Dodd, Physical review letters 76, 2073 (1996b).
  * Esarey et al. (1997a) E. Esarey, R. Hubbard, W. Leemans, A. Ting, and P. Sprangle, Physical Review Letters 79, 2682 (1997a).
  * Hemker et al. (1998) R. Hemker, K.-C. Tzeng, W. Mori, C. Clayton, and T. Katsouleas, Physical Review E 57, 5920 (1998).
  * Fubiani et al. (2004) G. Fubiani, E. Esarey, C. Schroeder, and W. Leemans, Physical Review E 70, 016402 (2004).
  * Kotaki et al. (2004) H. Kotaki, S. Masuda, M. Kando, J. Koga, and K. Nakajima, Physics of Plasmas 11, 3296 (2004).
  * Faure et al. (2006) J. Faure, C. Rechatin, A. Norlin, A. Lifschitz, Y. Glinec, and V. Malka, Nature 444, 737 (2006).
  * Davoine et al. (2009) X. Davoine et al., Phys. Rev. Lett. 102, 065001 (2009).
  * Lehe et al. (2013) R. Lehe, A. Lifschitz, X. Davoine, C. Thaury, and V. Malka, Physical review letters 111, 085005 (2013).
  * Chen et al. (2022) Q. Chen, D. Maslarova, J. Wang, S. X. Lee, V. Horný, and D. Umstadter, Phys. Rev. Lett. 128, 164801 (2022), URL https://link.aps.org/doi/10.1103/PhysRevLett.128.164801.
  * Vieira et al. (2011) J. Vieira, S. Martins, V. Pathak, R. Fonseca, W. Mori, and L. Silva, Physical review letters 106, 225001 (2011).
  * Bulanov et al. (1998) S. Bulanov, N. Naumova, F. Pegoraro, and J. Sakai, Phys. Rev. E 58, R5257 (1998).
  * Suk et al. (2001) H. Suk, N. Barov, J. B. Rosenzweig, and E. Esarey, Phys. Rev. Lett. 86, 1011 (2001).
  * Geddes et al. (2008) C. G. R. Geddes et al., Phys. Rev. Lett. 100, 215004 (2008).
  * Gonsalves et al. (2011) A. Gonsalves, K. Nakamura, C. Lin, D. Panasenko, S. Shiraishi, T. Sokollik, C. Benedetti, C. Schroeder, C. Geddes, J. Van Tilborg, et al., Nature Physics 7, 862 (2011).
  * Buck et al. (2013) A. Buck, J. Wenz, J. Xu, K. Khrennikov, K. Schmid, M. Heigoldt, J. M. Mikhailova, M. Geissler, B. Shen, F. Krausz, et al., Phys. Rev. Lett. 110, 185006 (2013).
  * Li et al. (2013a) F. Li, Z. Sheng, Y. Liu, J. Meyer-ter Vehn, W. Mori, W. Lu, and J. Zhang, Physical review letters 110, 135002 (2013a).
  * Moore et al. (1999) C. I. Moore et al., Phys. Rev. Lett. 82, 1688 (1999).
  * Chen et al. (2006) M. Chen, Z.-M. Sheng, Y.-Y. Ma, and J. Zhang, Journal of applied physics 99, 056109 (2006).
  * Oz et al. (2007) E. Oz, S. Deng, T. Katsouleas, P. Muggli, C. D. Barnes, I. Blumenfeld, F. J. Decker, P. Emma, M. J. Hogan, R. Ischebeck, et al., Phys. Rev. Lett. 98, 084801 (2007).
  * Rowlands-Rees et al. (2008) T. Rowlands-Rees, C. Kamperidis, S. Kneip, A. Gonsalves, S. Mangles, J. Gallacher, E. Brunetti, T. Ibbotson, C. Murphy, P. Foster, et al., Physical Review Letters 100, 105005 (2008).
  * Pak et al. (2010) A. Pak, K. A. Marsh, S. F. Martins, W. Lu, W. B. Mori, and C. Joshi, Phys. Rev. Lett. 104, 025003 (2010).
  * McGuffey et al. (2010) C. McGuffey, A. G. R. Thomas, W. Schumaker, T. Matsuoka, V. Chvykov, F. J. Dollar, G. Kalintchenko, V. Yanovsky, A. Maksimchuk, K. Krushelnick, et al., Phys. Rev. Lett. 104, 025004 (2010).
