# Electron energy gain due to a laser frequency modulation
experienced by electron during betatron motion

A. Arefiev<EMAIL_ADDRESS>Department of Mechanical and Aerospace
Engineering, University of California San Diego, La Jolla, CA 92093  I-L. Yeh
Department of Physics, University of California San Diego, La Jolla, CA 92093
L. Willingale  Center for Ultrafast Optical Science, University of Michigan,
Ann Arbor, Michigan 48109

###### Abstract

Direct laser acceleration of electrons is an important energy deposition
mechanism for laser-irradiated plasmas that is particularly effective at
relativistic laser intensities in the presence of quasi-static laser-driven
plasma electric and magnetic fields. These radial electric and azimuthal
magnetic fields provide transverse electron confinement by inducing betatron
oscillations of forward-moving electrons undergoing laser acceleration.
Electrons are said to experience a betatron resonance when the frequency of
betatron oscillations matches the average frequency of the laser field
oscillations at the electron position. In this paper, we show that the
modulation of the laser frequency caused by the betatron oscillation can be
another important mechanism for net energy gain that is qualitatively
different from the betatron resonance. Specifically, we show that the
frequency modulation experienced by the electron can lead to net energy gain
in the regime where the laser field performs three oscillations per betatron
oscillation. There is no net energy gain in this regime without the modulation
because the energy gain is fully compensated by the energy loss. The
modulation slows down the laser oscillation near transverse stopping points,
increasing the time interval during which the electron gains energy and making
it possible to achieve net energy gain.

##  I Introduction

In laser-plasma interactions, direct laser acceleration (DLA) of electrons is
a fundamental mechanism for transferring energy from the laser pulse to the
plasma. It is particularly important at relativistic laser intensities,
because then the laser is able to generate a forward-moving population of
relativistic electrons. These electrons are essential for the development of
compact laser-driven gamma-ray Stark, Toncian, and Arefiev [2016]; Wang  _et
al._ [2020]; Hadjisolomou  _et al._ [2023] and particle sources (e.g. sources
of ions Wilks  _et al._ [2001]; Dover  _et al._ [2020]; Gong  _et al._ [2022],
positrons Chen  _et al._ [2009, 2010], or neutrons Lancaster  _et al._ [2004];
Pomerantz  _et al._ [2014]; Günther  _et al._ [2022]).

Electron dynamics during DLA in a laser-irradiated plasma has been extensively
studied using experiments Gahn  _et al._ [1999]; Willingale  _et al._ [2013];
Peebles  _et al._ [2017]; Hussein  _et al._ [2021]; Rinderknecht  _et al._
[2021], simulations Pukhov, Sheng, and Meyer-ter Vehn [1999]; Robinson,
Arefiev, and Neely [2013]; Kemp and Wilks [2020]; Jirka  _et al._ [2020], and
analytical theory Arefiev  _et al._ [2012, 2016a]; Khudik  _et al._ [2016];
Huang  _et al._ [2017]; Gong  _et al._ [2020]; Li  _et al._ [2021] with the
goal of determining the key factors influencing electron energy gain. There
are several aspects that make electron acceleration inside a plasma
qualitatively different from electron acceleration in vacuum, which include
laser guiding/focusing, superluminal rather than luminal phase velocity, and
presence of quasi-static plasma fields. Radial plasma electric and azimuthal
plasma magnetic fields that naturally arise during laser propagation through a
plasma have attracted particularly attention due to their ability to enhance
electron energy gain from the laser. The fields provide transverse electron
confinement by deflecting outward moving electrons towards the axis of the
laser beam. If the deflections occur on the same time-scale as the
oscillations of the laser electric field, then the transverse electron
velocity $v_{\perp}$ can remain anti-parallel to the transverse laser electric
field $E_{\perp}$ over multiple laser field oscillations. As a result, the
electron energy gain can be increased. The extended anti-parallel alignment of
$v_{\perp}$ and $E_{\perp}$ is often referred to as a betatron resonance,
where the term ‘betatron’ refers to the transverse or betatron oscillations
induced by plasma fields.

The betatron resonance is not the only option for increasing electron energy
gain from the transverse laser electric field. During the betatron resonance,
the laser electric field performs one oscillation during one betatron
oscillation. It was shown in Ref. [Khudik  _et al._ , 2016] using analytical
theory that net energy gain is also possible in the case of three laser
oscillations. This phenomenon was named the third-harmonic resonance in Ref.
[Khudik  _et al._ , 2016]. The term itself and the emphasis on the similarity
between the third-harmonic resonance and the betatron resonance can perhaps
obscure a fundamental difference between the two regimes.

In this paper, we show that the modulation of the laser frequency induced by
the betatron motion is the underlying mechanism responsible for net energy
gain in the regime where the laser electric field performs three oscillation
during one betatron oscillation. In fact, there is no net energy gain in this
regime in the absence of the frequency modulation. We employ a test electron
model with prescribed laser and plasma fields to elucidate the electron
dynamics. We show that the frequency modulation experienced by the electron
stretches energy gain regions around stopping points (in the transverse
direction) and thus breaks the compensation that occurs without the
modulation. In contrast to that, the betatron resonance leads to net energy
gain with or without the frequency modulation experienced by the electron. We
also show that the period of the frequency modulation experienced by the
electron is the reason why no net energy gain occurs for an even number of
laser field oscillations.

The rest of the paper consists of seven sections. In Section II, we introduce
our test-electron model that we then use throughout the paper. In Section III
and IV, we provide derivations of the betatron frequency and the frequency of
laser field oscillations at the electron location. In Sections V, VI, and VII
we analyze three different regimes with one, three, and two laser field
oscillations during a single betatron oscillation. In Section VIII, we
summarize our key findings.

##  II Test-electron model

Electron dynamics during direct laser acceleration can be examined using a
test-electron model with prescribed laser and plasma fields. The purpose of
this section is to provide a fully self-contained description of the model, as
this model is then used to gain insights into the energy gain process. Similar
models have been employed in multiple publications on direct laser
acceleration Arefiev, Khudik, and Schollmeier [2014]; Khudik  _et al._ [2016];
Jirka  _et al._ [2020].

