# Stable radiation field positron acceleration in a micro-tube

Meiyu Si,1,2,3 Yongsheng Huang,2∗ Manqi Ruan,1 Baifei Shen,4 Zhangli Xu,4
Tongpu Yu,5 Xiongfei Wang,6,7 Yuan Chen,8

1Institute of High Energy Physics, Chinese Academy of Sciences, Beijing
100049, China
2School of Science, Shenzhen Campus of Sun Yat-sen University, Shenzhen
518107, China
3University of Chinese Academy of Sciences, Beijing 100049, China
4Department of Physics, Shanghai Normal University, Shanghai 200234, China
5Department of Physics, National University of Defense Technology, Changsha
410073, China
6School of Physical Science and Technology, Lanzhou University, Lanzhou
730000, China
7Lanzhou Center for Theoretical Physics, Laboratory of Theoretical Physics of
Gansu Province
and Frontiers Science Center for Rare Isotopes, Lanzhou University, Lanzhou
730000, China
8The Institute for Advanced Studies of Wuhan University, 299, Bayi Road, Wuhan
430072, China

∗ E-mail<EMAIL_ADDRESS>
> Nowadays, there is a desperate need for an ultra-acceleration-gradient
> method for antimatter particles, which holds great significance in exploring
> the origin of matter, CP violation, astrophysics, and medical physics.
> Compared to traditional accelerators with low gradients and a limited
> acceleration region for positrons in laser-driven charge separation fields,
> we propose an innovative high-gradient positron acceleration mechanism with
> implementation advantages. Injecting a relativistic electron beam into a
> dense plasma micro-tube generates a stable and periodic high-intensity mid-
> infrared radiation (mid-IR) field, reaching tens of $\mathrm{GV/m}$. This
> field, propagating synchronously with the electron beam, achieves a 1
> $\mathrm{GeV}$ energy gain for the positron bunch within 140 picoseconds
> with a minimal energy spread—approximately 1.56$\%$ during a stable phase.
> By utilizing continuous mid-IR, the efficiency of energy transfer from the
> electron beam to either a single positron bunch or three positron bunches
> simultaneously could reach up to 20$\%$ and 40$\%$, respectively. This
> acceleration scheme can achieve cascaded acceleration for a single positron
> bunch and series acceleration for multiple positron bunches in a continuous,
> stable, and efficient manner.

##  1 Introduction

The high-efficiency particle acceleration technology holds unique importance
in astroparticle physics, medical physics, and material sciences[1, 2, 3].
Traditional accelerators are limited by the ionization threshold of
materials[4, 5], typically reaching gradients of around hundreds of
$\mathrm{MV/m}$. Therefore, to obtain high-energy particle beams, larger
accelerator facilities are required, leading to increased costs. Laser and
particle beam-driven plasma charge separation wakefields can generate higher
acceleration gradients, typically ranging from several tens to several
hundreds of $\mathrm{GV/m}$, while allowing control over the energy spread and
beam emittance of the accelerated particles[6, 7, 8, 9, 10, 11, 12].
Significant progress has been made in electron beam acceleration in this
regard. Simultaneously, various positron acceleration schemes are continuously
being proposed. The fundamental plasma bubble acceleration, driven by a laser
or an electron beam[13, 14, 15, 16, 17, 18, 19], results in positrons
undergoing a shorter acceleration phase in comparison to electron plasma
acceleration, which can lead to instability. Concurrently, strong transverse
forces can also diminish the quality of the positron beam. Therefore, the
concept of electron beam driving within a hollow plasma channel is proposed,
aiming to achieve longitudinally stable acceleration for positrons[20, 21, 22,
23]. For proton beam driven acceleration, there is an increase in particle
divergence due to the strong Coulomb force, with additional four-stage iron
focusing guidance applied to the hollow channel[24]. When considering the
driving of electron beams, the density of the hollow plasma channel has a
different effect on positron acceleration. Positron acceleration in low-
density channels occurs during the phase formed between the first and second
bubbles. Addressing the stability of the acceleration phase and the
preparation of the given-density plasma channel is challenged.

Ultra-short electron beams passing through the copper film target interface
generate strong coherent transition radiation (CTR), which can be used to
capture and accelerate positron beams produced through the Bethe-Heitler (BH)
process in the target[25]. Additionally, the issues of radiation and electron
beam divergence are suggested to be solved by strengthening the magnetic field
and using a proper waveguide[26]. Simultaneously, single-cycle tunable
infrared pulses generated in density-tailored plasma can be used to drive the
acceleration of positrons[27, 28]. In this paper, we propose a novel mechanism
based on the continuous emission of high-intensity mid-infrared radiation
(mid-IR). This radiation is generated by the oscillations of surface-nanometer
electron films driven by a relativistic electron beam in a plasma and/or metal
micro-tube. This method allows for the construction of a highly efficient and
stable uniform field structure. The stable wavelength of the periodic
acceleration field is approximately 24 $\mathrm{\mu m}$ in the mid-IR range,
and it can be tuned by adjusting the micro-tube radius, for example, into the
terahertz range. The acceleration field can reach several tens of
$\mathrm{GV/m}$ and depends on the charge of the driving electron beam. For an
electron beam with a charge of several tens of $\mathrm{nC}$, the acceleration
field can even reach several hundreds of $\mathrm{GV/m}$, resulting in
positron acceleration energy gains ranging from tens to hundreds of
$\mathrm{GeV}$.

