# Electron Acceleration Using Twisted Laser Wavefronts

Yin Shi  Department of Plasma Physics and Fusion Engineering, University of
Science and Technology of China, Hefei 230026, China  Department of Mechanical
and Aerospace Engineering, University of California at San Diego, La Jolla, CA
92093, USA  David Blackman  Department of Mechanical and Aerospace
Engineering, University of California at San Diego, La Jolla, CA 92093, USA
Alexey Arefiev  Department of Mechanical and Aerospace Engineering, University
of California at San Diego, La Jolla, CA 92093, USA

###### Abstract

Using plasma mirror injection we demonstrate, both analytically and
numerically, that a circularly polarized helical laser pulse can accelerate
highly collimated dense bunches of electrons to several hundred MeV using
currently available laser systems. The circular-polarized helical (Laguerre-
Gaussian) beam has a unique field structure where the transverse fields have
helix-like wave-fronts which tend to zero on-axis where, at focus, there are
large on-axis longitudinal magnetic and electric fields. The acceleration of
electrons by this type of laser pulse is analysed as a function of radial mode
number and it is shown that the radial mode number has a profound effect on
electron acceleration close to the laser axis. Using three-dimensional
particle-in-cell simulations a circular-polarized helical laser beam with
power of 0.6 PW is shown to produce several dense attosecond bunches. The
bunch nearest the peak of the laser envelope has an energy of 0.47 GeV with
spread as narrow as 10%, a charge of 26 pC with duration of $\sim 400$ as, and
a very low divergence of 20 mrad. The confinement by longitudinal magnetic
fields in the near-axis region allows the longitudinal electric fields to
accelerate the electrons over a long period after the initial reflection. Both
the longitudinal E and B fields are shown to be essential for electron
acceleration in this scheme. This opens up new paths towards attosecond
electron beams, or attosecond radiation, at many laser facilities around the
world.

##  1 Introduction

Laser-driven electron accelerators have become a very active area of research
due to technological developments and improvements of high-power laser beams
[1]. There are usually two common approaches. One approach is laser wakefield
acceleration [2] that utilizes plasma electric fields whose strength is
related to the plasma density. Another approach is direct laser acceleration
[3] that relies on the fields of the laser for the energy transfer inside a
plasma (e.g. see [4]) or in vacuum [5]. In the vacuum regime, transverse
electron expulsion typically terminates the energy gain and leads to strong
electron divergence. This is the reason why direct laser acceleration in
vacuum has been deemed ineffective and most of the research has been focused
on the plasma regime where collective fields prevent the expulsion [6, 7, 8].
In an attempt to mitigate the expulsion in the vacuum regime, some studies
examined alternative approaches utilizing longitudinal electric fields of a
radially polarized beam [9] and higher-order Gaussian beams [10] for electron
acceleration.

Currently there are several multi-PW laser systems operational in the world,
with several more due to come online in the next few years [1]. The biggest of
the new systems recently proposed, at the Shanghai Superintense-Ultrafast
Laser Facility [11], is pushing the 100 PW limit and will be in development
over the next decade. At the same time, new optical techniques using helical
wave-fronts [12, 13] are being developed. One of the key advantages of the
helical, or Laguerre-Gaussian, laser mode is that it can be produced, at high
efficiency, from a standard Gaussian laser pulse in reflection from a fan-like
structure [13, 14]. Very significant differences can be seen when comparing
the laser-plasma interactions of conventional laser beams to those of helical
beams, some of which have been examined in simulations [15, 16, 17, 18, 14,
19, 20, 21, 22, 23], and some have begun to be explored in recent experiments
[12, 24, 25, 26, 27]. An electron acceleration scheme has recently been
proposed where a high-power high-intensity circular-polarized Laguerre-
Gaussian beam is reflected from a plasma mirror [28]. The unique field
structure of this beam both confines and accelerates tightly packed electron
bunches to GeV energies with a narrow energy spread.

In this article, we show using 3D particle-in-cell (PIC) simulations that the
same electron acceleration scheme can be successfully applied 1) using
significantly lower laser power than that used in [28] and 2) using oblique
incidence. Specifically, we show that a 600 TW laser beam generates several
dense attosecond bunches at intervals similar to the laser wavelength. The
bunch nearest the peak of the laser envelope gains an energy of 0.47 GeV (10%
FWHM energy spread), while maintaining a bunch charge as high as 26 pC.  The
tightly confined bunch has a duration of $\sim 400$ as and a remarkably low
divergence of just ${1.15}^{\circ}$ (20 mrad). In addition to this, the scheme
is demonstrated to be tenable with an angle of incidence as high as
${25}^{\circ}$. Our results show that the electron acceleration by helical
beams is not limited to high-power high-intensity lasers and can be
successfully explored at a wide range of laser facilities.

The rest of this paper is organized as follows. Section 2 examines the field
structure of linearly and circularly polarized laser beams with twisted
wavefronts. The near-axis structure of the longitudinal electric field for
different radial modes is explicitly derived. Section 3 is dedicated to
estimating the acceleration that the longitudinal fields can provide and the
effect of the radial mode structure of the laser on this acceleration. In
Section 4, we present results of a 3D PIC simulation for a 600 TW laser beam
with twisted wavefronts whose near-axis field structure is dominated by
longitudinal electric and magnetic fields. Section 5 is concerned with
exploring the role of the longitudinal magnetic field where the effect of this
field on the bunch density and particle trajectories is shown, including the
possible reliability of this model with an oblique angle of incidence. In
Section 6, we summarise the main results of this work.

##  2 Field topology of a laser beam with twisted wavefronts

In this section, we examine the field topology of linearly and circularly
polarized laser beams with twisted wavefronts. We show that, for a properly
chosen twist index, the field structure in the region close to the axis of the
beam can primarily consist of longitudinal electric and magnetic fields. The
differences between radial modes are explicitly emphasized.

###  2.1 Linearly polarized beam

We start by considering a linearly polarized laser beam with wavelength
$\lambda_{0}$ propagating in vacuum along the $x$-axis. We assume that the
diffraction angle, defined as $\theta_{d}=w_{0}/x_{R}$, is small, where
$w_{0}$ is the beam waist, and $x_{R}=\pi w_{0}^{2}/\lambda_{0}$ is the
Rayleigh range. Without any loss of generality, we assume that the laser
electric field is polarized along the $y$-axis. In this case, it is convenient
to describe the field structure using a vector potential $\bm{A}$ that
satisfies the Lorenz gauge condition and has only one non-zero component,
$A_{y}$. The paraxial wave equation for $A_{y}$ has the form

$\left[\frac{\partial^{2}}{\partial\widetilde{y}^{2}}+\frac{\partial^{2}}{\partial\widetilde{z}^{2}}+4i\frac{\partial}{\partial\widetilde{x}}\right]A_{y}=0,$
(1)

where, for compactness, the longitudinal coordinate $x$ is normalized to
$x_{R}$ and the transverse coordinates $y$ and $z$ are normalized to $w_{0}$:

$\displaystyle\widetilde{x}=x/x_{R},$ (2)
$\displaystyle\widetilde{y}=y/w_{0},$ (3)
$\displaystyle\widetilde{z}=z/w_{0}.$ (4)

Equation 1 implies the following form of the solution with $\omega=2\pi
c/\lambda_{0}$:

$\displaystyle
A_{y}=\Psi_{y}(\widetilde{x},\widetilde{y},\widetilde{z})g(\xi)\exp(i\xi),$
(5)

where $g$ is the envelope function with $\max(g)=1$ and

$\xi\equiv 2\widetilde{x}/\theta_{d}^{2}-\omega t$ (6)

is the phase variable.