  * Clayton et al. (2010) C. E. Clayton, J. E. Ralph, F. Albert, R. A. Fonseca, S. H. Glenzer, C. Joshi, W. Lu, K. A. Marsh, S. F. Martins, W. B. Mori, et al., Phys. Rev. Lett. 105, 105003 (2010).
  * Liu et al. (2011) J. S. Liu, C. Q. Xia, W. T. Wang, H. Y. Lu, C. Wang, A. H. Deng, W. T. Li, H. Zhang, X. Y. Liang, Y. X. Leng, et al., Phys. Rev. Lett. 107, 035001 (2011).
  * Pollock et al. (2011) B. B. Pollock, C. E. Clayton, J. E. Ralph, F. Albert, A. Davidson, L. Divol, C. Filip, S. H. Glenzer, K. Herpoldt, W. Lu, et al., Phys. Rev. Lett. 107, 045001 (2011).
  * Maier et al. (2020) A. R. Maier, N. M. Delbos, T. Eichner, L. Hübner, S. Jalas, L. Jeppe, S. W. Jolly, M. Kirchen, V. Leroux, P. Messner, et al., Physical Review X 10, 031039 (2020).
  * Xu et al. (2014a) X. L. Xu, J. F. Hua, F. Li, C. J. Zhang, L. X. Yan, Y. C. Du, W. H. Huang, H. B. Chen, C. X. Tang, W. Lu, et al., Phys. Rev. Lett. 112, 035003 (2014a).
  * Xu et al. (2017) X. Xu, F. Li, W. An, T. Dalichaouch, P. Yu, W. Lu, C. Joshi, and W. Mori, Physical Review Accelerators and Beams 20, 111303 (2017).
  * Hidding et al. (2012) B. Hidding, G. Pretzler, J. B. Rosenzweig, T. Königstein, D. Schiller, and D. L. Bruhwiler, Phys. Rev. Lett. 108, 035001 (2012).
  * Li et al. (2013b) F. Li, J. F. Hua, X. L. Xu, C. J. Zhang, L. X. Yan, Y. C. Du, W. H. Huang, H. B. Chen, C. X. Tang, W. Lu, et al., Phys. Rev. Lett. 111, 015003 (2013b).
  * Bourgeois et al. (2013) N. Bourgeois, J. Cowley, and S. M. Hooker, Phys. Rev. Lett. 111, 155004 (2013).
  * Martinez de la Ossa et al. (2013) A. Martinez de la Ossa, J. Grebenyuk, T. Mehrling, L. Schaper, and J. Osterhoff, Phys. Rev. Lett. 111, 245003 (2013).
  * Yu et al. (2014a) L.-L. Yu, E. Esarey, C. B. Schroeder, J.-L. Vay, C. Benedetti, C. G. R. Geddes, M. Chen, and W. P. Leemans, Phys. Rev. Lett. 112, 125001 (2014a).
  * Xu et al. (2014b) X. L. Xu, Y. P. Wu, C. J. Zhang, F. Li, Y. Wan, J. F. Hua, C.-H. Pai, W. Lu, P. Yu, C. Joshi, et al., Phys. Rev. ST Accel. Beams 17, 061301 (2014b).
  * Zeng et al. (2015) M. Zeng, M. Chen, L.-L. Yu, W. B. Mori, Z.-M. Sheng, B. Hidding, D. Jaroszynski, and J. Zhang, Physical review letters 114, 084801 (2015).
  * Xu et al. (2016a) X. Xu, C.-H. Pai, C. Zhang, F. Li, Y. Wan, Y. Wu, J. Hua, W. Lu, W. An, P. Yu, et al., Physical review letters 117, 034801 (2016a).
  * Tooley et al. (2017) M. Tooley, B. Ersfeld, S. Yoffe, A. Noble, E. Brunetti, Z. Sheng, M. Islam, and D. Jaroszynski, Physical review letters 119, 044801 (2017).
  * Deng et al. (2019) A. Deng, O. Karger, T. Heinemann, A. Knetsch, P. Scherkl, G. G. Manahan, A. Beaton, D. Ullmann, G. Wittig, A. F. Habib, et al., Nature Physics 15, 1156 (2019).
  * Dalichaouch et al. (2020) T. N. Dalichaouch, X. L. Xu, F. Li, A. Tableman, F. S. Tsung, W. An, and W. B. Mori, Phys. Rev. Accel. Beams 23, 021304 (2020).
  * Xu et al. (2022a) X. Xu, J. Vieira, M. J. Hogan, C. Joshi, and W. B. Mori, Phys. Rev. Accel. Beams 25, 011302 (2022a), URL https://link.aps.org/doi/10.1103/PhysRevAccelBeams.25.011302.