Fully self-consistent particle-in-cell (PIC) simulations Pukhov and Meyer-ter
Vehn [1996]; Jansen  _et al._ [2018]; Gong  _et al._ [2020]; Wang, Gong, and
Arefiev [2020]; Jirka  _et al._ [2020]; Hussein  _et al._ [2021] show that a
sufficiently long laser pulse propagating through a plasma generates quasi-
static electric and magnetic fields. There is a quasi-static radial electric
field caused by expulsion of some electrons out of the laser beam. There is
also an azimuthal magnetic field Stark, Toncian, and Arefiev [2016]; Jansen
_et al._ [2018] sustained by a longitudinal electron current that is generated
by the laser during its propagation. In our model, we treat both fields as
static. We assume that the azimuthal magnetic field $\bm{B}_{\rm{stat}}$ is
generated by a current with uniform current density $j_{x}=-|j_{0}|$ and that
the radial electric field $\bm{E}_{\rm{stat}}$ is generated by uniform charge
density $\rho_{ch}>0$.

We approximate the laser by a plane monochromatic electromagnetic wave with a
phase velocity $v_{ph}$ and frequency $\omega_{0}$. Under this approximation,
we neglect the longitudinal electric field component of the laser. We consider
a linearly polarized laser propagating in the positive direction along the
$x$-axis, such that

$\displaystyle\bm{E}_{\rm{plane}}=\bm{e}_{y}E_{0}\cos(\xi+\xi_{0}),$ (1)
$\displaystyle\bm{B}_{\rm{plane}}=\bm{e}_{z}E_{\rm{plane}}/u,$ (2)

where $E_{0}$ is the laser amplitude, $c$ is the speed of light,

$u\equiv v_{ph}/c$ (3)

is the normalized phase velocity,

$\xi=\omega_{0}t-\omega_{0}x/v_{ph}=\omega_{0}t-\omega_{0}x/uc$ (4)

is the laser phase, and $\xi_{0}$ is a phase offset. In general, the laser
amplitude $E_{0}$ varies with $\xi$. We neglect this dependence, so our
results are applicable to those regimes where the variation of $E_{0}$ at
electron location is insignificant on the time-scale of the electron energy
gain. We introduce a normalized laser amplitude

$a_{0}=|e|E_{0}/m_{e}c\omega_{0},$ (5)

where $m_{e}$ is the electron mass. We also introduce the vacuum laser
wavelength $\lambda_{0}=2\pi c/\omega_{0}$ that we use for normalization.

In this study, we limit our analysis to flat particle trajectories in the
$(x,y)$-plane ($z=0$ and $p_{z}=0$). Non-planar electron trajectories can
become unstable Arefiev  _et al._ [2016b], so they are not considered in this
work. We therefore only need the knowledge of $\bm{B}_{\rm{stat}}$ and
$\bm{E}_{\rm{stat}}$ in the $(x,y)$-plane. Under our assumptions, the radial
electric field in the $(x,y)$-plane is

$\bm{E}_{\rm{stat}}=2\kappa\frac{y}{\lambda_{0}^{2}}\frac{m_{e}c^{2}}{|e|}\bm{e}_{y}$
(6)

where

$\kappa\equiv\pi^{2}\omega_{ch}^{2}/\omega_{0}^{2}$ (7)

and

$\omega_{ch}^{2}\equiv 4\pi|e|\rho_{ch}/m_{e}$ (8)

is a plasma frequency calculated using $\rho_{ch}$. The plasma magnetic field
is

$\bm{B}_{\rm{stat}}=\frac{m_{e}c^{2}}{|e|}\left[\nabla\times\bm{a}_{\rm{stat}}\right],$
(9)

where

$\bm{a}_{\rm{stat}}=\bm{e}_{x}\alpha(y^{2}+z^{2})/\lambda_{0}^{2}.$ (10)

The dimensionless parameter

$\alpha\equiv\pi\lambda_{0}^{2}|j_{0}|/J_{A}.$ (11)

represents the ratio of the current flowing through a circular area with
radius $\lambda_{0}$ to the classical Alfvén current $J_{A}=m_{e}c^{3}/|e|$.
The field in the $(x,y)$-plane is

$\bm{B}_{\rm{stat}}=-\frac{m_{e}c^{2}}{|e|}\frac{2\alpha
y}{\lambda_{0}^{2}}\bm{e}_{z}.$ (12)

The electron dynamics is then described by the following system of coupled
differential equations:

$\displaystyle\frac{d\bm{p}}{dt}=-|e|\bm{E}-\frac{|e|}{\gamma
m_{e}c}\left[\bm{p}\times\bm{B}\right],$ (13)
$\displaystyle\frac{d\bm{r}}{dt}=\frac{c}{\gamma}\frac{\bm{p}}{m_{e}c},$ (14)

where $\bm{E}=\bm{E}_{\rm{plane}}+\bm{E}_{\rm{stat}}$,
$\bm{B}=\bm{B}_{\rm{plane}}+\bm{B}_{\rm{stat}}$, and

$\gamma=\sqrt{1+\bm{p}^{2}/m_{e}^{2}c^{2}}$ (15)

is the relativistic factor of the electron. We assume that $a_{0}$ is
sufficiently low to neglect the effect of radiation reaction Gong  _et al._
[2019]; Yeh  _et al._ [2021]; Jirka  _et al._ [2020]. Our test-electron model
consists of Eqs. (13) and (14) where the fields are the fields of the laser
given by Eqs. (1) and (2) and the fields of the plasma given by Eqs. (6) and
(12). In addition to initial conditions, the model requires four input
parameters: $a_{0}$, $\kappa$, $\alpha$, and $u$.

##  III Betatron oscillations

The plasma electric and magnetic fields are responsible for transverse
electron confinement. An electron with a transverse momentum stays near the
axis while performing transverse oscillations that we will refer to as
betatron oscillations. The oscillations typically take place in conjunction
with longitudinal electron motion in the direction of laser propagation.