The transverse electric field in the paraxial approximation, i.e. at
$\theta_{d}\ll 1$, is given by

$E_{y}\approx-\frac{1}{c}\frac{\partial A_{y}}{\partial
t}=\frac{i\omega}{c}A_{y}.$ (7)

We are interested in solutions of the form

$E_{y}=E_{0}g(\xi)\exp(i\xi)\psi_{l,p}(\widetilde{x},\widetilde{r},\phi)$ (8)

where

$\psi_{l,p}(\widetilde{x},\widetilde{r},\phi)=C_{p,l}f(\widetilde{x})^{|l|+1+2p}(1+\widetilde{x}^{2})^{p}L_{p}^{|l|}\left(\frac{2\widetilde{r}^{2}}{1+\widetilde{x}^{2}}\right)\left(\sqrt{2}\widetilde{r}\right)^{|l|}\exp\left[-\widetilde{r}^{2}f(\widetilde{x})\right]\exp\left(il\phi\right)$
(9)

is a mode with a twist index $l$ and radial index $p$. Here we introduced

$\widetilde{r}=\sqrt{\widetilde{y}^{2}+\widetilde{z}^{2}},\quad\phi=\arctan(\widetilde{z}/\widetilde{y}),\quad
f(\widetilde{x})=\frac{1-i\widetilde{x}}{1+\widetilde{x}^{2}}.$ (10)

The $L_{p}^{|l|}$ function is the generalized Laguerre polynomial and
$C_{p,l}$ is a normalization constant. The modes
$\psi_{l,p}(\widetilde{x},\widetilde{r},\phi)$ are orthonormal at a given
$\widetilde{x}$ [29], with

$C_{p,l}=\sqrt{\frac{2p!}{\pi(p+|l|)!}},$ (11)

such that

$\int_{0}^{2\pi}d\phi\int_{0}^{\infty}\psi_{l,p}(\widetilde{x},\widetilde{r},\phi)\psi^{*}_{l,p}(\widetilde{x},\widetilde{r},\phi)\widetilde{r}d\widetilde{r}=1.$
(12)

It is worth pointing out that $E_{0}$ in Eq. 8 is not the peak amplitude of
$E_{y}$. It is therefore convenient to introduce

$E_{0}^{*}\equiv\sqrt{\frac{2}{\pi}}E_{0},$ (13)

which is the peak amplitude of the transverse electric field in the beam with
$l=0$.

The transverse magnetic field has a $z$-component only, since
$\bm{B}=\nabla\times\bm{A}$. In the paraxial approximation, we have

$B_{z}=\frac{1}{x_{R}}\frac{\partial
A_{y}}{\partial\widetilde{x}}\approx\frac{2i}{\theta_{d}w_{0}}A_{y}.$ (14)

After taking into account that $\theta_{d}=\lambda_{0}/\pi w_{0}$, we find
that $B_{z}=E_{y}$. The longitudinal component of the Poynting vector $\bm{S}$
is then given by $S_{x}=(c/4\pi)\left(\mbox{Re}E_{y}\right)^{2}$ and the peak
period-averaged power for the linearly polarized beam is

$P_{lin}=\left\langle\int_{0}^{2\pi}d\phi\int_{0}^{\infty}S_{x}rdr\right\rangle=\frac{cw_{0}^{2}}{4\pi}\int_{0}^{2\pi}d\phi\int_{0}^{\infty}\left\langle\left(\mbox{Re}E_{y}\right)^{2}\right\rangle\widetilde{r}d\widetilde{r},$
(15)

where the angle brackets indicate the time-averaging. It is convenient to
compute the power in the focal plane, i.e. at $\widetilde{x}=0$. We use the
expression given by Eq. 8 to find that

$P_{lin}=\frac{cw_{0}^{2}}{8\pi}E_{0}^{2}=\frac{\pi}{2}\frac{a_{0}^{2}w_{0}^{2}m_{e}^{2}c^{5}}{\lambda_{0}^{2}e^{2}},$
(16)

where

$a_{0}=|e|E_{0}/m_{e}c\omega$ (17)

is a dimensionless parameter, and $e$ and $m_{e}$ are the electron charge and
mass. We have explicitly taken into account the normalization condition given
by Eq. 12. The advantage of the chosen normalization is that the power $P$ is
the same for different modes with the same $E_{0}$ or $a_{0}$.

The longitudinal laser electric and magnetic fields can be calculated from the
$(\nabla\cdot\bm{E})=0$ and $(\nabla\cdot\bm{B})=0$ conditions, respectively.
In the paraxial approximation, we have

$\displaystyle E_{x}\approx\frac{i\theta_{d}}{2}\frac{\partial
E_{y}}{\partial\widetilde{y}},$ $\displaystyle
B_{x}\approx\frac{i\theta_{d}}{2}\frac{\partial
E_{y}}{\partial\widetilde{z}}.$ (18)

It follows from Eqs. (8) - (10) that

$\displaystyle
E_{x}^{\pm}=\begin{cases}\frac{i\theta_{d}}{2}\left[\frac{|l|}{\widetilde{r}}e^{\mp
i\phi}-2f\widetilde{r}\cos\phi-\frac{2}{1+\widetilde{x}^{2}}\frac{L_{p-1}^{|l|+1}}{L_{p}^{|l|}}\cos\phi\right]E_{y};\mbox{
for }p\geq 1,\\\ \frac{i\theta_{d}}{2}\left[\frac{|l|}{\widetilde{r}}e^{\mp
i\phi}-2f\widetilde{r}\cos\phi\right]E_{y};\mbox{ for }p=0,\end{cases}$ (19)
$\displaystyle
B_{x}^{\pm}=\begin{cases}\frac{\theta_{d}}{2}\left[\mp\frac{|l|}{\widetilde{r}}e^{\mp
i\phi}-2if\widetilde{r}\sin\phi-\frac{2i}{1+\widetilde{x}^{2}}\frac{L_{p-1}^{|l|+1}}{L_{p}^{|l|}}\sin\phi\right]E_{y};\mbox{
for }p\geq 1,\\\ \frac{\theta_{d}}{2}\left[\mp\frac{|l|}{\widetilde{r}}e^{\mp
i\phi}-2if\widetilde{r}\sin\phi\right]E_{y};\mbox{ for }p=0,\end{cases}$ (20)

where the superscript on the left-hand side represents the sign of $l$.

The twist of the field represented by $l$ qualitatively changes the topology
of the transverse and longitudinal fields. We are particularly interested in
the field structure close to the central axis, i.e. at
$\widetilde{r}\rightarrow 0$. It is important to distinguish three cases based
on the value of the twist index: $l=0$, $|l|=1$, and $|l|>1$. In the near-axis
region, we have $E_{y}\propto\widetilde{r}^{|l|}\exp(il\phi)$. In the case of
$l=0$ or a beam without a twist, the longitudinal fields vanish on the central
axis, while the transverse fields reach their maximum value. In the case of
$|l|>1$, all laser fields vanish on the central axis. The most unusual is the
case of $|l|=1$, because in this case the longitudinal rather than transverse
fields peak on axis. As a result, the near-axis field structure is dominated
by longitudinal fields. Note that according to Eqs. (19) and (20) these fields
are not axis-symmetric.

Figure 1: Transverse and longitudinal electric field structure in the focal
plane, $\widetilde{x}=0$, of right-circularly polarized beams ($\sigma=1$)
with $l=0,\pm 1$, 2 and $p=$0, 1. All beams have the same $E_{0}$ and the same
diffraction angle, $\theta_{d}=8.5\times 10^{-2}$. The fields are normalized
to $E_{0}^{*}$, which is the peak amplitude of the transverse electric field
in the beam with $l=0$.

###  2.2 Circularly polarized beam

The results of the previous subsection can be readily generalized to the case
of a circularly polarized laser pulse. In addition to $E_{y}$, the laser beam
also has an $E_{z}$-component. We set $E_{z}=i\sigma E_{y}$, where $\sigma=1$
corresponds to the right-circularly polarized wave and $\sigma=-1$ corresponds
to the left-circularly polarized wave. The longitudinal electric and magnetic
fields can again be calculated using the $(\nabla\cdot\bm{E})=0$ and
$(\nabla\cdot\bm{B})=0$ equations, respectively. In the paraxial
approximation, we have

$\displaystyle E_{x}\approx\frac{i\theta_{d}}{2}\left(\frac{\partial
E_{y}}{\partial\widetilde{y}}+\frac{\partial
E_{z}}{\partial\widetilde{z}}\right),$ (21) $\displaystyle
B_{x}\approx\frac{i\theta_{d}}{2}\left(\frac{\partial
E_{y}}{\partial\widetilde{z}}-\frac{\partial
E_{z}}{\partial\widetilde{y}}\right).$ (22)

After substituting $E_{z}=i\sigma E_{y}$ into these equations, we find that

$\displaystyle
E_{x}^{\pm}=\begin{cases}i\theta_{d}E_{y}\left[\frac{|l|}{\widetilde{r}}\frac{1\mp\sigma}{2}e^{\mp
i\phi}-\widetilde{r}fe^{i\sigma\phi}-\frac{1}{1+\widetilde{x}^{2}}\frac{L_{p-1}^{|l|+1}}{L_{p}^{|l|}}e^{i\sigma\phi}\right];\mbox{
}p\geq 1,\\\
i\theta_{d}E_{y}\left[\frac{|l|}{\widetilde{r}}\frac{1\mp\sigma}{2}e^{\mp
i\phi}-\widetilde{r}fe^{i\sigma\phi}\right];\mbox{ }p=0,\end{cases}$ (23)
$\displaystyle
B_{x}^{\pm}=\begin{cases}\theta_{d}E_{y}\left[\frac{|l|}{\widetilde{r}}\frac{\sigma\mp
1}{2}e^{\mp i\phi}-\widetilde{r}f\sigma
e^{i\sigma\phi}-\frac{\sigma}{1+\widetilde{x}^{2}}\frac{L_{p-1}^{|l|+1}}{L_{p}^{|l|}}e^{i\sigma\phi}\right];\mbox{
}p\geq 1,\\\ \theta_{d}E_{y}\left[\frac{|l|}{\widetilde{r}}\frac{\sigma\mp
1}{2}e^{\mp i\phi}-\widetilde{r}f\sigma e^{i\sigma\phi}\right];\mbox{
}p=0,\end{cases}$ (24)

where the superscripts on the left-hand side again represent the sign of $l$.