  * Li et al. (2022) F. Li, T. N. Dalichaouch, J. R. Pierce, X. Xu, F. S. Tsung, W. Lu, C. Joshi, and W. B. Mori, Phys. Rev. Lett. 128, 174803 (2022), URL https://link.aps.org/doi/10.1103/PhysRevLett.128.174803.
  * Xu et al. (2022b) X. Xu, F. Li, F. S. Tsung, K. Miller, V. Yakimenko, M. J. Hogan, C. Joshi, and W. B. Mori, Nature communications 13, 1 (2022b).
  * Wang et al. (2022) J. Wang, M. Zeng, D. Li, X. Wang, W. Lu, and J. Gao, Matter and Radiation at Extremes 7, 054001 (2022).
  * Lu et al. (2006) W. Lu, C. Huang, M. Zhou, W. B. Mori, and T. Katsouleas, Phys. Rev. Lett. 96, 165002 (2006).
  * Tzoufras et al. (2008) M. Tzoufras et al., Phys. Rev. Lett. 101, 145002 (2008).
  * Kalmykov et al. (2009) S. Kalmykov, S. Yi, V. Khudik, and G. Shvets, Physical review letters 103, 135004 (2009).
  * Kalmykov et al. (2011) S. Y. Kalmykov, A. Beck, S. Yi, V. Khudik, M. C. Downer, E. Lefebvre, B. A. Shadwick, and D. Umstadter, Physics of Plasmas 18, 056704 (2011).
  * Zhang et al. (2016) Z. Zhang, W. Li, J. Liu, W. Wang, C. Yu, Y. Tian, K. Nakajima, A. Deng, R. Qi, C. Wang, et al., Physics of Plasmas 23, 053106 (2016).
  * Manahan et al. (2017) G. G. Manahan, A. Habib, P. Scherkl, P. Delinikolas, A. Beaton, A. Knetsch, O. Karger, G. Wittig, T. Heinemann, Z. Sheng, et al., Nature communications 8, 1 (2017).
  * Kirchen et al. (2021) M. Kirchen, S. Jalas, P. Messner, P. Winkler, T. Eichner, L. Hübner, T. Hülsenbusch, L. Jeppe, T. Parikh, M. Schnepp, et al., Physical review letters 126, 174801 (2021).
  * Pousa et al. (2019) A. F. Pousa, A. M. De La Ossa, R. Brinkmann, and R. Assmann, Physical review letters 123, 054801 (2019).
  * Ke et al. (2021) L. Ke, K. Feng, W. Wang, Z. Qin, C. Yu, Y. Wu, Y. Chen, R. Qi, Z. Zhang, Y. Xu, et al., Physical Review Letters 126, 214801 (2021).
  * Pukhov and Meyer-ter Vehn (2002) A. Pukhov and J. Meyer-ter Vehn, Applied Physics B 74, 355 (2002).
  * Tsung et al. (2004) F. S. Tsung, R. Narang, W. B. Mori, C. Joshi, R. Fonseca, and L. O. Silva, Physical review letters 93, 185002 (2004).
  * Xu et al. (2005) H. Xu, W. Yu, P. Lu, V. Senecha, F. He, B. Shen, L. Qian, R. Li, and Z. Xu, Physics of plasmas 12, 013105 (2005).
  * Lu et al. (2007) W. Lu et al., Phys. Rev. ST Accel. Beams 10, 061301 (2007).
  * Kostyukov et al. (2009) I. Kostyukov, E. Nerush, A. Pukhov, and V. Seredov, Physical review letters 103, 175003 (2009).
  * Martins et al. (2010) S. F. Martins, R. Fonseca, W. Lu, W. B. Mori, and L. Silva, Nature Physics 6, 311 (2010).
  * Leemans et al. (2006) W. Leemans et al., Nature Phys. 2, 696 (2006).
  * Kneip et al. (2009) S. Kneip et al., Phys. Rev. Lett. 103, 035002 (2009).
  * Froula et al. (2009) D. H. Froula et al., Phys. Rev. Lett. 103, 215006 (2009).
  * Hafz et al. (2008) N. A. Hafz, T. M. Jeong, I. W. Choi, S. K. Lee, K. H. Pae, V. V. Kulagin, J. H. Sung, T. J. Yu, K.-H. Hong, T. Hosokai, et al., Nature Photonics 2, 571 (2008).
  * Wang et al. (2013) X. Wang et al., Nature communications 4, 1988 (2013).