The self-consistent fields of a relativistic electron beam in a plasma micro-
tube and the high-intensity continuous mid-IR generated by surface-nanometer
film-plasma oscillations were analyzed. Field validation was performed by
injecting a positron witness bunch. The results indicate that positrons can
achieve energy gains at the GeV level within 140 $\mathrm{ps}$, with an
achievable energy spread of up to 1.56$\%$ and an efficiency of up to 20$\%$.
The quality of the positron beam can be adjusted by modifying the driving
parameters, allowing for positron energies ranging from $\mathrm{GeV}$ to even
hundreds of $\mathrm{GeV}$. In the uniform mid-IR acceleration field, the
transverse field ($E_{y}-cB_{z}$) focuses the positron bunch within the first
21 $\mathrm{ps}$, followed by weak defocusing until the transverse field
reduces to zero due to energy loss and beam break of the electron beam.
Through several 2-dimensional particle-in-cell (PIC) simulations, the
transverse spatial misalignment of positron injection was allowed to be a few
microns if a 10$\%$ relative loss in energy transfer efficiency is acceptable,
where the accelerated energy and energy spread are hardly affected. The design
scheme for cascaded acceleration of a single positron bunch and series
acceleration of multiple positron bunches is presented in the ’Discussion’
section.

##  2 Analysis of $e^{+}$ acceleration process

###  2.1 Self-fields of relativistic $e^{-}$ beam in a plasma micro-tube

For a relativistic electron beam propagating in the $x$ direction in free
space with a normalized velocity $\beta_{0}=v_{x}/c$ and a density
distribution $\rho_{b}$, the initial current is given as
$J(x,t=0)=\beta_{0}\rho_{b}$. The relativistic Poisson’s equation in the
laboratory frame can be written as [29, 30, 31]

$\bigg{[}\frac{1}{\gamma^{2}_{0}}\frac{\partial^{2}}{\partial^{2}_{x^{2}}}+\frac{\partial^{2}}{\partial^{2}_{y^{2}}}+\frac{\partial^{2}}{\partial^{2}_{z^{2}}}\bigg{]}\Phi=-\rho_{b},$
(1)

where $\gamma_{0}=1/\sqrt{1-\beta^{2}_{0}}$ is the Lorentz factor of the
relativistic electron beam, and $\Phi$ is the electrostatic potential. To
solve the initial potential $\Phi$ in free space, the standard fast Fourier
transformation (FFT) algorithm is used to perform the cosine transform.
Therefore, the electric fields of the initial electron beam can be computed as
$E_{x}=-\frac{1}{\gamma_{0}^{2}}\partial_{x}\Phi,E_{y}=-\partial_{y}\Phi$. Due
to the $\frac{1}{\gamma_{0}^{2}}$ factor, the longitudinal electric field
$E_{x}$ has a much smaller magnitude than the transverse electric field
$E_{y}$.

In this study, the initial electric field distribution was obtained through a
PIC simulation for a Gaussian electron bunch with a charge of 3.402
$\mathrm{nC}$ and a Lorentz factor of $\gamma_{0}\approx 1957$ corresponding
to an electron beam energy of 1 $\mathrm{GeV}$. Detailed parameters can be
found in the Methods section. The initial $E_{x}$ and $E_{y}$ of the
relativistic electron beam are shown in Figs. 1(A) and (B), respectively, with
their subplots displaying the generated electrostatic fields. In the case of a
plasma micro-tube with a radius of 50 $\mathrm{\mu m}$, the $E_{y}$ of the
relativistic electron beam was applied to the inner walls due to its larger
value (approximately 200 $\mathrm{GV/m}$) compared to that of a static
electron bunch. The field decays exponentially as it passes through the dense
plasma film (now denoted as $E_{yp}$), and can be expressed as

$\begin{split}E_{yp}=E_{0}e^{-\sqrt{k_{x}^{2}-1+\omega^{2}_{pe}}\delta y},\\\
\delta y=y^{\prime}-r,\end{split}$ (2)

where $r=\sqrt{y^{2}+z^{2}}$ is the radius of the plasma micro-tube,
$y^{\prime}$ is the penetration depth of the electric field in the plasma, and
$\delta y$ is the charge separation distance. The plasma frequency was
calculated as $\omega_{pe}=\sqrt{4\pi e^{2}\rho/m}$, which is approximately
$5.6\times 10^{15}\mathrm{Hz}$, where $\rho$ is the density of dense plasma
micro-tube. The longitudinal length of the electron beam was fixed at
$\lambda$ $\sim 7$ $\mathrm{\mu m}$, and the normalization parameter was set
as $L=c/\omega_{L}\times 10^{6}$ $\mathrm{\mu m}$, where $\omega_{L}=2\pi
c/\lambda$. A strong charge separation field $E_{cs}$ proportional to the
separation distance $\delta y$, expressed as