We start by deriving a conservation law for a general case where the electron
experiences the fields of the plasma and the fields of the laser. The work
done on the electron per unit time is a scalar product of Eq. (13) with
$\bm{v}=\bm{p}/\gamma m_{e}$. We take into account the specific dependence of
$\bm{E}_{\rm{stat}}$ on $y$ given by Eq. (6) to find from the resulting
equation that

$\frac{d}{dt}\left(\gamma+\kappa\frac{y^{2}}{\lambda_{0}^{2}}\right)=-\frac{|e|E_{\rm{plane}}}{m_{e}c}\frac{p_{y}}{\gamma
m_{e}c}.$ (16)

Equation 16 quantifies the energy exchange between the electron and the
electric fields of the laser and plasma. These fields change the transverse
momentum $p_{y}$, whereas the magnetic fields redistribute the energy between
the transverse and longitudinal motion. It is convenient to re-write the
longitudinal component of Eq. (13) using the expression for
$\bm{B}_{\rm{stat}}$ given by Eq. (12) as

$\frac{d}{dt}\left(-\frac{p_{x}}{m_{e}c}+\alpha\frac{y^{2}}{\lambda_{0}^{2}}\right)=\frac{|e|B_{\rm{plane}}}{m_{e}c}\frac{p_{y}}{\gamma
m_{e}c}.$ (17)

This equation quantifies changes in $p_{x}$ due to the electron motion in the
magnetic fields of the plasma and laser. We multiply both sides of Eq. 17 by
$u$ and add the resulting equation to Eq. 16. After taking into account that
$B_{\rm{plane}}=E_{\rm{plane}}/u$, the terms on the right-hand side cancel
each other out and we obtain

$\frac{d}{dt}\left(\gamma-u\frac{p_{x}}{m_{e}c}+\left[u\alpha+\kappa\right]\frac{y^{2}}{\lambda_{0}^{2}}\right)=0.$
(18)

The derived equation shows that

$S=\gamma-u\frac{p_{x}}{m_{e}c}+\left[u\alpha+\kappa\right]\frac{y^{2}}{\lambda_{0}^{2}}$
(19)

is conserved during electron betatron oscillations. This is a well-known
result in DLA research.

It might appear odd that the derived conservation law involves $u$ without
involving the laser fields. Indeed, in the limit of $E_{\rm{plane}}\rightarrow
0$ and $B_{\rm{plane}}\rightarrow 0$, $S$ still involves the laser phase
velocity even though there is no longer any laser field acting on the
electron. In order to understand what is happening at
$E_{\rm{plane}}\rightarrow 0$ and $B_{\rm{plane}}\rightarrow 0$, we need to go
back to Eqs. (16) and (17). In this limit, these two equations turn into

$\frac{d}{dt}\left(\gamma+\kappa\frac{y^{2}}{\lambda_{0}^{2}}\right)=0$ (20)

and

$\frac{d}{dt}\left(-\frac{p_{x}}{m_{e}c}+\alpha\frac{y^{2}}{\lambda_{0}^{2}}\right)=0,$
(21)

which means that, in the absence of the laser, we have two conservation laws
rather than one. At $E_{\rm{plane}}=0$ and $B_{\rm{plane}}=0$, Eq. (18) is
simply a sum of these two equations without providing any additional
information, with the terms that involve $u$ canceling each other and the
terms without $u$ canceling each other as well.

Figure 1:  Betatron oscillation of an ultra-relativistic electron moving to
the right along the $x$-axis.

We now consider an ultra-relativistic electron that performs betatron
oscillations in the absence of the laser, as shown in Fig. 1. We focus on a
scenario where the electron crosses the $x$-axis at a small angle
$|\theta_{0}|\ll 1$. To find the amplitude of betatron oscillations $y_{*}$,
we note that the maximum transverse displacement is achieved at a turning
point where $p_{y}=0$. Equations (20) and (21) at $y=y_{*}$ give a system of
equations that determines $y_{*}$ in terms of electron parameters at $y=0$:

$\displaystyle\gamma+\kappa\frac{y_{*}^{2}}{\lambda_{0}^{2}}=\gamma_{0},$ (22)
$\displaystyle-\frac{p_{x}}{m_{e}c}+\alpha\frac{y_{*}^{2}}{\lambda_{0}^{2}}=-\frac{p_{0}\cos\theta_{0}}{m_{e}c},$
(23)

where $\gamma_{0}$ and $p_{0}$ are the electron $\gamma$-factor and the total
momentum at $y=0$. This system has a simple approximate solution for
$|\theta_{0}|\gg 1/\gamma_{0}$. We add the two equations, which yields

$\left(\kappa+\alpha\right)\frac{y_{*}^{2}}{\lambda_{0}^{2}}=\left(\gamma_{0}-\frac{p_{0}\cos\theta_{0}}{m_{e}c}\right)-\left[\gamma-\frac{p_{x}}{m_{e}c}\right].$
(24)

The expression in the round brackets on the right hand side of Eq. 24 can be
approximated by

$\displaystyle\gamma_{0}-\frac{p_{0}\cos\theta_{0}}{m_{e}c}$ $\displaystyle=$
$\displaystyle\gamma_{0}-\frac{p_{0}}{m_{e}c}+\frac{p_{0}}{m_{e}c}\left(1-\cos\theta_{0}\right)$
(25) $\displaystyle\approx$
$\displaystyle\frac{1}{2\gamma_{0}}+\gamma_{0}\frac{\theta_{0}^{2}}{2}\approx\gamma_{0}\theta_{0}^{2}/2,$

where it is taken into account that the electron is relativistic and that
$\theta_{0}\gg 1/\gamma_{0}$. The expression inside the square brackets in Eq.
24 can be roughly estimated as $1/2\gamma_{0}$, where we disregard the change
in $\gamma$. This quantity is much smaller than $\gamma_{0}\theta_{0}^{2}/2$.
We thus neglect the terms inside the square brackets in Eq. 24 and obtain the
following expression for the amplitude of betatron oscillations:

$\frac{y_{*}}{\lambda_{0}}\approx\left(\frac{\gamma_{0}\theta_{0}^{2}}{2(\kappa+\alpha)}\right)^{1/2}.$
(26)

The change in electron kinetic energy during a betatron oscillation is
relatively small for $|\theta_{0}|\ll 1$. We find directly from Eq. 20 that
the maximum change in $\gamma$ is given by

$\Delta\gamma=|\gamma-\gamma_{0}|=\kappa y_{*}^{2}/\lambda_{0}^{2}.$ (27)

Using the expression for $y_{*}$ given by Eq. 26, we find that

$\frac{\Delta\gamma}{\gamma_{0}}=\frac{\kappa\theta_{0}^{2}}{2(\kappa+\alpha)}<\theta_{0}^{2}.$
(28)

For $|\theta_{0}|\ll 1$, we have $\Delta\gamma\ll\gamma_{0}$, so that the
relative energy change is indeed small. The underlying reason for the small
change can be understood by recognizing that it is only the transverse plasma
electric field that can reduce the electron $\gamma$. An electron with
$|\theta_{0}|\ll 1$ has most of the energy associated with the longitudinal
rather than transverse motion when it crosses the axis. It therefore does not
have a lot of energy that it can give up when it starts to lose $p_{y}$ during
its outward motion.