  * Kim et al. (2013) H. T. Kim et al., Phys. Rev. Lett. 111, 165002 (2013).
  * Leemans et al. (2014) W. Leemans, A. Gonsalves, H.-S. Mao, K. Nakamura, C. Benedetti, C. Schroeder, C. Tóth, J. Daniels, D. Mittelberger, S. Bulanov, et al., Physical review letters 113, 245002 (2014).
  * Gonsalves et al. (2019) A. J. Gonsalves, K. Nakamura, J. Daniels, C. Benedetti, C. Pieronek, T. C. H. de Raadt, S. Steinke, J. H. Bin, S. S. Bulanov, J. van Tilborg, et al., Phys. Rev. Lett. 122, 084801 (2019).
  * Miao et al. (2022) B. Miao, J. Shrock, L. Feder, R. Hollinger, J. Morrison, R. Nedbailo, A. Picksley, H. Song, S. Wang, J. Rocca, et al., Physical Review X 12, 031038 (2022).
  * Aniculaesei et al. (2022) C. Aniculaesei, T. Ha, S. Yoffe, E. McCary, M. M. Spinks, H. J. Quevedo, L. Labun, O. Z. Labun, R. Sain, A. Hannasch, et al., arXiv preprint arXiv:2207.11492 (2022).
  * Corde et al. (2013) S. Corde, C. Thaury, A. Lifschitz, G. Lambert, K. T. Phuoc, X. Davoine, R. Lehe, D. Douillet, A. Rousse, and V. Malka, Nature communications 4, 1 (2013).
  * Davidson et al. (2018) A. Davidson, A. Tableman, P. Yu, W. An, F. Tsung, W. Lu, R. Fonseca, and W. Mori, arXiv preprint arXiv:1805.08761 (2018).
  * Davidson et al. (2015) A. Davidson, A. Tableman, W. An, F. S. Tsung, W. Lu, J. Vieira, R. A. Fonseca, L. O. Silva, and W. B. Mori, Journal of Computational Physics 281, 1063 (2015).
  * Fonseca et al. (2002) R. Fonseca et al., Lecture notes in computer science 2331, 342 (2002).
  * Li et al. (2017) F. Li, P. Yu, X. Xu, F. Fiuza, V. K. Decyk, T. Dalichaouch, A. Davidson, A. Tableman, W. An, F. S. Tsung, et al., Computer Physics Communications 214, 6 (2017).
  * Xu et al. (2020) X. Xu, F. Li, F. S. Tsung, T. N. Dalichaouch, W. An, H. Wen, V. K. Decyk, R. A. Fonseca, M. J. Hogan, and W. B. Mori, Journal of Computational Physics p. 109451 (2020).
  * Esarey et al. (1997b) E. Esarey, P. Sprangle, J. Krall, and A. Ting, IEEE journal of quantum electronics 33, 1879 (1997b).
  * Mori (1997) W. Mori, IEEE Journal of Quantum Electronics 33, 1942 (1997).
  * Grebenyuk et al. (2014) J. Grebenyuk, A. M. de la Ossa, T. Mehrling, and J. Osterhoff, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 740, 246 (2014).
  * Dalichaouch et al. (2021) T. N. Dalichaouch, X. L. Xu, A. Tableman, F. Li, F. S. Tsung, and W. B. Mori, Physics of Plasmas 28, 063103 (2021), eprint https://doi.org/10.1063/5.0051282, URL https://doi.org/10.1063/5.0051282.
  * Rosenzweig et al. (2005) J. Rosenzweig, A. Cook, A. Scott, M. Thompson, and R. Yoder, Physical review letters 95, 195002 (2005).
  * An et al. (2017) W. An, W. Lu, C. Huang, X. Xu, M. J. Hogan, C. Joshi, and W. B. Mori, Physical Review Letters 118, 244801 (2017).
  * Godfrey (1974) B. B. Godfrey, Journal of Computational Physics 15, 504 (1974).
  * Godfrey and Vay (2013) B. B. Godfrey and J.-L. Vay, Journal of Computational Physics 248, 33 (2013).
  * Xu et al. (2013) X. Xu, P. Yu, S. F. Martins, F. S. Tsung, V. K. Decyk, J. Vieira, R. A. Fonseca, W. Lu, L. O. Silva, and W. B. Mori, Computer Physics Communications 184, 2503 (2013).