$E_{cs}=\frac{e}{\varepsilon_{0}}\rho\delta y.$ (3)

Fig. 1(C) displays the distribution of $E_{yp}$ within a dense plasma micro-
tube with a radius of $r=$ 50 $\mathrm{\mu m}$. The resultant force for this
setup is given an $F=F_{yp}-F_{cs}=-e(E_{yp}-E_{cs})$. For $y>0$, the electron
film was pushed back and forth between regions of $F>0$ and $F<0$ to realize
the oscillation of the surface-nanometer film - plasma electrons. Fig. 1(D)
displays the distribution of the electric field $E_{yp}$ and the charge
separation field $E_{cs}$ within the dense plasma micro-tube. For the surface-
nanometer electron film with $\delta_{y}=1.1$ $\mathrm{nm}$, the resultant
force $F$ nullifies, and the plasma electrons gain the maximum momentum.

Figure 1: Self-consistent fields and the high-intensity continuous mid-IR
acceleration field formed in the dense plasma micro-tube: (A) The initial
longitudinal electric field $E_{x}$, and (B) the initial transverse electric
field $E_{y}$ of the relativistic electron beam. The subplots display the
electrostatic field of a static electron beam; (C) The strong transverse field
of a relativistic electron beam at the inner-surface of the plasma micro-tube;
(D) The trend of the resultant force $F$ of the electric field $E_{yp}$ in the
dense plasma, and the charge separation field $E_{cs}$; (E) The momentum
distribution of the surface-nanometer plasma electron film. The maximum
momentum of the plasma, obtained as $\sim$0.016 $\mathrm{m_{e}c}$, corresponds
to the region where the driven beam is located; and (F) The acceleration
(radiation) field distribution in the plasma micro-tube with a radius of $50$
$\mathrm{\mu m}$ at $t=0.14$ $ps$ and $t=0.28$ $ps$.

###  2.2 High-intensity mid-IR acceleration field

The resultant force $F$ applied by the electric field $E_{yp}$ and the charge-
separation field $E_{cs}$ within the dense plasma causes oscillations of the
surface-nanometer electron film, moving in and out of the inner-surface. The
energy of these electrons can be obtained as

$\int_{0}^{\delta_{y}}{(eE_{yp}-eE_{cs}})dy=(\gamma_{p}-1)m_{e}c^{2},$ (4)

where $\gamma_{p}$ is the Lorentz factor of the plasma electrons, and $m_{e}$
is electron mass. For $\delta y=1.1$ $\mathrm{nm}$, $\gamma_{p}$ is computed
to be $\sim$1.0002, and the normalized electron velocity in the plasma,
$\beta_{p}$, is computed to be 0.0206. Fig. 1(E) shows the momentum
distribution of the oscillating electrons from the PIC simulation. Only the
electrons on the surface-nanometer film gain kinetic energy. As the electron
beam passes through, the oscillations weaken, and the velocity decreases.
Initially, the surface-nanometer electron film starts to oscillate and emit
continuous mid-IR radiation. The electric field radiated through the
oscillations of an electron, $E_{rad}$, is then obtained as

$E_{rad}=\frac{e}{4\pi\varepsilon_{0}}\bigg{[}\frac{\vec{n}\times(\vec{n}\times\dot{\vec{\beta_{p}}})}{Rc(1-\beta_{p}\cos\theta)^{3}}\bigg{]},$
(5)

where $\dot{\vec{\beta_{p}}}=e\vec{E}/(cm_{e}\gamma_{p}^{3})$ and $\theta$ is
the angle between the solid angle direction $\vec{n}$ and $\vec{\beta_{p}}$.
For the interaction region, the total electric field $E_{rad}^{tot}$ is then
calculated through the 3-dimensional integration with the micro-tube density
as

$E_{rad}^{tot}=\int E_{x,rad}\rho xd\varphi
dxdy=\frac{e\dot{\beta_{p}}\rho\delta_{y}p}{2\varepsilon_{0}c}\int_{0}^{\theta}\frac{\sqrt{1-\cos^{2}\theta}\sin^{2}\theta}{(1-\beta_{p}\cos\theta)^{3}\cos^{2}\theta}d\theta,$
(6)

where $E_{x,rad}=E_{rad}\sin\theta$ is the longitudinal radiation field,
$p=R\cos\theta$ is the distance from the reference point to the micro-tube
wall, and $L_{y}=50$ $\mathrm{\mu m}$.

Using Eq. (6), the total radiation field was obtained for different parameter
ranges. For example, with $p=1$ $\mathrm{\mu m}$, we obtained
$E_{rad}^{tot}\approx 38$ $\mathrm{GV/m}$. Fig. 1(F) shows the radiation field
distribution in the plasma micro-tube obtained from the PIC simulation at
$t=0.14$ $ps$ and $t=0.28$ $ps$, respectively. The radiation field produced by
the oscillations of the electron film propagates from the inner wall to the
center to catch up with the driving electron beam. Therefore, in Fig. 2(A), a
high intensity acceleration field is formed in a stable TM mode with a mid-IR
wavelength of $\sim$24 $\mathrm{\mu m}$. The relevant characteristics of the
mid-infrared field inside the micro-tube have been discussed in reference
[28], including radiation intensity, wavelength, micro-tube radius, and so on.