We conclude this section by deriving an expression for the frequency of
betatron oscillations. We describe betatron oscillations using a new variable
$\psi$ that we call the betatron phase, with

$y=y_{*}\sin\psi.$ (29)

We define the frequency of betatron oscillations as

$\omega_{\beta}\equiv d\psi/dt.$ (30)

To obtain an equation for $\omega_{\beta}$, we start by adding Eq. 20 to Eq.
21. After integrating the resulting equation, we obtain

$\gamma-\frac{p_{x}}{m_{e}c}+(\kappa+\alpha)\frac{y^{2}}{\lambda_{0}^{2}}=\gamma_{0}-\frac{p_{0}\cos\theta_{0}}{m_{e}c}.$
(31)

It follows from the definition of $\gamma$ given by Eq. 15 that

$\gamma-\frac{p_{x}}{m_{e}c}\approx\frac{p_{y}^{2}}{2p_{x}m_{e}c}\approx\frac{\gamma}{2}\frac{v_{y}^{2}}{c^{2}},$
(32)

where we took into account that $v_{y}/c=p_{y}/\gamma m_{e}c$ and then
replaced $p_{x}$ with $\gamma m_{e}c$. We differentiate Eq. 29 with respect to
$t$ to find that

$v_{y}=y_{*}\frac{d\psi}{dt}\cos\psi=y_{*}\omega_{\beta}\cos\psi.$ (33)

As a result, we have

$\gamma-\frac{p_{x}}{m_{e}c}\approx\frac{\gamma}{2}\frac{y_{*}^{2}\omega_{\beta}^{2}}{c^{2}}\cos^{2}\psi.$
(34)

The next step is to substitute this expression and the expression for $y$
given by Eq. 29 into Eq. 31, which yields:

$\frac{\gamma}{2}\frac{y_{*}^{2}\omega_{\beta}^{2}}{c^{2}}\cos^{2}\psi+(\kappa+\alpha)\frac{y_{*}^{2}}{\lambda_{0}^{2}}\sin^{2}\psi=\gamma_{0}-\frac{p_{0}\cos\theta_{0}}{m_{e}c}.$
(35)

After taking into account that the right-hand side is approximately given by
Eq. (25) and using the expression for $y_{*}$ given by Eq. 26, we obtain

$\left(\frac{\gamma}{2}\frac{\omega_{\beta}^{2}}{c^{2}}\frac{\lambda_{0}^{2}}{\kappa+\alpha}\right)\cos^{2}\psi+\sin^{2}\psi=1.$
(36)

Equation 36 must be satisfied for any $\psi$, which requires the combination
inside the brackets to be equal to unity. This condition provides an
expression for the frequency of betatron oscillations:

$\frac{\omega_{\beta}}{\omega_{0}}=\frac{1}{\sqrt{2}\pi}\left[\frac{\kappa+\alpha}{\gamma}\right]^{1/2},$
(37)

where it is taken into account that $\lambda_{0}=2\pi c/\omega_{0}$.

##  IV Frequency of laser field oscillations experienced by electron

In the presence of a laser, the forward moving electron we are considering
sees the laser fields oscillate with frequency

$\omega^{\prime}\equiv d\xi/dt,$ (38)

where $\xi$ is the laser phase at the electron location. This frequency is
much lower than $\omega_{0}$ because of the forward relativistic motion. More
importantly, the betatron oscillations strongly modulate $\omega^{\prime}$ by
changing the longitudinal velocity of the electron.

Our goal in this section is to derive an expression for $\omega^{\prime}$ in
terms of the betatron phase $\psi$. This requires re-deriving the expression
for the amplitude of betatron oscillations $y_{*}$ while taking into account
that the electron can exchange energy with the laser. We again consider a
forward-moving electron with $p_{x}\gg|p_{y}|\gg m_{e}c$. We focus on a regime
where the superluminosity is weak ($\delta u\ll 1$), treating $\delta u$ as a
small parameter. Our derivation leverages the fact that the quantity $S$ given
by Eq. (19) remains constant during the electron motion. We substitute
$u=1+\delta u$ into the second term on the right-hand side of Eq. (19) to
obtain the following approximate expression that we will use:

$S\approx\left(\gamma-\frac{p_{x}}{m_{e}c}\right)-\gamma\delta
u+[u\alpha+\kappa]\frac{y^{2}}{\lambda_{0}^{2}}.$ (39)

Here we took into account that the difference between $\gamma$ and
$p_{x}/m_{e}c$ is relatively small. We neglect this difference in the $\delta
u$-term by replacing $p_{x}/m_{e}c$ with $\gamma$.

We use the conservation of $S$ to find the amplitude of betatron oscillations.
We consider a single betatron oscillation and write down the expression for
$S$ at a turning point, $y=y_{*}$. We neglect the contribution from the
expression in the round brackets in Eq. (39) to obtain

$S\approx-\gamma\delta u+[u\alpha+\kappa]\frac{y_{*}^{2}}{\lambda_{0}^{2}}.$
(40)

As discussed in Sec. III, the contribution from the neglected terms is
relatively small compared to the $y_{*}^{2}$-term if $\theta_{0}\gg
1/\gamma_{0}$, where $\gamma_{0}$ and $\theta_{0}$ are the electron
relativistic factor and the angle between $\bm{p}$ and the $x$-axis at $y=0$.
We find directly from Eq. (40) that

$\frac{y_{*}}{\lambda_{0}}\approx\left[\frac{S+\gamma\delta
u}{\kappa+u\alpha}\right]^{1/2}\approx\left[\frac{S+\gamma\delta
u}{\kappa+\alpha}\right]^{1/2},$ (41)

where we set $u\approx 1$ in the denominator due to the smallness of $\delta
u$. The derived expression is particularly meaningful when the change in
$\gamma$ is relatively small over one betatron oscillation. The convenience of
Eq. (41) is that it enables us to track the evolution of $y_{*}$ over many
betatron oscillations as the electron energy changes due to the interaction
with the laser. In the case of $\delta u=0$, the amplitude of betatron
oscillations remains unchanged. In the superluminal case ($\delta u>0$), the
behavior of $y_{*}$ is qualitatively different. The amplitude starts to grow
with the increase of electron energy once the relativistic factor reaches the
level at which $\gamma\gg S/\delta u$.