  * Yu et al. (2014b) P. Yu, X. Xu, V. K. Decyk, W. An, J. Vieira, F. S. Tsung, R. A. Fonseca, W. Lu, L. O. Silva, and W. B. Mori, Journal of Computational Physics 266, 124 (2014b).
  * Yu et al. (2015a) P. Yu, X. Xu, V. K. Decyk, F. Fiuza, J. Vieira, F. S. Tsung, R. A. Fonseca, W. Lu, L. O. Silva, and W. B. Mori, Computer Physics Communications 192, 32 (2015a).
  * Yu et al. (2015b) P. Yu, X. Xu, A. Tableman, V. K. Decyk, F. S. Tsung, F. Fiuza, A. Davidson, J. Vieira, R. A. Fonseca, W. Lu, et al., Computer Physics Communications 197, 144 (2015b).
  * Yu et al. (2016) P. Yu, X. Xu, A. Davidson, A. Tableman, T. Dalichaouch, F. Li, M. D. Meyers, W. An, F. S. Tsung, V. K. Decyk, et al., Journal of Computational Physics 316, 747 (2016).
  * Godfrey and Vay (2015) B. B. Godfrey and J.-L. Vay, Computer Physics Communications 196, 221 (2015).
  * Li et al. (2021) F. Li, K. G. Miller, X. Xu, F. S. Tsung, V. K. Decyk, W. An, R. A. Fonseca, and W. B. Mori, Computer Physics Communications 258, 107580 (2021).
  * Götzfried et al. (2020) J. Götzfried, A. Döpp, M. F. Gilljohann, F. M. Foerster, H. Ding, S. Schindler, G. Schilling, A. Buck, L. Veisz, and S. Karsch, Phys. Rev. X 10, 041015 (2020), URL https://link.aps.org/doi/10.1103/PhysRevX.10.041015.
  * Dickson et al. (2022) L. T. Dickson, C. I. D. Underwood, F. Filippi, R. J. Shalloo, J. B. Svensson, D. Guénot, K. Svendsen, I. Moulanier, S. D. Dufrénoy, C. D. Murphy, et al., Phys. Rev. Accel. Beams 25, 101301 (2022), URL https://link.aps.org/doi/10.1103/PhysRevAccelBeams.25.101301.
  * Emma et al. (2021) C. Emma, X. Xu, A. Fisher, R. Robles, J. MacArthur, J. Cryan, M. Hogan, P. Musumeci, G. White, and A. Marinelli, APL Photonics 6, 076107 (2021).
  * Zhang et al. (2019) C. Zhang, C.-K. Huang, K. A. Marsh, X. L. Xu, F. Li, M. Hogan, V. Yakimenko, S. Corde, W. B. Mori, and C. Joshi, Phys. Rev. Accel. Beams 22, 111301 (2019).
  * Glinec et al. (2008) Y. Glinec, J. Faure, A. Lifschitz, J. M. Vieira, R. A. Fonseca, L. O. Silva, and V. Malka, Europhysics Letters 81, 64001 (2008), URL https://dx.doi.org/10.1209/0295-5075/81/64001.
  * Mangles et al. (2009) S. P. D. Mangles, G. Genoud, S. Kneip, M. Burza, K. Cassou, B. Cros, N. P. Dover, C. Kamperidis, Z. Najmudin, A. Persson, et al., Applied Physics Letters 95, 181106 (2009), eprint https://doi.org/10.1063/1.3258022, URL https://doi.org/10.1063/1.3258022.
  * Popp et al. (2010) A. Popp, J. Vieira, J. Osterhoff, Z. Major, R. Hörlein, M. Fuchs, R. Weingartner, T. P. Rowlands-Rees, M. Marti, R. A. Fonseca, et al., Phys. Rev. Lett. 105, 215001 (2010), URL https://link.aps.org/doi/10.1103/PhysRevLett.105.215001.
  * Beaurepaire et al. (2015) B. Beaurepaire, A. Vernier, M. Bocoum, F. Böhle, A. Jullien, J.-P. Rousseau, T. Lefrou, D. Douillet, G. Iaquaniello, R. Lopez-Martens, et al., Phys. Rev. X 5, 031012 (2015), URL https://link.aps.org/doi/10.1103/PhysRevX.5.031012.
  * Stupakov (2017) G. Stupakov, Physics of Plasmas 24, 113110 (2017), eprint https://doi.org/10.1063/1.4990048, URL https://doi.org/10.1063/1.4990048.