###  2.3 Results of $e^{+}$ beam acceleration

As the relativistic electron beam propagates in the micro-tube, the continuous
mid-IR evolves into a high-intensity stable period structure of the
longitudinal acceleration field in Fig. 2(A). Figs. 2(A1) and (A2) provide a
magnified view of the first period region and the phase of the electrons and
positrons. The acceleration of the positron bunch in the dense plasma micro-
tube was observed and analyzed through PIC simulations.

We analyzed the relationship between positron acceleration time, positron
energy, and relative energy spread in Fig. 2(B). The positron bunch gains an
energy increase of 1 $\mathrm{GeV}$ within 140 picoseconds, with a relative
energy spread of $1.56\%$. Fig. 2(C) shows the decrease in the total charge of
the electron beam and two cases of positron bunches (with initial energies of
$1$ $\mathrm{GeV}$ and $5$ $\mathrm{GeV}$, both having a beam charge of
$0.111$ ${\mathrm{nC}}$) during the acceleration process. In Fig. 2(D), the
final energy spectrum of positron bunches with different initial energies and
the electron beam at $t=140$ $\mathrm{ps}$ is displayed.

The energy transfer efficiency from the driving electron beam to the positron
bunch is defined as $\eta=(\delta E_{p}\cdot Q_{p})/(\delta E_{e}\cdot
Q_{e})$, where $\delta E_{e/p}$ represents the change in the energy of the
electron or positron bunch, and $Q_{e/p}$ is the charge of the electron beam
or positron bunch. The energy transfer efficiency was approximately $4\%$ and
$5\%$ for positron bunches with initial energies of $1$ GeV and $5$ GeV,
respectively. The energy transfer efficiency, $\eta$, increased with the
initial charge of the positron bunch, $n_{p}$, and even reached up to $50\%$
for $n_{p}=1.85$ nC, as shown in Fig. 2(E).

The acceleration field exhibits a stable period distribution, with the field
intensity and stabilization time being correlated with the charge and energy
of the driving electron beam. We can adjust the parameters of the injected
electron and positron beams to meet various application requirements: (1) A
positron beam with an initial energy of 50 $\mathrm{MeV}$ can be accelerated
to several $\mathrm{GeV}$ within 140 $\mathrm{ps}$ using a driving electron
beam with 1 $\mathrm{GeV}$ and 3 $\mathrm{nC}$. (2) The acceleration mechanism
allows for cascaded acceleration, enabling the positron beam to achieve higher
energy levels under less demanding initial conditions. (3) Positrons
undergoing acceleration over a one-meter with a 45 $\mathrm{GeV}$ driving
electron beam can gain an energy increase of 100 $\mathrm{GeV}$.

Figure 2: A novel scheme for positron bunch acceleration in a micro-tube: (A)
The periodically distributed mid-IR acceleration field is generated by the
relativistic electron beam propagation within the dense plasma micro-tube. (B)
The relationship between the acceleration time and the positron energy, and
the relative energy spread. At the acceleration distance of $4.2$
$\mathrm{cm}$, the positron bunch gains an average energy of $1$
$\mathrm{GeV}$. The relative energy spread of the positron bunch was to be
$1.56\%$; (C) The decrease in the total charge of the electron beam $Q_{e}$
and positron bunch $Q_{p}$ during acceleration process; (D) The final energy
spectrum of the positron bunches with different initial energies and the
electron beam at $t=140$ $\mathrm{ps}$; and (E) The energy transfer efficiency
$\eta$ increases with the initial charge of positron bunch, $n_{p}$, and
reaches up to 50$\%$ at $n_{p}=1.85\mathrm{nC}$.

###  2.4 Evolution of transverse electromagnetic fields

The stable acceleration of the positron bunch is achieved using the high-
intensity longitudinal acceleration field generated by the oscillation of
surface-nanometer electron film in the plasma micro-tube. Simultaneously, the
transverse self-field generated by the relativistic electron beam also
provides a transverse focusing force to the positron bunch. The transverse
fields in the y- and z-directions, specifically, $E_{y}-cB_{z}$ and
$E_{z}+cB_{y}$, were symmetrically configured in the PIC simulation.

Fig. 3(A) displays the transverse field distribution of the positron bunch at
$0.1$ $\mathrm{ps}$, $1.9$ $\mathrm{ps}$, $21.0$ $\mathrm{ps}$, and $93.0$
$\mathrm{ps}$. In the early stage of the longitudinal acceleration ($0<t<21$
$\mathrm{ps}$), the transverse force remains focused. After $21$
$\mathrm{ps}$, the transverse field becomes uniform at the edges, and the
positron bunch remains stable in the transverse direction for several tens of
picoseconds. Beyond 21 $\mathrm{ps}$, the field gradually weakens and becomes
weakly defocused as the energy of the electron beam is consumed. Fig. 3(B)
shows the corresponding phase-space distribution of the positron bunch,
corresponding to the transverse field distribution in Fig. 3(A). Therefore,
the analysis of the transverse field distribution and the transverse motion of
the positron bunch in the y-direction revealed an initial weak focusing stage,
followed by a weak defocusing stage, until the magnitudes of the transverse
fields reduced to zero.