It is worth pointing out that, over a single betatron oscillation, Eq. (41) is
approximately the same as Eq. (26) for small $\delta u$. We first make a
comparison for a case without the laser. We take into account that

$\displaystyle S$ $\displaystyle=$
$\displaystyle\gamma_{0}-u\frac{p_{0}}{m_{e}c}\cos\theta_{0}\approx\frac{p_{0}}{m_{e}c}-u\frac{p_{0}}{m_{e}c}\left(1-\frac{\theta_{0}^{2}}{2}\right)$
(42) $\displaystyle\approx$ $\displaystyle-\delta
u\gamma_{0}+\gamma_{0}\theta_{0}^{2}/2$

and find that

$S+\gamma\delta u\approx\gamma_{0}\theta_{0}^{2}/2+(\gamma-\gamma_{0})\delta
u.$ (43)

In the absence of the laser, $|\gamma-\gamma_{0}|\leq\Delta\gamma$, where
$\Delta\gamma$ is the quantity given by Eq. 28. The amplitude of the last term
on the right-hand side can then be estimated as $\delta
u\gamma_{0}\theta_{0}^{2}$, which shows that this term is negligible compared
to the first term on the right-hand side. After neglecting the last term, Eq.
41 becomes identical to Eq. 26. In the presence of the laser, the same
argument holds if the second term remains negligible.

We are interested in a regime where the accelerating electron is able to
continue gaining energy from the laser over many betatron oscillations. In
this case, the change in $\gamma$ and thus the change in $y_{*}$ in a single
betatron oscillation are relatively small. We can then again describe betatron
oscillations as

$y\approx y_{*}\sin\psi,$ (44)

where $y_{*}$ is assumed to be independent of $\psi$.

To derive an expression for $\omega^{\prime}$, we start with the definition
given by Eq. (38) that we re-write using the expressions for $\xi$ and $v_{x}$
given by Eqs. (4) and (14):

$\displaystyle\omega^{\prime}$ $\displaystyle=$
$\displaystyle\omega_{0}\left(1-\frac{v_{x}}{v_{ph}}\right)=\frac{\omega_{0}}{\gamma}\left(\gamma-\frac{1}{u}\frac{p_{x}}{m_{e}c}\right).$
(45)

After substituting $u=1+\delta u$ and retaining only the linear $\delta
u$-term, we obtain

$\displaystyle\omega^{\prime}$ $\displaystyle\approx$
$\displaystyle\frac{\omega_{0}}{\gamma}\left(\gamma-\frac{p_{x}}{m_{e}c}+\gamma\delta
u\right).$ (46)

In the $\delta u$-term we replaced $p_{x}/m_{e}c$ with $\gamma$ neglecting the
small difference between the two. The next step is to obtain an expression for
$\gamma-p_{x}/m_{e}c$ from Eq. (39). We substitute $y=y_{*}\sin\psi$ and use
the expression for $y_{*}$ given by Eq. 41 to find that

$\gamma-\frac{p_{x}}{m_{e}c}=\left(S+\gamma\delta u\right)\cos^{2}\psi.$ (47)

After substituting this expression into Eq. (46), we obtain an expression for
$\omega^{\prime}$ in terms of the electron $\gamma$-factor and the betatron
phase $\psi$:

$\frac{\omega^{\prime}}{\omega_{0}}=\frac{1}{\gamma}\left[(S+\gamma\delta
u)\cos^{2}\psi+\gamma\delta u\right].$ (48)

During a single betatron oscillation ($-\pi\leq\psi<\pi$), $\omega^{\prime}$
performs two oscillations. It has two maxima that correspond to $y=0$: one at
$\psi=-\pi$ and one at $\psi=0$. It also has two minima that correspond to
$v_{y}=0$: one at $\psi=-\pi/2$ and one at $\psi=\pi/2$. The modulation of
$\omega^{\prime}$ is a direct consequence of the fact that
$\omega^{\prime}\propto 1-v_{x}/v_{ph}$ [see Eq. (45)]. In the course of a
betatron oscillation, the velocity amplitude $v$ remains almost unchanged,
whereas the velocity vector $\bm{v}$ experiences an oscillation with respect
to the $x$-axis during which $v_{y}$ changes its sign. The longitudinal
velocity $v_{x}$ remains positive and this is why $v_{x}$ performs two
oscillations during a single betatron oscillation. In order to make the
discussed dependence more apparent, we re-write Eq. 48 as

$\frac{\omega^{\prime}}{\omega_{0}}=\frac{1}{2\gamma}\left[(S+3\gamma\delta
u)+(S+\gamma\delta u)\cos(2\psi)\right].$ (49)

To conclude this section, we show that the expression for $\omega_{\beta}$
given by Eq. 37 is still valid in the presence of the laser. In this section,
we used the conservation of $S$ to derive the expression for $\gamma-
p_{x}/m_{e}c$ given by Eq. (47). This expression should agree with the
expression given by Eq. 34 that was derived using the relation
$y=y_{*}\sin\psi$. The two expressions are consistent if

$\omega_{\beta}=\left[\frac{2}{\gamma}(S+\delta
u\gamma)\right]^{1/2}\frac{c}{y_{*}}.$ (50)

To obtain the final expression, we substitute the definition of $y_{*}$ given
by Eq. 41 into Eq. 50 and express $\lambda_{0}$ in terms of $\omega_{0}$. The
result reads

$\frac{\omega_{\beta}}{\omega_{0}}=\frac{1}{\sqrt{2}\pi}\left[\frac{\kappa+\alpha}{\gamma}\right]^{1/2}.$
(51)

This result matches the betatron frequency derived in the absence of the
laser.