  * Ma et al. (2018) Y. Ma, D. Seipt, S. J. D. Dann, M. J. V. Streeter, C. A. J. Palmer, L. Willingale, and A. G. R. Thomas, Physics of Plasmas 25, 113105 (2018), eprint https://doi.org/10.1063/1.5054807, URL https://doi.org/10.1063/1.5054807.
  * Mehrling et al. (2012) T. Mehrling et al., Phys. Rev. ST Accel. Beams 15, 111303 (2012).
  * Antici et al. (2012) P. Antici, A. Bacci, C. Benedetti, E. Chiadroni, M. Ferrario, A. R. Rossi, L. Lancia, M. Migliorati, A. Mostacci, L. Palumbo, et al., Journal of Applied Physics 112, 044902 (2012), URL http://scitation.aip.org/content/aip/journal/jap/112/4/10.1063/1.4740456.
  * Migliorati et al. (2013) M. Migliorati et al., Phys. Rev. ST Accel. Beams 16, 011302 (2013).
  * Floettmann (2014) K. Floettmann, Phys. Rev. ST Accel. Beams 17, 054402 (2014), URL https://link.aps.org/doi/10.1103/PhysRevSTAB.17.054402.
  * Dornmair et al. (2015) I. Dornmair, K. Floettmann, and A. R. Maier, Phys. Rev. ST Accel. Beams 18, 041302 (2015), URL https://link.aps.org/doi/10.1103/PhysRevSTAB.18.041302.
  * Xu et al. (2016b) X. L. Xu, J. F. Hua, Y. P. Wu, C. J. Zhang, F. Li, Y. Wan, C.-H. Pai, W. Lu, W. An, P. Yu, et al., Phys. Rev. Lett. 116, 124801 (2016b).
  * Zhao et al. (2020) Y. Zhao, W. An, X. Xu, F. Li, L. Hildebrand, M. J. Hogan, V. Yakimenko, C. Joshi, and W. B. Mori, Phys. Rev. Accel. Beams 23, 011302 (2020), URL https://link.aps.org/doi/10.1103/PhysRevAccelBeams.23.011302.
  * Ariniello et al. (2019) R. Ariniello, C. E. Doss, K. Hunt-Stone, J. R. Cary, and M. D. Litos, Phys. Rev. Accel. Beams 22, 041304 (2019), URL https://link.aps.org/doi/10.1103/PhysRevAccelBeams.22.041304.
  * Labat et al. (2018) M. Labat, A. Loulergue, T. Andre, I. Andriyash, A. Ghaith, M. Khojoyan, F. Marteau, M. Valléau, F. Briquez, C. Benabderrahmane, et al., Physical Review Accelerators and Beams 21, 114802 (2018).
  * Xu et al. (2023) X. Xu, J. Liu, T. Dalichaouch, F. S. Tsung, Z. Zhang, Z. Huang, M. J. Hogan, X. Yan, C. Joshi, and W. B. Mori, arXiv preprint arXiv:2302.08864 (2023).
  * Benedetti et al. (2018) A. Benedetti, M. Tamburini, and C. H. Keitel, Nature Photonics 12, 319 (2018).
  * Sampath et al. (2021) A. Sampath, X. Davoine, S. Corde, L. Gremillet, M. Gilljohann, M. Sangal, C. H. Keitel, R. Ariniello, J. Cary, H. Ekerfelt, et al., Phys. Rev. Lett. 126, 064801 (2021), URL https://link.aps.org/doi/10.1103/PhysRevLett.126.064801.
  * Sahai et al. (2020) A. A. Sahai, T. Tajima, and V. D. Shiltsev, arXiv preprint arXiv:2004.09452 (2020).
  * Yakimenko et al. (2019) V. Yakimenko, S. Meuren, F. Del Gaudio, C. Baumann, A. Fedotov, F. Fiuza, T. Grismayer, M. J. Hogan, A. Pukhov, L. O. Silva, et al., Phys. Rev. Lett. 122, 190404 (2019), URL https://link.aps.org/doi/10.1103/PhysRevLett.122.190404.
  * Matheron et al. (2022) A. Matheron, P. Claveria, R. Ariniello, H. Ekerfelt, F. Fiuza, S. Gessner, M. Gilljohann, M. Hogan, C. Keitel, A. Knetsch, et al., arXiv preprint arXiv:2209.14280 (2022).
  * Yee (1966) K. Yee, IEEE Transactions on antennas and propagation 14, 302 (1966).
  * Lifschitz et al. (2009) A. F. Lifschitz, X. Davoine, E. Lefebvre, J. Faure, C. Rechatin, and V. Malka, Journal of Computational Physics 228, 1803 (2009).