Figure 3: Distribution of the transverse field ($E_{y}-cB_{z}$) as experienced
by the positron bunch while being accelerated in the plasma micro-tube: (A)
The transverse field distribution of positron bunch at $0.1$ $\mathrm{ps}$,
$1.9$ $\mathrm{ps}$, $21.0$ $\mathrm{ps}$, and $93.0$ $\mathrm{ps}$,
respectively; and (B) The phase-space distribution of the positron bunch
corresponding to the field distribution cases in (A).

###  2.5 Transverse off-axis $e^{+}$ bunch injection

The proposed scheme enables stable and efficient acceleration of an externally
injected positron bunch. Given the uniformity of the acceleration field,
experimental misalignment of the injected positron bunch is inevitable but
allowed within specific regions. To determine acceptable misalignment, several
2-dimensional PIC simulations were conducted with varying transverse injection
deviations from the central axis. The simulations used the following key
parameters: electron beam longitudinal and transverse lengths of $3.3$
$\mathrm{\mu m}$ and $26$ $\mathrm{\mu m}$, respectively, with a number
density of $10^{24}$ $\mathrm{m^{-3}}$; positron bunch longitudinal and
transverse lengths of $2$ $\mathrm{\mu m}$ and $22$ $\mathrm{\mu m}$, with a
number density of 7$\times 10^{21}$ $\mathrm{m^{-3}}$; and initial injection
positions of the positron bunch at $y_{0}=$ 0, 5$\mathrm{\mu m}$, and 10
$\mathrm{\mu m}$. Other simulation parameters were the same as those of the
cylindrical coordinate simulations mentioned in the Methods section. Fig. 4(A)
shows the asymmetric phase-space distribution of a positron bunch with a $10$
$\mathrm{\mu m}$ positive y-axis deviation compared to the symmetric one. Fig.
4(B) shows the density distribution of the positron bunch in the $x-y$ plane
after a long period of acceleration in both the asymmetric and symmetric
cases. Due to the initial injection deviation in the y-direction, although the
phase-space distribution of the positron bunch maintains this asymmetry during
the acceleration process, the central energy and the energy spread of the
accelerated positron bunch are hardly affected in the uniform longitudinal
acceleration field. Fig. 4(C) displays the influence of the different
transverse injection misalignments ($y_{0}=$ 0, 5$\mathrm{\mu m}$, and 10
$\mathrm{\mu m}$) on the accelerated energy spectrum of positron bunch. In
each case, the positron bunch gained the same amount of energy at a given time
$t$. However, the difference was the peak charge at the central energy which
affects the energy transfer efficiency $\eta$ of the accelerated positron
bunch; it decreases by $\sim 10\%$ for the deviation of 10 $\mathrm{\mu m}$
relative to that without deviation. Therefore, the excellent tolerance for the
injection deviation of the positron bunch greatly reduces the implementation
difficulty of the experiment.

Figure 4: Impact of positron bunch injection deviation: (A) The symmetric and
asymmetric phase-space distribution of positron bunch; (B) The symmetric and
asymmetric density distribution of positron bunch in the $x-y$ plane after a
long period of acceleration; and (C) Effect of transverse injection
misalignments at $y_{0}=$ 0, 5$\mathrm{\mu m}$, and 10 $\mathrm{\mu m}$, on
the accelerated energy spectrum of positron bunch. $\eta$ is the energy
transfer efficiency.

##  3 Dicussions

In our model, the density of the plasma micro-tube was fixed to be of the same
order as the free-electron density in metals. Therefore, we propose a similar
application of the method in metal micro-tubes. The transverse self-field
$E_{y}$ of the relativistic electron beam (several hundreds of
$\mathrm{GV/m}$) can directly ionize the surface of the metal micro-tube. It
can also interact with the space-separation field to drive the oscillation of
the ionized electron film, and the production of high-intensity, continuous
mid-IR. This high-efficiency scheme has a high acceleration gradient and
stable acceleration structure, and can also realize cascaded acceleration,
allowing for a stable, continuous, and efficient positron acceleration. Fig.
5(A) displays the cascade design schematic diagram illustrating the
acceleration of a positron bunch by electron beams, demonstrating a three-step
acceleration process. The driving electron beam is extracted to the trash
after completing a single acceleration. A cone-shaped coupling tube surrounded
by a magnetic field. For a magnetic field of $B=1$ $\mathrm{T}$ and initial
energy of 1 $\mathrm{GeV}$ of the electron or positron bunch, the deflection
radius of particle bunch ($R=E/qvB$) was computed as $3.3$ $\mathrm{m}$. After
setting the deflection angle to $\theta\approx 1.8\times 10^{-4}$
$\mathrm{rad}$ for $t=2$ $\mathrm{ps}$, the positron bunch was accelerated for
$4.2$ $\mathrm{cm}$ in a mid-IR acceleration field through the micro-tube to
obtain a $1$ $\mathrm{GeV}$ energy gain. Due to the stable periodic structure
of the mid-IR acceleration field in the micro-tube, the series acceleration
can also be considered to obtain multiple high-energy positron bunches at the
same time. This process is demonstrated by the 2-dimensional PIC simulation
(Fig. 5(B)) of the acceleration field driven by a single electron beam in a
micro-tube, accelerating three positron bunches, $x_{positrons}^{(1)}$,
$x_{positrons}^{(2)}$, and $x_{positrons}^{(3)}$, simultaneously, starting
from initial positions of $178$ $\mathrm{\mu m}$, $156$ $\mathrm{\mu m}$, and
$126$ $\mathrm{\mu m}$, respectively. The final energy spectrum of the
positron bunches is shown in Fig. 5(C) with the initial energy of $5$
$\mathrm{GeV}$. The energy transfer efficiency was found to be about 2-3 times
higher than that of a single positron bunch acceleration. For an injected
positron bunch with a long longitudinal dimension, such as 20 $\mathrm{\mu
m}$, at a phase of $3\pi/2$, where the front and rear of the positron bunch
are in an accelerating phase, and the center is in a decelerating phase.
During long-distance propagation, the positrons in the central portion will be
decelerated to the second acceleration phase. This allows the single bunch to
split into two separate bunches for subsequent individual acceleration.