Only for $|l|=1$ the longitudinal rather than transverse fields peak on axis.
However, the two circular polarizations (right and left) are not equivalent.
In the case of $\sigma=-l$, the longitudinal fields reach their highest
amplitude at $\widetilde{r}\rightarrow 0$. On the other hand, in the case of
$\sigma=l$, the longitudinal fields vanishes on axis. The transverse fields
vanish on axis in both cases. It is worth pointing out that, in contrast to
the linearly polarized beam, the longitudinal fields of the circularly
polarized beam with $\sigma=-l$ are axis-symmetric.

Figure 1 shows the field structure for right-circularly polarized beams
($\sigma=1$) with different values of the twist index $l$ and radial index
$p$. In agreement with our analysis, the transverse field in Fig. 1(a) only
peaks on axis for the beams without a twist ($l=0$). In all other cases, the
transverse field vanishes at $\widetilde{r}\rightarrow 0$. The longitudinal
field, shown in Fig. 1(b), only peaks on axis for the beam with $l=-1$. As
predicted, the longitudinal field vanishes at $\widetilde{r}\rightarrow 0$ for
$l=1$.

It must be pointed out that the circularly polarized beams with $|l|=1$ have
two, rather than one, rotations to consider: the rotation of the electric
field maxima about the central axis due to the wavefronts twist and the
rotation of the electric field vector due to the choice of polarization. In
the case with $|l|=1$ and $\sigma=-l$, the rotation of the transverse electric
field vector due to the polarization and the twist of the transverse field
wavefronts have opposing chiralities. Figure 2 (bottom row) provides schematic
diagrams showing the two rotations. For comparison, the upper row in Fig. 2)
shows the field topology for a linearly polarized beam, with
$E_{\perp}=E_{y}$. Recall that the circularly polarized beam is a
superposition of two linearly polarized beams.

Figure 2: Schematic representation of the transverse electric field topology
for a linearly polarized beam (top row) with $l=-1$ and $p=0$ and a circularly
polarized beam (bottom row) with $l=-1$, $p=0$, and $\sigma=-l$. The diagrams
show one wavelength near the focal plane of each beam. The color-coded
surfaces are the surfaces of the constant amplitude $|\bm{E}_{\perp}|=E_{*}$:
blue is positive $E_{y}$, red is negative $E_{y}$, green is positive $E_{z}$,
and purple is negative $E_{z}$. The large arrows show the direction of
$\bm{E}_{\perp}$ at a given location. The small black arrows placed on the
large circles represent the motion of $|\bm{E}_{\perp}|$ peaks over time. The
small black arrows placed on the small circles in the bottom row represent the
direction of rotation of the electric field over time. The red structure at
the centre of each image represents the the central axis of each beam. The
images from left to right are at times incremented by a quarter period $T/4$,
where $T=2\pi/\omega$.

In the sections that follow, we focus on electron acceleration in the near-
axis region by $E_{x}$ of a circularly polarized laser beam with $l=-1$ and
$\sigma=1$. In order to obtain a compact expression for $E_{x}$, we set
$\widetilde{r}=0$ in Eq. (23) and substitute the expression for $E_{y}$ given
by Eq. (8), which yields

$E_{x}{(\widetilde{r}=0)}=iE_{\parallel}^{\max}f^{2(1+p)}(1+\widetilde{x}^{2})^{p}g(\xi)e^{i\xi}=\frac{ig(\xi)E_{\parallel}^{\max}}{1+\widetilde{x}^{2}}\exp\left[i\xi-i2(1+p)\tan^{-1}\widetilde{x}\right],$
(25)

where

$E_{\parallel}^{\max}=2\sqrt{\frac{p+1}{\pi}}\theta_{d}E_{0}=\frac{2}{\pi}\sqrt{\frac{p+1}{\pi}}\frac{\lambda_{0}}{w_{0}}E_{0}$
(26)

is the peak value of the longitudinal electric field. A feature that is
important for electron acceleration is the explicit dependence of the phase on
$p$. This will be discussed in more detail in Section 3. The peak period-
averaged power $P$ for the circularly-polarized beam is twice the power
$P_{lin}$ for a single linearly polarized beam with the same amplitude
$E_{0}$. Taking this into account and using the expression for $P_{lin}$ given
by Eq. 16, we obtain the following expression for $P$ in terms of the
amplitude of the longitudinal field:

$P=\frac{\pi^{4}}{4(p+1)}\frac{w_{0}^{4}}{\lambda_{0}^{4}}\frac{m_{e}^{2}c^{5}}{e^{2}}a_{\parallel}^{2},$
(27)

where $a_{\parallel}\equiv|e|E_{\parallel}^{\max}/m_{e}c\omega$. Equation 27
can be recast as an expression for normalized amplitudes of longitudinal
electric and magnetic fields for a given power $P$ in the laser beam:

$a_{\parallel}\equiv\frac{|e|E_{\parallel}^{\max}}{m_{e}c\omega}=\frac{|e|B_{\parallel}^{\max}}{m_{e}c\omega}\approx
71\sqrt{p+1}\left(\frac{\lambda_{0}}{w_{0}}\right)^{2}P^{1/2}[\mbox{PW}].$
(28)

We conclude this section by pointing out that the longitudinal fields can be
strong even for $\theta_{d}\ll 1$. Let us take a $P=0.6$ PW circularly-
polarized beam with $l=-1$, $\sigma=1$, and $p=0$. The focal spot size is
$w_{0}=3.0~{}{\upmu\mathrm{m}}$ and the wavelength is
$\lambda_{0}=0.8~{}{\upmu\mathrm{m}}$. In this case, $\theta_{d}\approx
8.5\times 10^{-2}$, so our analysis that was performed in the paraxial
approximation is applicable to this beam. It follows from Eq. 26 that
$E_{\parallel}^{\max}\approx 0.1E_{0}$. Taking into account that
$E_{y}^{\max}=E_{0}C_{0,l}|l|^{|l|/2}\exp{(-|l|/2)}$ for $p=0$, we find that
$E_{\parallel}^{\max}/E_{y}^{\max}=B_{\parallel}^{\max}/B_{y}^{\max}\approx
0.2$. We also find from Eq. 28 that $a_{\parallel}\approx 3.8$. The
corresponding dimensional field amplitudes are $E_{\parallel}^{\max}\approx
1.5\times 10^{13}$ V/m and $B_{\parallel}^{\max}\approx 51$ kT.

##  3 Preliminary estimates for electron acceleration

In this section, we perform preliminary estimates for electrons accelerated in
the near-axis region by a helical laser beam whose field structure in this
region predominantly consists of longitudinal electric and magnetic fields. We
assume that the electrons are injected into the laser beam near the focal
plane located at $\widetilde{x}=0$. The injection is implied to occur when an
incident beam is reflected off a mirror at normal incidence. This process is
examined self-consistently in Section 4 and Section 5 using kinetic
simulations.

The phase velocity, $v_{ph}$, of $E_{\parallel}$ wavefronts plays a key role
in electron acceleration along the central axis. For simplicity, we limit our
analysis to the part of the pulse that is near the peak of the envelope, which
means that we can set $g(\xi)\approx 1$ in Eq. 25. In order to determine
$v_{ph}$, it is convenient to re-write the expression for the longitudinal
electric field given by Eq. 25 as

$E_{\parallel}=-\frac{E_{\parallel}^{\max}\sin\left(\Phi+\Phi_{0}\right)}{1+x^{2}/x_{R}^{2}},$
(29)

where the phase $\Phi$ is given by

$\Phi=2\left[\theta_{d}^{-2}(x/x_{R})-(p+1)\tan^{-1}(x/x_{R})\right]-\omega
t.$ (30)

The constant $\Phi_{0}$ can be interpreted as the injection phase for an
electron that starts its acceleration at $x\approx 0$ at $t\approx 0$. We
define the phase velocity as $v_{ph}=dx/dt$ for $\Phi=\mbox{const}$. We
differentiate Eq. 30, where $\Phi=\mbox{const}$, to obtain

$0=\frac{v_{ph}}{c}-\frac{v_{ph}}{c}\frac{(p+1)\theta_{d}^{2}}{1+x^{2}/x^{2}_{R}}-1,$
(31)

where it was taken into account that $\theta_{d}^{2}=\lambda_{0}/\pi x_{R}$.
In the paraxial approximation ($\theta_{d}\ll 1$ ), the second term on the
right-hand side is small. We neglect this term to find that $v_{ph}/c\approx
1$. In order to find the correction associated with $\theta_{d}$, we set
$v_{ph}/c=1$ in the second term on the right-hand side of Eq. 31 and obtain
the following expression for the relative degree of superluminosity along the
central axis:

$\frac{v_{ph}-c}{c}\approx\frac{(p+1)\theta_{d}^{2}}{1+x^{2}/x^{2}_{R}}.$ (32)

The key feature here is the explicit dependence on the radial index $p$, with
the superluminosity being higher for higher-order radial modes.