##  V Betatron resonance

In the test-electron model of Section II, the electron gains kinetic energy
only from the transverse electric field that consists of the static plasma
field and the oscillating field of the laser. The electron continuously
exchanges energy with the static field of the plasma, but there is no net
energy gain over one betatron oscillation. The laser field is time-dependent
and, therefore, a net energy gain over one oscillation becomes possible.

We calculate the energy gain over one betatron oscillation by integrating Eq.
16 over $t$. We first express $p_{y}$ in terms of $v_{y}$ using the relation
$p_{y}/\gamma m_{e}=v_{y}$. We then use the expression for $E_{\rm{plane}}$
given by Eq. (1), the expression for $v_{y}$ given by Eq. 33, and the
definition of $a_{0}$ given by Eq. 5 to re-write the right-hand side of Eq. 16
as

$-\frac{|e|E_{\rm{plane}}}{m_{e}c}\frac{p_{y}}{\gamma
m_{e}c}=-a_{0}\omega_{0}\cos(\xi+\xi_{0})\frac{y_{*}\omega_{\beta}}{c}\cos(\psi).$
(52)

We assume that the relative energy gain over one betatron oscillation is
small, which allows us to treat $y_{*}$ as constant. It is convenient to re-
write the time integral as an integral over the betatron phase $\psi$ using
the definition of $\omega_{\beta}$ given by Eq. 51, which yields

$\Delta\gamma=-a_{0}\frac{y_{*}\omega_{0}}{c}\int_{-\pi}^{\pi}\cos(\xi+\xi_{0})\cos(\psi)d\psi.$
(53)

To compute the integral in Eq. 53, we need to know the wave phase $\xi$ as a
function of the betatron phase $\psi$. We find the corresponding dependence
from Eq. 38. We multiply both sides of Eq. 38 by $dt$ and integrate the
resulting equation. After replacing the time integration with an integral over
$\psi$, we obtain

$\xi=\int_{0}^{\psi}\frac{\omega^{\prime}}{\omega_{\beta}}d\psi.$ (54)

Without any loss of generality, we set $\xi=0$ at $\psi=0$.

In contrast to the betatron frequency, the frequency of the laser oscillations
perceived by the electron $(\omega^{\prime})$ is modulated. Its average over
one betatron oscillation is

$\langle\omega^{\prime}\rangle=\frac{1}{2\pi}\int_{-\pi}^{\pi}\omega^{\prime}d\psi.$
(55)

Using the expression given by Eq. 49, we find that

$\frac{\langle\omega^{\prime}\rangle}{\omega_{0}}=\frac{1}{2\gamma}\left[S+3\gamma\delta
u\right].$ (56)

We now re-write the expression for $\xi$ as

$\xi=\frac{\langle\omega^{\prime}\rangle}{\omega_{\beta}}\psi+\int_{0}^{\psi}\frac{\omega^{\prime}-\langle\omega^{\prime}\rangle}{\omega_{\beta}}d\psi.$
(57)

We substitute $\omega^{\prime}$ from Eq. 49 and perform the integration to
obtain

$\xi=\frac{\langle\omega^{\prime}\rangle}{\omega_{\beta}}\psi+\frac{\omega_{0}}{\omega_{\beta}}\frac{S+\gamma\delta
u}{4\gamma}\sin(2\psi).$ (58)

The second term on the right-hand side results from the modulation of
$\omega^{\prime}$ caused by the betatron oscillations.

Figure 2: Betatron resonance with
$\langle\omega^{\prime}\rangle=\omega_{\beta}$ and $\xi_{0}=-\pi$. The
amplitude of the frequency modulation defined by Eq. 63 is set at $C_{1}=0.3$.
The circles mark turning points in the transverse direction.

One should expect net energy gain for
$\langle\omega^{\prime}\rangle=\omega_{\beta}$, because the average frequency
of the laser oscillations experienced by the electron matches the frequency of
betatron oscillations. Indeed, if we disregard the modulation of $\xi$, we
readily find from Eq. 53 that

$\Delta\gamma=-\cos(\xi_{0})\Delta_{\max},$ (59)

where

$\Delta_{\max}=\pi a_{0}y_{*}\omega_{0}/c.$ (60)

The highest gain is achieved for $\xi_{0}=-\pi$, because then the laser field
remains anti-parallel to the electron velocity over the entire betatron
oscillation. This is the so-called betatron resonance.

The modulation of $\omega^{\prime}$ caused by the betatron oscillation is
detrimental to the energy gain during the betatron resonance. To find the
energy gain without neglecting the modulation, we first re-write Eq. 58 for a
specific case of $\langle\omega^{\prime}\rangle=\ell\omega_{\beta}$, where
$\ell$ is an integer. Since $\langle\omega^{\prime}\rangle$ is given by Eq.
56, the condition $\langle\omega^{\prime}\rangle=\ell\omega_{\beta}$ requires
a specific value of the betatron frequency:

$\frac{\omega_{\beta}}{\omega_{0}}=\frac{S+3\gamma\delta u}{2\ell\gamma}.$
(61)

Taking this expression into account, we find from Eq. 58 that

$\xi=\ell\psi+C_{\ell}\sin(2\psi),$ (62)

where

$C_{\ell}=\frac{\ell}{2}\frac{S+\gamma\delta u}{S+3\gamma\delta u}.$ (63)

We next substitute the expression for $\xi$ with $\ell$ set to unity
($\ell=1$) into Eq. 53 and use the Hansen-Bessel formula, i.e. the integral
Bessel function representation, to find that

$\Delta\gamma=\Delta_{\max}\left[J_{0}(C_{1})-J_{1}(C_{1})\right]=\Delta_{\max}\left[1-\delta_{mod}\right],$
(64)

where $J_{0}$ and $J_{1}$ are the Bessel functions of the first kind and

$\delta_{mod}=1+J_{1}(C_{1})-J_{0}(C_{1})$ (65)

is the relative reduction in the energy gain due to the modulation of
$\omega^{\prime}$. The modulation amplitude given by $C_{1}$ reduces with the
increase of $\gamma$ and $\delta u$. It has the highest value of $C_{1}=1/2$
in the luminal regime ($\delta u=0$). It reaches its lowest value of
$C_{1}=1/6$ in the limit of $\gamma\rightarrow\infty$. The corresponding range
for the relative reduction in the energy gain is $0.30\geq\delta_{mod}\geq
0.09$.