Figure 5: The schematic diagram of the cascade acceleration of a single
positron bunch, and the series acceleration of multiple positron bunches: (A)
The cascade design for the injection of electron beams and a positron bunch. A
cone-shaped coupling tube surrounded by a magnetic field. R is the radius of
deflection; (B) The mid-IR acceleration field accelerating three positron
bunches, $x_{positrons}^{(1)}$, $x_{positrons}^{(2)}$, and
$x_{positrons}^{(3)}$, simultaneously, with the initial positions of $178$
$\mathrm{\mu m}$, $156$ $\mathrm{\mu m}$, and $126$ $\mathrm{\mu m}$,
respectively; and (C) The final energy spectrum of the positron bunches with
the initial energy of $5$ $\mathrm{GeV}$ at $t=140$ $\mathrm{ps}$.

##  4 Conclusion

In this paper, we proposed a novel scheme for the acceleration of positron
bunches using high-intensity, continuous mid-IR field. The injected positron
bunch could gain energy of $1$ $\mathrm{GeV}$ after accelerating for a
distance of $\sim 4.2$ $\mathrm{cm}$, and could maintain a low energy spread
of $<2\%$. The energy transfer efficiencies ($\eta$) for two positron bunches
with a charge of $0.111$ $\mathrm{nC}$ and $0.560$ $\mathrm{nC}$ were observed
to be $\sim 5\%$ and $\ 22\%$, respectively. In the uniform mid-IR
acceleration field, the transverse field in the $y$-direction ($E_{y}-cB_{z}$)
weakly focused on the positron bunch within the first $21$ $\mathrm{ps}$, post
which it began weakly defocusing, till it finally reduced to zero due to the
energy loss and beam break of the driving electron beam. The ease of
performing the experiment was increased due to the excellent tolerance of the
positron bunch injection deviation. Through several 2-dimensional PIC
simulations, the transverse spatial misalignment of the positron injection is
allowed to be a few microns if $10\%$ relative loss of the energy transfer
efficiency is accepted, where the accelerated energy and the energy spread are
hardly affected. The novel high-efficiency scheme had a high acceleration
gradient and a stable acceleration structure, allowing for a stable,
efficient, and continuous acceleration of multiple positron bunches
simultaneously.