The electron unavoidably slips with respect to the wavefronts as it moves
forward, which limits its energy gain. The slipping is determined by the
difference $v_{ph}-v_{x}$, where $v_{ph}\geq c$ and $v_{x}<c$. As the electron
becomes ultra-relativistic due to the acceleration by $E_{\parallel}$, it
enters a regime where $c-v_{x}\ll v_{ph}-c$. In this regime, the slipping, or
dephasing, is primarily determined by the relative degree of superluminosity
given by Eq. 32. The lowest estimate for the phase slip experienced by an
electron that has travelled from $x_{0}$ to $x$ is obtained by setting
$x=x_{0}+c(t-t_{0})$ in Eq. 30, which yields

$\Delta\Phi=2(p+1)\left[\tan^{-1}(x_{0}/x_{R})-\tan^{-1}(x/x_{R})\right]$ (33)

For $x_{0}\ll x_{R}$, this expression reduces to
$\Delta\Phi=-2(p+1)\tan^{-1}(x/x_{R})$. The phase velocity is superluminal
near $x=0$, but it decreases to $c$ at $x\gg x_{R}$. As a result, the total
phase slip is finite and it approaches $\Delta\Phi=-(p+1)\pi$ at $x\gg x_{R}$.
There is a significant difference between the $p=0$ mode and higher order
radial modes. At $p=0$, the phase slip is $\Delta\Phi=-\pi$, which means that
some electrons (this depends on the initial phase) can remain in the
accelerating phase of $E_{\parallel}$ until the laser defocuses ($x\gg x_{R}$)
and $E_{\parallel}$ becomes very weak. In contrast to that, all electrons
experience deceleration by $E_{\parallel}$ at $p\geq 1$, because they slip
into the decelerating phase prior to strong defocusing at $x\gg x_{R}$.

In order to estimate the electron energy gain from $E_{\parallel}$, we assume
that the electron is ultra-relativistic with $c-v_{x}\ll v_{ph}-c$. In this
case, the phase $\Phi$ in Eq. 29 can be replaced by
$\Phi\approx\Phi_{0}+\Delta\Phi$, where $\Delta\Phi$ is given by the already
derived Eq. 33. The change in electron momentum during the acceleration is
obtained by integrating the momentum balance equation
$dp_{\parallel}/dt=-|e|E_{\parallel}$, which yields

$\Delta
p_{\parallel}=|e|E_{\parallel}^{\max}\int_{t_{0}}^{t}\frac{\sin(\Delta\Phi+\Phi_{0}+\pi)dt^{\prime}}{1+(x^{\prime})^{2}/x_{R}^{2}},$
(34)

where $x^{\prime}$ is the electron location at time $t^{\prime}$. We note that
$dx^{\prime}/dt^{\prime}\approx c$ in the considered regime, so we can switch
from integration over time to integration over the longitudinal coordinate by
replacing $dt^{\prime}$ with $dx^{\prime}/c$. After substituting the
expression for Eq. 33, we obtain

$\displaystyle\Delta p_{\parallel}$ $\displaystyle=$
$\displaystyle\frac{|e|E_{\parallel}^{\max}}{c}\int_{x_{0}}^{x}\frac{\sin(\Delta\Phi+\Phi_{0})dx^{\prime}}{1+(x^{\prime})^{2}/x_{R}^{2}}$
(35) $\displaystyle=$
$\displaystyle-\frac{|e|E_{\parallel}^{\max}x_{R}}{2(p+1)c}\left(\cos\Phi_{0}-\cos\left[\Phi_{0}+2(p+1)\tan^{-1}(x_{0}/x_{R})-2(p+1)\tan^{-1}(x/x_{R})\right]\right).$

This result can be further simplified by assuming that $x_{0}\ll x_{R}$, so
that $\tan^{-1}(x_{0}/x_{R})\approx 0$ and

$\Delta
p_{\parallel}=-\frac{|e|E_{\parallel}^{\max}x_{R}}{2(p+1)c}\left(\cos\Phi_{0}-\cos\left[\Phi_{0}-2(p+1)\tan^{-1}(x/x_{R})\right]\right).$
(36)

Equation 36 represents an important qualitative result, as it shows that the
electrons can retain a significant portion of the energy they gain from
$E_{\parallel}$. We find the terminal momentum gain by taking the limit of
$x/x_{R}\rightarrow\infty$ in Eq. 36. There is a profound difference between
odd and even radial radial modes, i.e. odd and even radial indices $p$. In the
case of even modes with (e.g. $p=0,2,4$), we have

$\Delta
p^{term}_{\parallel}=\frac{|e|E_{\parallel}^{\max}x_{R}}{(p+1)c}\cos(\Phi_{0}-\pi).$
(37)

The energy gain occurs for $0.5\pi<\Phi_{0}<1.5\pi$ regardless of the radial
mode structure. Our assumption that the electron is moving forward with an
ultra-relativistic velocity breaks down for $1.5\pi/<\Phi_{0}<2.5\pi$, which
invalidates Eq. 37 for these injection phases. In contrast to the even radial
modes, there is no terminal momentum gain, $\Delta p^{term}_{\parallel}=0$,
for the odd radial modes regardless of the injection phase. It is worth
pointing out that this estimate was obtained under the assumption that the
electron is ultra-relativistic, so the analysis has to be revised along the
parts of the trajectory where $c-v_{x}\geq v_{ph}-c$. However, the value of
$\Delta p^{term}_{\parallel}$ is unlikely to increase dramatically as a
result.

Figure 3: The momentum gain predicted by Eq. 36 for electrons injected at
phase $\Phi_{0}$ into a right-circularly polarized beam ($\sigma=1,l=-1$). The
solid curves are for a beam with the radial index $p=0$, whereas the green and
black dashed curves correspond to beams with $p=1$ and $p=2$, respectively.

Figure 3 illustrates the results of our analysis for three different radial
modes, $p=0,1,$ and 2. The solid curves show electron acceleration by a beam
with $p=0$, as predicted by Eq. 36. Both electrons are injected at a phase
that leads to a net momentum gain. However, the delayed injection of the
electron shown with the solid blue curve ($\Phi_{0}=0.6\pi$) means that it
slips into the decelerating phase before the amplitude of $E_{\parallel}$
becomes negligibly small due to the beam diffraction. As a result, the net
momentum gain is two times lower than for the case of $\Phi_{0}=0.9\pi$.
Higher-order modes speed up the electron slip into the decelerating phase,
because the relative degree of superluminosity given by Eq. 32 increases with
$p$. This trend is clearly shown by the dashed curves, representing modes with
$p=1$ and $p=2$, respectively. In both cases, the injection phase is
$\Phi_{0}=0.9\pi$, which is the same phase as that for the solid red curve
($p=0$). The green dashed curve, corresponding to $p=1$, rolls over, after a
peak in momentum gain, at $\widetilde{x}\approx 1$, whereas the black dashed
curve, corresponding to $p=2$, rolls over even sooner when the electron
reaches $\widetilde{x}\approx 0.5$. In the case of $p=1$, the electron remains
in the decelerating phase until the beam experiences significant diffraction.
As a result, there is no net momentum gain. In the case of $p=2$, the
dephasing is faster, so the electron enters another accelerating region of the
beam when it reaches $\widetilde{x}\approx 1.5$. The acceleration continues
until the diffraction eliminates $E_{\parallel}$. In this case there is a net
momentum gain, but it is smaller than in the case of $p=0$ because the
acceleration time is shorter and the accelerating field (due to the
diffraction) is weaker.