Figure 2 illustrates a betatron resonance with $C_{1}=0.3$. The upper panel
shows the electron velocity $v_{y}$ and laser field $E_{y}$ as functions of
the betatron phase $\psi$ during a single betatron oscillation. The lower
panel of Fig. 2 shows how the laser phase $\xi$ increases with $\psi$ during
the same betatron oscillation. The modulation of $\omega^{\prime}$ modulates
$d\xi/d\psi$, but the net change of the laser phase $\xi$ between the turning
points $\psi=\pm\pi/2$ (shown with circles) remains equal to $\pi$. As a
result, the laser field remains anti-parallel to $v_{y}$ during the entire
betatron oscillation despite the frequency modulation.

Figure 3: Electron energy gain for
$\langle\omega^{\prime}\rangle=\omega_{\beta}$ and $\xi_{0}=-\pi$. The
constant $C_{1}$ is defined by Eq. 63 with $\ell=1$. The circles mark the
boundaries of the possible range, $1/6\leq C_{1}\leq 1/2$.

The modulation of $\omega^{\prime}$ reduces the energy gain by inducing the
modulation of $\xi$ that is visible in the lower panel of Fig. 2. Indeed,
$E_{y}$ in the upper panel of Fig. 2 looks markedly different from
$\cos(\psi)$: the low field regions around $\psi=\pm\pi/2$ are extended and
strong field regions around $\psi=0$ and $\psi=\pm\pi$ are reduced. Note that
the betatron phase $\psi$ increases linearly with time $t$. Therefore, we
conclude that the modulation of $\xi$ reduces the time the electron spends
interacting with the strongest field while also increasing the time the
electron spends interacting with $|E_{y}|\ll E_{0}$.

To explain the origin of the modulation of $\xi$, we take a closer look at the
electron dynamics. The points with $\psi=\pm\pi/2$ are the transverse turning
points. At the turning points, $v_{y}$ goes to zero and $v_{x}$ reaches its
highest value. As a result, $d\xi/d\psi\propto d\xi/dt\propto 1-v_{x}/v_{ph}$
reaches its lowest value. We find from Eq. 62 that $d\xi/d\psi=1-2C_{1}$ at
$\psi=\pm\pi/2$. The points with $\psi=0$ and $\psi=\pm\pi$ are the points
where the electron crosses the axis. Here $|v_{y}|$ reaches its highest value
and $v_{x}$ reaches its lowest value. As a result, $d\xi/d\psi\propto
d\xi/dt\propto 1-v_{x}/v_{ph}$ reaches its highest value. We find from Eq. 62
that $d\xi/d\psi=1+2C_{1}$ at $\psi=0$ and $\psi=\pm\pi$.

Figure 3 summarizes our findings by showing the dependence of the energy gain
over a single betatron oscillation on the frequency modulation amplitude
$C_{1}$ for $\xi_{0}=-\pi$. The circles mark the boundaries of the possible
range, $1/6\leq C_{1}\leq 1/2$. According to Eq. 63, $C_{1}$ decreases with
the increase of $\gamma$, which means that electron energy gain reduces the
amplitude of the modulation. We can therefore conclude from Fig. 3 that
electron energy gain becomes more efficient (more energy per betatron
oscillation) with the increase of electron energy.

##  VI Energy gain at $\langle\omega^{\prime}\rangle=3\omega_{\beta}$ due to
experienced frequency modulation

We have already shown that an electron experiences a betatron resonance if the
laser field performs one full oscillation during a betatron oscillation. This
condition is equivalent to $\langle\omega^{\prime}\rangle=\omega_{\beta}$. If
the laser field performs two or more full oscillations, then no net energy
gain is possible in the absence of $\omega^{\prime}$ modulations. Indeed, it
follows directly from Eq. 53 that $\Delta\gamma=0$ for
$\xi=\psi\langle\omega^{\prime}\rangle/\omega_{\beta}$, where
$\langle\omega^{\prime}\rangle=\ell\omega_{\beta}$ and $\ell$ is an integer
greater than one ($\ell>1$). In what follows, we show that the
$\omega^{\prime}$ modulation induced by the betatron motion leads to net
energy gain for $\ell=3$.

Figure 4: Electron energy gain for
$\langle\omega^{\prime}\rangle=3\omega_{\beta}$ and $\xi_{0}=0$. The constant
$C_{3}$ is defined by Eq. 63 with $\ell=3$.

To find the energy gain due to the frequency modulation, we substitute the
expression for $\xi$ given by Eq. 62 into Eq. 53 and set $\ell=3$. We use the
Hansen-Bessel formula, i.e. the integral Bessel function representation, to
find that, for a given phase offset $\xi_{0}$, the energy gain over one
betatron oscillation is

$\Delta\gamma=\cos(\xi_{0})\Delta_{\max}\left[J_{1}(C_{3})-J_{2}(C_{3})\right],$
(66)

where $\Delta_{\max}$ is defined by Eq. 60. According to Eq. 63, $C_{3}$
varies between 0.5 and 1.5, reaching its highest value in the luminal regime
($\delta u=0$). The expression inside the square brackets in Eq. 66 remains
positive for this range of $C_{3}$, which means that the highest energy gain
for a given value of $C_{3}$ occurs for the case with no phase offset, i.e.
$\xi_{0}=0$.

Figure 4 shows how the energy gain given by Eq. 66 depends on $C_{3}$ for
$\xi_{0}=0$. In the luminal regime ($\delta u=0$), the value of $C_{3}$ is
fixed ($C_{3}=1.5$). However, in the superluminal regime ($\delta u>0$),
$C_{3}$ decreases with the increase of $\gamma$. As a consequence, the energy
gain becomes less efficient after $C_{3}$ exceeds 1.25 due to the increase of
$\gamma$ ($C_{3}\approx 1.25$ is the location of the maximum). The energy gain
per betatron oscillation is the lowest for $\gamma\delta u\gg S$, because, in
this limit, $C_{3}$ approaches its lowest value of 0.5. The discussed behavior
of $\Delta\gamma$ differs from that given by Eq. 64 for $\ell=1$. In the case
of the betatron resonance, the energy gain becomes more efficient with the
increase of $\gamma$ because of the reduced frequency modulation amplitude, as
seen in Fig. 3.