##  5 Methods

The proposed scheme is confirmed by cylindrical geometry with a decomposition
in azimuthal modes using the SMILEI code[32], which is an open-source, user-
friendly, high-performance and multi-purpose electromagnetic particle-in-cell
(PIC) code for plasma simulation. The grid coordinates are two-dimensional
$(x,r)$, while particle coordinates are expressed in the three-dimensional
Cartesian frame $(x,y,z)$. A moving window of 51.2 $\mathrm{\mu m}$ in the $x$
direction and 100 $\mathrm{\mu m}$ in the $r$ direction was employed, which
was sampled by 512 cells in the $x$ direction and 400 cells in the $r$
direction. Electromagnetic boundary conditions were implemented using the
Perfectly Matched Layer (PML) to absorb fields at the boundary. Additionally,
a binomial filter module was applied to the currents to reduce noise. The
plasma critical number density $n_{c}$ is $2.82\times 10^{25}$
$\mathrm{m^{-3}}$ for the $\lambda=2\pi$ $\mathrm{\mu m}$. The density of the
dense plasma is $10^{28}$ $\mathrm{m^{-3}}$ with the radius of 50 $\mathrm{\mu
m}$ and the thickness of 30 $\mathrm{\mu m}$. The initial energy of the
driving electron bunch is 1 $\mathrm{GeV}$, with a 3.402 $\mathrm{nC}$ beam
charge. The initial momentum is Maxwell-Juttner distribution with temperature
0.1 $\mathrm{m_{e}c^{2}}$ in the x directron and 0.001 $\mathrm{m_{e}c^{2}}$
in the y and z direction. The density profile of the electron bunch is a
Gaussian distribution in the x and r direction,
$n_{e}=n_{e0}e^{-(x-x_{0})^{2}/\sigma_{xe}^{2}}\cdot
e^{-r^{2}/\sigma_{re}^{2}}$, where $n_{e0}=4\times 10^{24}$ $\mathrm{m^{-3}}$,
$x_{0}=5$ $\mathrm{\mu m}$, longitudinal rms-length $\sigma_{xe}=3.3$
$\mathrm{\mu m}$, transverse rms-length $\sigma_{re}=26$ $\mathrm{\mu m}$. The
witness positron bunch has an initial energy of 5 $\mathrm{GeV}$ or 1
$\mathrm{GeV}$ and a beam charge of 0.111 $\mathrm{nC}$. The density of the
positron bunch is $n_{p}=3\times 10^{23}$ $\mathrm{m^{-3}}$ with longitudinal
rms-length $\sigma_{xp}=2$ $\mathrm{\mu m}$, transverse rms-length
$\sigma_{rp}=22$ $\mathrm{\mu m}$, and the initial positron is 33.5
$\mathrm{\mu m}$. The initial energy spread, angular distribution and the
normalized emittances of the positron bunch is neglected. There are 20 macro-
particles per cell (PPC) for the driving electron bunch and the witness
positron bunch, and 4 PPC for the plasma species. The results have been
compared with PPC=50 for particles bunch and PPC=4/100/200/400 for the plasma
particles of micro-tube, which are almost no difference. The analysis of
transverse off-axis injection of positron bunch and series acceleration of
multiple positron bunches are simulated with 2-dimensions PIC. A moving window
of 204.8 $\mathrm{\mu m}$ ($L_{x}$) $\times$ 160 $\mathrm{\mu m}$ ($L_{y}$) is
used and sampled by 2048 $(x)$ $\times$ 640 $(y)$ cells.

## Acknowledgments

Funding: This work is supported in part by National Natural Science Foundation
of China (11655003); Innovation Project of IHEP (542017IHEPZZBS11820,
542018IHEPZZBS12427); the CAS Center for Excellence in Particle Physics
(CCEPP); IHEP Innovation Grant (Y4545170Y2); Chinese Academy of Science
Focused Science Grant (QYZDY-SSW-SLH002); Chinese Academy of Science Special
Grant for Large Scientific Projects (113111KYSB20170005); National 1000
Talents Program of China; the National Key Research and Development Program of
China (No.2018YFA0404300); National Natural Science Foundation of China (Grant
No. 11975252); Youth Innovation Promotion Association CAS (No. 2021012).

Author contributions: M.Y.S. contributed to investigation, validation, formal
analysis, data curation, simulation, writing—original draft. Y.S.H.
contributed to conceptualization, methodology, formal analysis, supervision,
project administration, funding acquisition, writing—review and editing.
M.Q.R. contributed to formal analysis. B.F.S. contributed to formal analysis.
Z.L.X. helped in formal analysis. T.P.Y. helped in formal analysis,
writing—review and editing. X.F.W. contributed to formal analysis. Y.C.
contributed to formal analysis.

Competing interests:The authors declare no conflicts of interest.

Data and materials availability: This manuscript has no associated data or the
data will not be deposited.