It is useful to recast our results for the momentum gain in terms of electron
energy. The energy of an ultra-relativistic electrons with momentum $\bm{p}$
is $\varepsilon\approx c|\bm{p}|$ and, in our case, $|\bm{p}|\approx
p_{\parallel}$. If the electron experiences a significant momentum gain, then
the terminal longitudinal electron momentum is
$p_{\parallel}^{term}\approx\Delta p_{\parallel}^{term}$. Putting all these
estimates together, we find that the terminal electron energy is given by
$\varepsilon^{term}\approx c\Delta p_{\parallel}^{term}$. For modes with an
even $p$ index, the momentum gain is given by Eq. 37. We also take into
account the relation between $a_{\parallel}$ and $P$ given by Eq. 28. As a
result we arrive to the following expression for the electron energy gain:

$\varepsilon^{term}[\mbox{GeV}]\approx\frac{0.72}{p+1}\cos(\Phi_{0}-\pi)P^{1/2}[\mbox{PW}],$
(38)

The result is independent of the spot size $w_{0}$ and wavelength
$\lambda_{0}$. At the end of Section 3, we examined the field structure of a
circularly-polarized 0.6 PW laser beam with $l=-1$, $\sigma=1$, and $p=0$.
According to Eq. 38, we expect this beam to generate electrons with hundreds
of MeV in energy, since  $\varepsilon^{term}\approx 0.56\cos(\Phi_{0}-\pi)$
GeV.

##  4 Simulation results for a normally incident 600 TW laser

Figure 4: Field structure of an incident circularly-polarized Laguerre-
Gaussian laser beam from a 3D PIC simulation. The snapshots are taken at
$t=-9$ fs. The laser beam and simulation parameters are given in Table 1.

In Section 3, we provided preliminary estimates showing that a properly chosen
beam with twisted wavefronts can generate forward-directed ultra-relativistic
electrons. In this section, we present a self-consistent analysis, performed
using a 3D PIC simulation, of electron injection and subsequent acceleration
by a laser beam with dominant $E_{\parallel}$ and $B_{\parallel}$ in the near-
axis region.

Laser parameters
---
Peak power (period averaged) |  0.6 PW
Twist and radial index |  $l=-1,p=0$
Right-circular polarization |  $\sigma=1$
Wavelength |  $\lambda_{0}=0.8~{}{\upmu\mathrm{m}}$
Pulse duration ($\sin^{2}$ electric field) |  $\tau_{g}=20$ fs
Focal spot size ($1/\mathrm{e}$ electric field) |  $w_{0}=3~{}{\upmu\mathrm{m}}$
Location of the focal plane |  $x=0~{}{\upmu\mathrm{m}}$
Direction of the incident laser |  $-x$
Other parameters
Position of the bulk target |  $-1.0~{}{\upmu\mathrm{m}}\leq x\leq-0.3~{}{\upmu\mathrm{m}}$
Position of the pre-plasma |  $-0.3~{}{\upmu\mathrm{m}}<x\leq 0.0~{}{\upmu\mathrm{m}}$
Electron and C+6 density |  $n_{e}=500\;n_{c}$ and $n_{Carbon}=83.3\;n_{c}$
Gradient length |  $L=\lambda_{0}$/20
Simulation box ($x\times y\times z$) |  $10~{}{\upmu\mathrm{m}}\times 30~{}{\upmu\mathrm{m}}\times 30~{}{\upmu\mathrm{m}}$
Moving window start time |  11 fs
Moving window velocity |  $c$
Cell number ($x\times y\times z$) |  400 cells $\times$ 800 cells $\times$ 800 cells
Macroparticles per cell for electrons |  300 at $r<2.5~{}{\upmu\mathrm{m}}$, 36 at $r>2.5{\upmu\mathrm{m}}$
Macroparticles per cell for C+6 |  24
Table 1: 3D PIC simulation parameters.

In our simulation, a 600 TW circularly-polarized Laguerre-Gaussian beam with
$l=-1$, $\sigma=1$, and $p=0$ is normally incident on a mirror that is
initialized as a fully ionized plasma with a sharp density gradient. The
incident pulse propagates in the negative direction along the $x$-axis. The
laser envelope function $g({\xi})$ has a temporal profile such that
$g(t)=\sin^{2}(0.5\pi*t/\tau_{g})$ with a total duration of $\tau_{g}=20$ fs.
The beam width is $w_{0}=3~{}{\upmu\mathrm{m}}$, the laser wavelength is
$\lambda_{0}=0.8~{}{\upmu\mathrm{m}}$, and the focal plane is located at
$x=0~{}{\upmu\mathrm{m}}$. The mirror is a carbon plasma with an electron
density profile
$n_{e}=500n_{c}\exp[-20(x+0.3~{}{\upmu\mathrm{m}})/\lambda_{0}]$ at
$x\geq-0.3~{}{\upmu\mathrm{m}}$, where $n_{c}=1.8\times 10^{27}$ m-3 is the
critical density for $\lambda_{0}=0.8~{}{\upmu\mathrm{m}}$. While ion mobility
does not appear to affect the simulation results, the ions are left mobile so
as to ensure a more realistic scenario. The initial kinetic energy of all
particles (electrons and ions) is set to zero. In order to follow the
electrons bunches over a long period of time a moving window is employed. The
window size is set to encompass the entire simulation box and moves at a
velocity $c$, beginning at a time $t=$11 fs . Additional simulation details
are provided in Table 1.

The field structure of the incident laser beam in the $(x,z)$-plane is shown
in Fig. 4. The time snapshots are taken at $t=-9$ fs. We define $t=0$ as the
time when, in the absence of the mirror, the peak of the laser envelope
reaches at $x=0$. As can be seen in Fig. 4, the longitudinal fields $E_{x}$
and $B_{x}$ are strongest on-axis where the transverse field $E_{z}$ tends to
zero. The field structure in these snapshots agrees with the paraxial analysis
presented in Section 2 for beams with twisted wavefronts.

Figure 5: (a) Accelerated electron bunches at $t=261$ fs. The color shows the
distribution of electrons with $r<1.5~{}{\upmu\mathrm{m}}$ in the
$(\varepsilon_{e},x)$-space. The inset shows the structure of the third bunch.
(b) The energy spectrum $dN/d\varepsilon_{e}$ in the third bunch at $t=261$ fs
for electrons with $r<1.5~{}{\upmu\mathrm{m}}$ for cases with (blue line) or
without (black line, multiplied by 10 to aid the comparison) the force from
$B_{x}$ in the electron equation of motion. (c) The time evolution of the
energy spectrum of the third bunch. The black dashed curve is the prediction
of Eq. 36 with $\Phi_{0}=0.8\pi$. (d) The energy-angle distribution in the
third bunch at $t=261$ fs.

The laser-plasma interaction that occurs after the laser reaches the mirror
can be sorted into two stages: the injection stage and the acceleration stage.
The injection stage occurs during laser reflection off the plasma mirror. At
this stage, plasma electrons are drawn out of the mirror by laser fields and
injected into the near-axis region of the laser beam in front of the mirror.
The electron injections stops after the laser beam is fully reflected off the
mirror. Results from tracking the particles in the third bunch back to the
target surface show a complex relationship between the transverse and
longitudinal field structures. This process requires significant further
research that is beyond the scope of this article. During the acceleration
stage, the injected electrons travel with the reflected laser beam away from
the surface of the mirror. Their longitudinal motion is caused by
$E_{\parallel}$ of the reflected laser beam. Only those electrons that are
injected into a region with $E_{\parallel}<0$ can accelerate while moving away
from the mirror. On the other hand, no acceleration is possible when
$E_{\parallel}>0$ in the vicinity of the mirror surface. As a result,
oscillations of $E_{\parallel}>0$ near the mirror surface generate electron
bunches rather than a continuous electron beam. The maximum areal density of a
bunch, $\rho_{e}$, (integral of $n_{e}$ along the bunch) can be estimated by
taking into account that the injection process during one laser period stops
once the space-charge of the extracted electrons shields $E_{\parallel}$ of
the laser. This yields $\rho_{e}\approx a_{\parallel}n_{c}c/\omega$ or

$\rho_{e}[\mbox{m}^{-2}]\approx 1.3\times
10^{22}P^{1/2}~{}[\mbox{PW}]~{}\lambda_{0}~{}[\mathrm{\mu
m}]~{}w^{-2}~{}[\mathrm{\mu m}].$ (39)

The implication is that a beam with a strong longitudinal electric field is
expected to produce high density bunches.

Figure 5(a) shows that the accelerated electrons indeed travel in bunches. The
snapshot is taken at $t=261$ fs after the injected electrons have travelled
roughly $100\lambda_{0}$ along the $x$ axis with the laser beam. The plot
shows the distribution of the electrons located in the region near the axis
with $r<1.5~{}{\upmu\mathrm{m}}$ in the $(\varepsilon_{e},x)$-space. The
compactness of the bunches, both in transverse [see Fig. 6(c)] and
longitudinal directions [see Fig. 5(a)], ensures that the electrons in each
bunch essentially experience the same accelerating field. This explains the
narrow energy spread within each bunch in Fig. 5(a).