## References

>

>
> ## References
>
>   * [1] L. Marchut, C. J. Mcmahon, Electron and positron spectroscopies in
> materials science and engineering. Electron Positron Spectroscopies in
> Materials Science Engineering, 1–33 (1979).
>   * [2] Guessoum, Nidhal, Positron astrophysics and areas of relation to
> low-energy positron physics. European Physical Journal D 68, 1-6 (2014).
>   * [3] K. R. Hogstrom, P. R. Almond, Review of electron beam therapy
> physics. Physics in Medicine, Biology 51, R455 (2006).
>   * [4] E. Gschwendtner, P. Muggli, Plasma wakefield accelerators. Nature
> Reviews Physics 1, 246-248 (2019).
>   * [5] X. J. Jiao, J. M. Shaw, T. Wang et al, A tabletop, ultrashort pulse
> photoneutron source driven by electrons from laser wakefield acceleration.
> Matter and Radiation at Extremes 2, 296-302 (2017).
>   * [6] T. Tajima, J. M. Dawson, Laser electron accelerator. Physical Review
> Letters 43, 267 (1979).
>   * [7] Katsouleas, Thomas, Electrons hang ten on laser wake. Nature 431,
> 515-516 (2004).
>   * [8] J. Faure, Y. Glinec, A. Pukhov et al, A laser–plasma accelerator
> producing monoenergetic electron beams. Nature 431, 541-544 (2004).
>   * [9] S. P. Mangles, C. D. Murphy, Z. Najmudin et al, Monoenergetic beams
> of relativistic electrons from intense laser–plasma interactions. Nature
> 431, 535-538 (2004).
>   * [10] C. G. R. Geddes, C. Toth, J. Van Tilborg et al, High-quality
> electron beams from a laser wakefield accelerator using plasma-channel
> guiding. Nature 431, 538-541 (2004).
>   * [11] Wang, J., Zeng, M., Li, D., Wang, X., Lu, W., Gao, J, Injection
> induced by coaxial laser interference in laser wakefield accelerators.
> Matter and Radiation at Extremes 7, 054001 (2022).
>   * [12] Dai, Y. N., Shen, B. F., Li, J. X., Shaisultanov, R.,
> Hatsagortsyan, K. Z., Keitel, C. H., Chen, Y. Y. , Photon polarization
> effects in polarized electron–positron pair production in a strong laser
> field. Matter and Radiation at Extremes 7, 014401 (2022).
>   * [13] T. J. Xu, B. F. Shen et al, Ultrashort megaelectronvolt positron
> beam generation based on laser-accelerated electrons. Physics of Plasmas 23,
> 033109 (2016).
>   * [14] X. L. Zhu, M. Chen, T. P. Yu et al, Collimated GeV attosecond
> electron–positron bunches from a plasma driven by 10 PW lasers. Matter and
> Radiation at Extremes 4, 014401 (2019).
>   * [15] K. V. Lotov, Acceleration of positrons by electron beam-driven
> wakefields in a plasma. Physics of Plasmas 14, 023101 (2007).
>   * [16] C. Du, Z. Xu, Positron acceleration by a laser pulse in a plasma.
> Physics of Plasmas 7, 1582-1585 (2000).
>   * [17] H. Hasegawa, S. Usami, Y. Ohsawa, Positron acceleration to
> ultrarelativistic energies by a shock wave in a magnetized
> electron–positron–ion plasma. Physics of Plasmas 10, 3455-3458 (2003).
>   * [18] X. Wang, R. Ischebeck et al, Positron Injection and Acceleration on
> the Wake Driven by an Electron Beam in a Foil-and-Gas Plasma. Physical
> Review Letters 101, 124801 (2008).
>   * [19] J. Vieira, J.T. Mendonça, Nonlinear laser driven donut wakefields
> for positron and electron acceleration. Physical Review Letters 112,
> 215001(2014).
>   * [20] C. B. Schroeder, D. H. Whittum, J. S. Wurtele, Multimode analysis
> of the hollow plasma channel wakefield accelerator. Physical Review Letters
> 82, 1177 (1999).
>   * [21] C. B. Schroeder et al, Beam loading in a laser-plasma accelerator
> using a near-hollow plasma channel. Physics of Plasmas 20, 123115 (2013).
>   * [22] S. Gessner, E. Adli et al, Demonstration of a positron beam-driven
> hollow channel plasma wakefield accelerator. Nature communications 7, 1-6
> (2016).
>   * [23] S. Zhou, J. Hua et al, High efficiency uniform wakefield
> acceleration of a positron beam using stable asymmetric mode in a hollow
> channel plasma. Physical Review Letters 127, 174801(2021).
>   * [24] L. Yi, B. Shen et al, Positron acceleration in a hollow plasma up
> to TeV regime. Scientific Reports 4, 4171 (2014).
>   * [25] Z. Xu, L. Yi et al, Driving positron beam acceleration with
> coherent transition radiation. Communications Physics 3, 191 (2020).
>   * [26] Xu, Z., Shen, B., Si, M., Huang, Y. S.. Positron acceleration by
> terahertz wave and electron beam in plasma channel. New Journal of Physics
> (2023).
>   * [27] Zhu, X. L., Liu, W. Y., Weng, S. M., Chen, M., Sheng, Z. M., Zhang,
> J, Generation of single-cycle relativistic infrared pulses at wavelengths
> above 20 $\mathrm{\mu m}$ from density-tailored plasmas. Matter and
> Radiation at Extremes 7, 014403 (2022).
>   * [28] Meiyu Si, Yongsheng Huang, Manqi Ruan, Baifei Shen, Zhangli Xu,
> Tongpu Yu, Xiongfei Wang, and Yuan Chen, ”Relativistic-guided stable mode of
> few-cycle 20 $\mathrm{\mu m}$ level infrared radiation,” Optics. Express 31,
> 40202-40209 (2023).
>   * [29] P. Londrillo, C. Gatti, M. Ferrario, Numerical investigation of
> beam-driven PWFA in quasi-nonlinear regime. Nuclear Instruments and Methods
> in Physics Research Section A: Accelerators, Spectrometers, Detectors and
> Associated Equipment 740, 236-241 (2014).
>   * [30] F. Massimo, A. Marocchino, A. R. Rossi, Electromagnetic self-
> consistent field initialization and fluid advance techniques for hybrid-
> kinetic pwfa code architect. Nuclear Instruments and Methods in Physics
> Research Section A: Accelerators, Spectrometers, Detectors and Associated
> Equipment 829, 378-382 (2016).
>   * [31] A. Marocchino, E. Chiadroni et al, Design of high brightness Plasma
> Wakefield Acceleration experiment at SPARC-LAB test facility with particle-
> in-cell simulations. Nuclear Instruments and Methods in Physics Research
> Section A: Accelerators, Spectrometers, Detectors and Associated Equipment
> 909, 408-413 (2018).
>   * [32] J. Derouillat et al, SMILEI: A collaborative, open-source, multi-
> purpose particle-in-cell code for plasma simulation. Computer Physics
> Communications 222, 351-373 (2017).
>