The evolution of the energy spectrum within a single bunch is plotted in Fig.
5(c). The selected bunch is marked as bunch #3 in Fig. 5(a). The bunch retains
a relatively narrow energy spread during the acceleration process. The dashed
curve shows the prediction given by Eq. 36 for an injection phase of
$\Phi_{0}=0.8\pi$. The good agreement indicates that the accelerated electrons
are likely injected into the same phase of the reflected beam. There is also a
strong correlation between electrons which have a high energy and those with a
low divergence angle. This is best seen in Fig. 5(d) which shows the
distribution of energy versus divergence angle
$\theta=\arctan{\left(p_{\perp}/p_{\parallel}\right)}$ of the third bunch.

The presented simulation demonstrates that a normally incident 600 TW beam can
generate several dense bunches of ultra-relativistic electrons. Specifically,
the terminal energy of the electrons in the third bunch is 0.47 GeV with a
FWHM of $\sim 10\%$ [see Fig. 5(b) for a snapshot at $t=261$ fs]. The bunch
has a duration of roughly 400 as and a total electron charge of 26 pC, while
the normalized transverse emittance is $9.5~{}{\upmu\mathrm{m}}$.

##  5 The impact of the longitudinal magnetic field on electron dynamics

The most distinctive feature of the considered laser beam with twisted
wavefronts is the strong longitudinal magnetic field $B_{\parallel}$ in the
near-axis region. In what follows, we examine the profound impact of this
field on electron dynamics.

Figure 6: A comparison between 3D PIC simulations with (upper row) and without
$B_{x}$ (bottom row) in the electron equations of motion. Both simulations use
the same 600 TW laser beam with parameters given in Table 1. All snapshots are
taken at $t=9$ fs. (a) and (b): Electron density on a log-scale, with the
color representing $\log(n_{e}/n_{c})$. The blue, red, and green contours
denote $n_{e}=0.1n_{c}$, $0.5n_{c}$, and $n_{c}$. The dashed rectangle shows
the radial and longitudinal extent of what is referred to as the third bunch
in the remaining panels. (c) and (d): Areal density of the electrons in the
third bunch. (e) and (f): Cell-averaged electron divergence angle
$\langle\theta\rangle$ in the third bunch. The angle
$\theta\equiv\arctan(p_{\bot}/p_{x})$ of an individual electron is averaged
over the electrons located within the cells with the same $y$ and $z$
coordinates.

In addition to $E_{\parallel}$, the electrons in the near-axis region
experience $B_{\parallel}$ that can provide transverse confinement at an early
stage of the acceleration before the electron momentum becomes predominantly
forward-directed. The importance of $B_{\parallel}$ can be assessed by
estimating the Larmor radius, $r_{L}$, for an electron injected with
transverse relativistic momentum $p_{\perp}$ into the field near the focal
plane, whose strength is given by Eq. 28. We find that

$r_{L}/w_{0}\approx\frac{2.2\times
10^{-3}}{\sqrt{1+p}}\left(\frac{p_{\perp}}{m_{e}c}\right)\left(\frac{w_{0}}{\lambda_{0}}\right)P^{-1/2}~{}[\mbox{PW}].$
(40)

In the case of the 600 TW laser beam considered in Section 4, we have
$r_{L}/w_{0}\approx 10^{-2}p_{\perp}/m_{e}c$. Even for those electrons that
are injected with
$p_{\perp}/m_{e}c\approx|e|E_{\perp}^{\max}/m_{e}c\omega\approx 19$, the
Larmor radius, $r_{L}\approx 0.2w_{0}$, is significantly smaller than the beam
waist. By keeping the injected electrons close to the axis, $B_{\parallel}$
ensures that the electrons are unable to sample strong $E_{\perp}$ and gain
additional transverse momentum. As a result, they are predominantly
accelerated by $E_{\parallel}$, which leads to a strongly collimated beam. It
is important to point out that $B_{\parallel}$ is shifted by $\pi/2$ with
respect to $E_{\parallel}$, which means that the direction of the electron
rotation induced by the magnetic field changes during the acceleration
process.

At the injection stage, plasma electrons experience a complicated interplay of
transverse and longitudinal fields that is not captured by the presented
estimate. The impact of $B_{x}$ during this process can be elucidated by
removing its effects from the electron dynamics even though this does lead to
a somewhat non-physical scenario. To remove the effect of $B_{x}$ in the PIC
simulation, we multiply the $B_{x}$ variable by zero in the relevant section
of the particle pusher. Figure 6(b) shows the electron density obtained using
this approach for the laser-mirror interaction examined in Section 4. For
comparison, Fig. 6(a) also shows the electron density from the original
simulation where the effect of $B_{x}$ is included. The bunches in Fig. 6(b)
form in a ring-like structure away from the central axis. The structure of the
bunches appears superficially similar to that observed for radially polarized
beams at lower intensities [9]. In contrast to that, the electron bunches in
Fig. 6(a) are formed close to the axis in the region where the longitudinal
field dominates. The bunches become more compact as they move away from the
mirror, with the electron density exceeding $n_{c}$.

Figure 7: (a) Time evolution of the electron distribution over the divergence
angle $\theta$ in the third bunch ($r<1.5~{}{\upmu\mathrm{m}}$). (b)
Trajectories of electrons randomly chosen from the third bunch at $t=261$ fs.
The trajectories correspond to $6\mbox{ fs}\leq t\leq 261~{}\mbox{ fs}$, with
the circles marking the electron positions at $t=261$ fs. The transverse
coordinates are normalized to the local width of the laser beam
$w(x)=w_{0}\sqrt{1+x^{2}/x_{R}^{2}}$.

Figures 6(c) - (f) compare the areal density and divergence angle between the
two simulations. The comparison is performed for the electrons within the
third bunch in Figs. 6(a) and (b). The radial and longitudinal extend of the
third bunch is shown with the the black dashed rectangles in Figs. 6(a) and
(b). The difference in collimation is striking even at this injection stage
($t=9~{}\mathrm{fs}$). The higher areal density point-like structure seen in
Fig. 6(c) is complimented by a corresponding small cell-averaged electron
divergence angle $\langle\theta\rangle$ seen in Fig. 6(e). The averaging,
indicated by the angular brackets, is performed by taking all the electrons
located within the cells with the same $y$ and $z$ coordinates, i.e. the cells
with the same projection onto the transverse plane. The averaging accounts for
the difference in the numerical weight $w_{\alpha}$ of individual macro-
particles caused by the initial density gradient:
$\langle\theta\rangle=\sum_{\alpha}w_{\alpha}\theta_{\alpha}/\sum_{\alpha}w_{\alpha}$.
In the case where the influence of $B_{x}$ is removed [with the corresponding
plots of areal density in Fig. 6(d), and cell-averaged divergence in Fig.
6(f)], we see a ring-like ‘bunch’ with divergent profile indicating that it
will continue to diverge long after the injection stage is over. The
contrasting pictures between the two simulations show the important role of
the $B_{x}$.

Figure 5(b) shows the energy spectrum of the third bunch in the simulation
without the $B_{x}$ forces at a much later time ($t=261~{}\mathrm{fs}$). The
same figure shows the spectrum for the third bunch in the original simulation
at the same time instant. Both spectra are calculated for the electrons in the
near-axis region with $r<1.5~{}{\upmu\mathrm{m}}$. Not surprisingly, the
electrons moving close to the central axis achieve similar energies in both
simulations. However, the number of such energetic electrons in the simulation
without the $B_{x}$ forces is an order of magnitude lower. During the
injection process and at the very early acceleration stage the longitudinal
magnetic field provides a significant amount of transverse electron
confinement. This keeps electrons in the region with a strong $E_{x}$ but weak
$E_{\perp}$, which ensures good collimation of the electron bunches.

To further examine the transverse electron motion during the acceleration
process, we have tracked several electrons. The electrons are randomly
selected from the third bunch ($r<1.5~{}{\upmu\mathrm{m}}$) at $t=261$ fs. The
projection of the electron trajectories onto the transverse plane for $6\mbox{
fs}\leq t\leq 261~{}\mbox{fs}$ is shown in Fig. 7(b), where the circles
indicate the electron positions at $t=261$ fs. The transverse coordinates are
intentionally normalized to the local width of the laser beam
$w(x)=w_{0}\sqrt{1+x^{2}/x_{R}^{2}}$ to correlate the electron location with
the strength of the transverse electric field $E_{\perp}$ that increases away
from the central axis. As expected, there is a pronounced rotation induced by
$B_{\parallel}$ that prevents electrons from expanding and reaching a region
with a strong $E_{\perp}$. This rotation is also observed to reverse,
effectively twisting the trajectory of the electrons. This feature is caused
by the $\pi/2$ shift between $E_{\parallel}$ and $B_{\parallel}$. The sign of
$B_{\parallel}$ changes during the acceleration process, which causes the
electrons to rotate in the opposite direction and manifests itself as the
twist seen in Fig. 7(b) for each individual trajectory.

Another important metric of the electron dynamics is the divergence angle
$\theta$. Figure 7(a) shows how the electron distribution over
$\theta=\arctan{\left(p_{\perp}/p_{\parallel}\right)}$ changes with time in
the third bunch. We find that the bunch becomes less divergent over time,
which is a clear indicator of the dominant role played by $E_{\parallel}$.
Indeed, $\theta$ decreases because $p_{\parallel}$ increases at a much faster
rate than $p_{\perp}$. The increase in $p_{\parallel}$ is caused by
$E_{\parallel}$, as already discussed in Section 3. The changes in $p_{\perp}$
are caused by $E_{\perp}$. The longitudinal magnetic field has no direct
impact on the divergence angle since it rotates $\bm{p}_{\perp}$ without
changing its amplitude. However, by keeping the electrons close to the central
axis, the magnetic field prevents them from experiencing a strong $E_{\perp}$
that peaks off axis and, as a result, it prevents $p_{\perp}$ from increasing.

Figure 8: 3D PIC simulation results for a $25^{\circ}$ incidence angle. The
snapshots are take at $t=39$ fs and plotted in an $(x^{\prime},y^{\prime},z)$
system of coordinates whose $x^{\prime}$-axis points in the specular
direction. (a) Areal electron density in a third bunch moving in the specular
direction. (b) Cell-averaged electron divergence angle
$\langle\theta^{\prime}\rangle$ in the third bunch. The angle
$\theta^{\prime}\equiv\arctan\left(\left.\sqrt{p_{y^{\prime}}^{2}+p_{z}^{2}}\right/p_{x^{\prime}}\right)$
is averaged on every mesh cell of the $(y^{\prime},z)$ plane.

We conclude this section by presenting results for a case of oblique
incidence. Experimental facilities often require that laser pulses are not
shot at normal incidence onto a reflecting surface so as to avoid damage to
optical systems. This was our primary motivation to examine the oblique
incidence case. In our simulation, the laser beam is still incident in the
negative direction along the $x$-axis, but the target is now titled by
$25^{\circ}$. In order to make the simulation manageable, we perform it in a
window moving along the $x$-axis. The simulation time is limited compared to
the case of normal incidence, because the electron bunches, that now travel at
an angle to the $x$-axis, leave the simulation window prematurely. This setup
does not allow enough time for the acceleration process to reach a plateau as
in the normal incidence case and so the final distribution function is left
for future research. Despite this limitation this simulation will allow
testing the early time conditions that set up the acceleration process and
bunch formation.

We find that the laser reflection again generates a series of dense collimated
bunches, which indicates the robustness of the considered approach to electron
acceleration. The areal density $\rho_{e}$ and the cell-averaged divergence
angle $\langle\theta^{\prime}\rangle$ in the third bunch are shown in Fig.
8(a, b). These snapshots are taken at $t=52~{}\mathrm{fs}$. We use an
orthogonal $(x^{\prime},y^{\prime},z)$ system of coordinates whose
$x^{\prime}$-axis points in the specular direction. The divergence angle
$\theta^{\prime}\equiv\arctan\left(\left.\sqrt{p_{y^{\prime}}^{2}+p_{z}^{2}}\right/p_{x^{\prime}}\right)$
is the angle between the momentum vector $\bm{p}$ and the $x^{\prime}$-axis.
Even though the axial symmetry is broken, there is still strong evidence of a
dense non-divergent bunch close to the laser axis. The high degree of
collimation following the injection suggests that the reflected laser pulse
would generate highly energetic bunches with similar characteristics to the
case presented at normal incidence.

##  6 Summary and discussion

In summary, this manuscript presents a detailed analysis of the topology of
linearly and circularly polarized Laguerre-Gaussian laser beams. It is shown
that the beams with a twist index $|l|=1$ have a distinct field structure in
the near-axis region with dominant $E_{\parallel}$ and $B_{\parallel}$. In the
case of circularly polarized beams, the rotation of $\bm{E}_{\perp}$ should be
in the opposite direction to the rotation of the wavefronts, i.e. $\sigma=-l$,
to achieve such a structure. The manuscript also presents kinetic 3D PIC
simulations for a 600 TW circularly polarized beam ($p=0$, $l=-1$, and
$\sigma=1$) reflected off a plasma mirror. The dominant $E_{\parallel}$ and
$B_{\parallel}$ combine to inject dense electron bunches upon reflection. The
bunches are effectively accelerated by $E_{\parallel}$ while being confined by
$B_{\parallel}$. The magnetic field prevents the bunch electrons from
travelling too far radially outwards, so they sample a relatively weak
transverse electric field and remain well-collimated. The bunch with the
largest energy has a distinctly narrow energy spread with a FWHM of just 10%.
The terminal energy gain for an individual bunch is well predicted by the
analytical model developed in the manuscript. The charge of a single bunch in
the simulation is as high as 26 pC. The bunches have a duration of $\sim 400$
as and a remarkably low divergence of just ${1.15}^{\circ}$ (20 mrad).

The nearest analogue to the discussed mechanism is the acceleration by a
radially polarized laser beam. Such a beam also has a strong longitudinal
electric field that dominates the field structure in the region close to the
central axis. However, in contrast to the beams considered in this manuscript,
the radially polarized beam lacks a strong longitudinal magnetic field in the
region close to the axis. As shown in Appendix A, the absence of
$B_{\parallel}$ has a profound impact on the electron acceleration even though
the amplitude of $E_{\parallel}$ is the same as in the case of the radially
polarized beam considered in Section 4. In the absence of $B_{\parallel}$,
there is no mechanism confining electrons in the region with $r/w(x)\ll 1$.
The lack of confinement also manifests itself in the case of oblique
incidence. We have shown some results that indicate the robustness of the
electron injection by a beam with a strong $B_{\parallel}$. In contrast to
that, the injection by radially polarized beams is not as robust [30], which
makes experimental implementation extremely challenging.

The transverse fields do have a role where they are, in-part, responsible for
detaching electrons from the pre-plasma. The exact dynamical process relating
to the transverse fields is likely of less importance than that of the
longitudinal fields. Previous studies have discussed the ponderomotive force
as a mechanism for keeping electrons [31] (and ions [32]) within the central
region of the beam. However, as can be seen in the comparison made in Fig. 6,
this does not appear to be the dominant mechanism confining the electrons
close to the axis in our case, at least at early times.

While this study is limited to a relatively short laser pulse with 3 peaks at
high amplitude, a TiSaphire laser with a FWHM of around 30 fs will have a
train of roughly 15 peaks. The energy of each bunch is strongly tied to the
amplitude at the position in the envelope. After being injected, electrons are
accelerated directly by laser field in vacuum, implying that they are not
likely subjected to instabilities that occur in bulk plasmas. The density of
the bunches is also related to the longitudinal field, with higher densities
recorded with higher amplitudes[28]

Using the scheme detailed in the manuscript, it may be possible to construct a
source of highly collimated dense attosecond bunches of ultra-relativistic
electrons suitable for potential applications. This can be achieved using a
combination of high-power laser systems, similar to those already in use
today. In addition to this, optical techniques that have already been put to
experimental use can be used to create twisted wave-fronts necessary for this
mechanism. Given the research presented here, with some further studies, a
working design for a small-scale high-energy electron accelerator may be
within grasping distance.

## Acknowledgements

Y.S. acknowledges the support by grant of KY2140000018 and Newton
International Fellows Alumni follow-on funding. D.R.B. and A.A. acknowledge
the support by the National Science Foundation (Grant No. PHY 1903098).
Simulations were performed with EPOCH (developed under UK EPSRC Grants
EP/G054950/1, EP/G056803/1, EP/G055165/1 and EP/ M022463/1). The simulations
and numerical calculations in this paper have been done on the supercomputing
system in the Supercomputing Center of University of Science and Technology of
China. This work also used the Extreme Science and Engineering Discovery
Environment (XSEDE), which is supported by National Science Foundation grant
number ACI-1548562.

##  Appendix A 3D PIC simulation results for electron acceleration by a
radially polarized laser beam

The acceleration scheme presented in this manuscript is superficially similar
to that associated with radially polarized beams [9, 30]. Near the axis,
radially polarized beams also have a strong longitudinal electric field, so a
comparison of the two schemes is worth discussing.

Figure 9: Electric field components of a radially polarized laser beam
_before_ it encounters the mirror . Panels (a) and (d) show $E_{z}$; panels
(b) and (e) show $E_{y}$; panels (c) and (f) show $E_{x}$. The left column
[(a), (b), and (c)] shows the field structure in the $(x,z)$-plane at $y=0$.
The right column [(d), (e), and (f)] shows the field structure in the
$(y,z)$-plane at the $x$-position indicated with the dashed line. All the
snapshots are taken a time $t\approx-9$ fs.

The difference in the field topology can be illustrated by constructing a
radially polarized beam using two circularly-polarized Laguerre-Gaussian beams
from Section 2: the first with $l=1$, $\sigma=-1$ and the second, a mirror
image of the first, with $l=-1$, $\sigma=1$. We set $E_{0}=E_{*}/2$ for each
of the beams in Eq. 8 that describes the structure of $E_{y}$. The resulting
beam structure will be compared to a circularly polarized beam with
$E_{0}=E_{*}$ and $l=-1$, $\sigma=1$, and $p=0$. The superposition of the two
beams has

$E_{y}=\frac{1}{2}E_{*}\left(e^{i\phi}+e^{-i\phi}\right)D=E_{*}\cos\phi D,$
(41)

where $D(\widetilde{x},\widetilde{r},\xi)=g(\xi)\exp(i\xi)\psi_{\pm
1,p}(\widetilde{x},\widetilde{r},\phi)\exp(\mp i\phi)$ is a function that, for
compactness, incorporates all the remaining dependencies besides the
dependence on $\phi$. We next take into account that $E_{z}=i\sigma E_{y}$ in
each of the circularly-polarized beams and find that the superposition of the
two beams has

$E_{z}=\frac{1}{2}E_{*}\left(-ie^{i\phi}+ie^{-i\phi}\right)D=E_{*}\sin\phi D.$
(42)

These $E_{z}$ and $E_{y}$ represent a radially polarized laser beam:

$\bm{E}_{\perp}=\bm{e}_{r}E_{*}D.$ (43)

In the context of the electron acceleration mechanism, the biggest difference
between the radially polarized beam and the circularly polarized beam with
twisted wavefronts is the absence of $B_{\parallel}$ near the axis. In order
to find the structure of the longitudinal fields, we use Eqs. (19) and (20).
Close to the axis, the longitudinal electric fields of the two beams are the
same, so the longitudinal field of the radially polarized beam is given by

$E_{x}=\frac{i\theta_{d}}{2}\frac{1}{\widetilde{r}}e^{-i\phi}\left[\frac{E_{*}}{2}e^{i\phi}\right]+\frac{i\theta_{d}}{2}\frac{1}{\widetilde{r}}e^{i\phi}\left[\frac{E_{*}}{2}e^{-i\phi}\right]=\frac{i\theta_{d}}{2}\frac{1}{\widetilde{r}}E_{*}.$
(44)

It follows directly from Eq. (19) that it is equal to $E_{x}$ of the
circularly polarized beam with $E_{0}=E_{*}$ and $l=-1$, $\sigma=1$, and $p=0$
that we are using for our comparison. On the other hand, the longitudinal
magnetic fields of the two circularly-polarized beams cancel each other out,
so that there is no strong $B_{x}$ close to the axis of the resulting radially
polarized beam:

$B_{x}=-\frac{\theta_{d}}{2}\frac{1}{\widetilde{r}}e^{-i\phi}\left[\frac{E_{*}}{2}e^{i\phi}\right]+\frac{\theta_{d}}{2}\frac{1}{\widetilde{r}}e^{i\phi}\left[\frac{E_{*}}{2}e^{-i\phi}\right]=0.$
(45)

In contrast to this, $B_{x}$ of the circularly polarized beam with $l=-1$ and
$\sigma=1$ that we use for our comparison has the same amplitude as $E_{x}$.

Even though $E_{x}$ has the same amplitude in the two beams that are being
compared, the power in the radially polarized beam is two times lower. In
order calculate the power of the radially polarized beam, we note that
$B_{z}=E_{y}$ and $B_{y}=-E_{z}$. The longitudinal component of the Poynting
vector is then given by

$S_{x}=\frac{c}{4\pi}\left[\left(\mbox{Re}E_{y}\right)^{2}+\left(\mbox{Re}E_{z}\right)^{2}\right]=\frac{c}{4\pi}E_{*}^{2}\left(\mbox{Re}D\right)^{2}.$
(46)

The peak period-averaged power is

$P_{rad}=\left\langle\int_{0}^{2\pi}d\phi\int_{0}^{\infty}S_{x}rdr\right\rangle=\frac{cw_{0}^{2}}{4\pi}\int_{0}^{2\pi}d\phi\int_{0}^{\infty}\left\langle\left(\mbox{Re}D\right)^{2}\right\rangle\widetilde{r}d\widetilde{r}$
(47)

with $g(\xi)=1$. After time-averaging, the expression on the right-hand side
is identical to the expression on the right-hand side of Eq. 15. Thus the peak
period-averaged power of the radially polarized beam is

$P_{rad}=\frac{cw_{0}^{2}}{8\pi}E_{0}^{2}.$ (48)

Not that it is equal to $P_{lin}$, the power of a linearly polarized beam with
$E_{0}=E_{*}$, given by Eq. 16. On the other hand, the peak period-averaged
power of a circularly polarized beam, $P_{CP}$, with $E_{0}=E_{*}$, $l=-1$,
$\sigma=1$, and $p=0$ is $2P_{lin}$, where $P_{lin}$ is given by Eq. 16. We
thus have

$P_{CP}=\frac{cw_{0}^{2}}{4\pi}E_{0}^{2}=2P_{rad}.$ (49)

If both types of beams can be generated at the same power, then a potential
advantage of using the radially polarized beam would be its ability to
generate higher $E_{\parallel}$. However, very different optical techniques
are employed to generate the two types of beams. The radially polarized beams
are produced using transmissive optical elements, which limits the incident
power. In contrast to that, the beams with twisted wavefronts can be produced
by adding an etched mirror [25, 13] at some point into a traditional laser-
system. This method relies of laser reflection, so it does not have the same
power limitation. This means that it can potentially be used to generate the
desired beams at very high power and reach very strong $E_{\parallel}$.

To compare the acceleration by the two beams with the same strength of
$E_{\parallel}$, we performed an additional simulation for a 300 TW radially
polarized beam. With the exception of the laser power, all other simulation
parameters are the same as those listed in Table 1 and used to obtain the
results presented in Section 4. The electric field structure in the
$(x,z)$-plane at $t=-9$ fs is shown in Fig. 9. The longitudinal electric field
structure in the region close to the axis is nearly identical to that shown in
Fig. 4 for the circularly-polarized Laguerre-Gaussian beam. There is no strong
$B_{\parallel}$ near the axis, which agrees with the analytical analysis given
by Eq. 44 and Eq. 45.

To obtain a comparable picture of the phenomena in the new simulation, we
again focus on the third bunch formed during the reflection process (see Fig 6
for the location of this in the circularly-polarized Laguerre-Gaussian case).
The evolution of the energy spectrum of this bunch is shown for the radial
polarized case in Fig. 10(a) whereas the energy-angle distribution is shown in
Fig. 10(b). These plots can be compared to similar plots in Fig. 5. In the
radially polarized case, we see a slightly lower peak kinetic energy at later
times. In addition to this, the bunch has a wider energy spectrum. When
looking at the energy-angle distribution, the bunch appears to be spread over
a slightly wider divergence angle. The spectra in the radially-polarized case
do not show the highly narrow energy spread features present in the
circularly-polarized case.

Figure 10: Sensitivity of the electron acceleration by a 300 TW radially
polarized beam to simulation parameters. The top row [subplots (a) and (b)]
shows results for the parameters from Table 1. The bottom row [subplots (c)
and (d)] shows results from a simulation with a reduced cell size of 20 nm and
a higher-order field solver. The black dashed curve is prediction of Eq. 36
for $\Phi_{0}=0.8\pi$. The left column is the energy distribution as a
function of time for the third bunch of electrons. The right column is energy-
angle distribution at $t=261$ fs.

Previous studies [30] have shown that the acceleration by radially polarized
beams can be affected by the high-harmonic radiation emitted during the
reflection. To test for this sensitivity, we performed two more simulations,
one for each beam scenario. In these simulations, the cell size is reduced
from 25 nm to 20 nm and the default second order Yee scheme is changed to the
fourth order version. The results for the circularly-polarized Laguerre-
Gaussian beam are unchanged, which is unsurprising given the extensive
resolution tests previously run [28, 33]. The new results for the radially-
polarized case are visibly different compared to those from the original
simulation: there are differences in the maximum energy, energy spread, and
the energy-angle distribution. There is also some evidence, which is not
plotted here, of a second harmonic forming on reflection of the beam in the
radially polarized case. No such high-harmonic generation is evident in the
circularly-polarized Laguerre-Gaussian case. The studies focused on radially
polarized beams typically use the field solvers specifically designed for
accurate propagation of high-harmonic generation [9, 30, 34]. While the
results of Fig. 10 primarily show the sensitivity of the radially polarized
mechanism to resolving the high-harmonic generation, there is experimental
evidence [30] that high-harmonics do have a significant impact when reflecting
this type of beam at an angle away from normal off a plasma mirror.

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