Figure 5: Electron dynamics and field evolution at
$\langle\omega^{\prime}\rangle=3\omega_{\beta}$ and $\xi_{0}=0$. The amplitude
of the frequency modulation defined by Eq. 63 is set at $C_{3}=1.25$. The
circles mark turning points in the transverse direction. The upper shaded
regions show the range of $\psi$ where $v_{y}$ is parallel to $E_{y}$. The
lower shaded regions show the range of $\psi$ where $v_{y}$ is parallel to
$\cos(3\psi)$.

Figure 5 shows the electron dynamics and field evolution for $C_{3}=1.25$ and
$\xi_{0}=0$. The net change of the laser phase $\xi$ between the turning
points $\psi=\pm\pi/2$ (shown with circles) is $3\pi$. This is also the case
when there is no frequency modulation, as indicated with the dashed curve in
the upper panel of Fig. 5. In both cases, $v_{y}$ is parallel to $E_{y}$ near
$\psi=0$ and $\psi=\pm\pi$. This is the part of the trajectory where the
electron loses energy (upper shaded regions in the upper panel of Fig. 5). On
the other hand, $v_{y}$ is anti-parallel to $E_{y}$ near the turning points.
This is the part of the trajectory where the electron gains energy (these
regions are not shaded in the upper part of the upper panel of Fig. 5). In the
absence of the modulation, the gain is fully compensated by the energy loss.
For reference, the lower shaded regions show the range of $\psi$ where $v_{y}$
is parallel to $\cos(3\psi)$ and thus the electron loses energy in the absence
of the modulation ($\cos(3\psi)$ represents the field without the modulation
of $\omega^{\prime}$). The modulation of $\omega^{\prime}$ extends the range
of $\psi$ where $v_{y}$ is anti-parallel to $E_{y}$ and reduces the range of
$\psi$ where $v_{y}$ is parallel to $E_{y}$ (the shaded regions become
narrower), which causes net energy gain over one betatron oscillation.

The upper panel in Fig. 6 illustrates the impact of the frequency modulation
induced by the betatron motion on the evolution of the transverse laser
electric field $E_{y}$. The amplitude of the modulation is quantified by the
parameter $C_{3}$ that can vary between 0.5 and 1.5. As a reference, the
dashed curve shows the case without the modulation ($C_{3}=0$). In this case,
there is no net energy gain or, equivalently, the integral of $E_{y}v_{y}$
over $\psi$ goes to zero. The lower panel in Fig. 6 shows the variation of
$E_{y}v_{y}$ over one half of the betatron oscillation (the other half looks
identical). The filled circles indicate where $E_{y}v_{y}$ changes its sign.
The energy gain occurs between the two filled circles. It is fully compensated
by the energy loss outside of this interval.

Figure 6: Impact of the frequency modulation on the evolution of the laser
electric field for $\langle\omega^{\prime}\rangle=3\omega_{\beta}$ and
$\xi_{0}=0$. The solid curves in the upper panel show $E_{y}/E_{0}$ for three
different values of $C_{3}$. The dashed curve is the field profile without the
frequency modulation ($C_{3}=0$). The lower panel shows the normalized value
of $-E_{y}v_{y}$ versus $\psi$ for the same field profiles. The empty circle
marks a transverse turning point ($v_{y}=0$). The filled circles show where
$E_{y}v_{y}$ changes its sign.

The role of the experienced frequency modulation can be understood by tracking
the location of the filled circles in the lower panel of Fig. 6. These circles
indicate where $E_{y}v_{y}$ changes its sign. At $C_{3}=0.5$, the circles are
further apart than in the case without the modulation ($C_{3}=0$). This is
because $d\xi/d\psi$ is lowered near the turning point located at
$\psi=\pi/2$. As a result, the region with $-E_{y}v_{y}>0$ is wider and the
electron gains more energy than it loses. An increase of $C_{3}$ from 0.5 to
1.25 pushes the filled circles even further apart, extending the region with
$-E_{y}v_{y}>0$ and increasing the energy gain. Increasing $C_{3}$ above 1.25
is counterproductive, because the width of the region with $-E_{y}v_{y}>0$
increases insignificantly. At the same time, the derivative of $E_{y}/E_{0}$
near the turning point becomes noticeably smaller. This stretches the region
of low values of $E_{y}/E_{0}$ and thus reduces the energy gain.

##  VII Lack of net energy in the presence of the frequency modulation at
$\langle\omega^{\prime}\rangle=2\omega_{\beta}$

In Section VI, we showed that the frequency modulation caused by the betatron
motion leads to net energy gain for
$\langle\omega^{\prime}\rangle=3\omega_{\beta}$. In this section, we examine
the case with $\langle\omega^{\prime}\rangle=2\omega_{\beta}$ to show that net
energy gain requires for the number of laser field oscillations during a
betatron oscillation to be odd.

Figure 7: Electron dynamics and field evolution at
$\langle\omega^{\prime}\rangle=2\omega_{\beta}$ and $\xi_{0}=0$. The amplitude
of the frequency modulation defined by Eq. 63 is set at $C_{2}=0.5$. The
circles mark turning points in the transverse direction.

Figure 7 shows the evolution of the laser field $E_{y}$ and the phase variable
$\xi$ for $\langle\omega^{\prime}\rangle=2\omega_{\beta}$ and $\xi_{0}=0$. A
fundamental difference with the regimes where
$\langle\omega^{\prime}\rangle=\omega_{\beta}$ or
$\langle\omega^{\prime}\rangle=3\omega_{\beta}$ is that $E_{y}$ no longer
changes its sign at the turning points. Instead, the turning points are the
locations of the minima of $E_{y}$. This behavior remains unaltered by the
frequency modulation.

As seen in the upper panel of Fig. 8, the frequency modulation elongates the
region with a strong $E_{y}$ near the turning point (marked with a circle).
The elongation is symmetrical with respect to $\psi=\pi/2$, so that
$E_{y}(-\psi^{\prime})=E_{y}(\psi^{\prime})$, where
$\psi^{\prime}=\psi-\pi/2$. Since the velocity changes its sign at
$\psi=\pi/2$, we have
$v_{y}(-\psi^{\prime})E_{y}(-\psi^{\prime})=-v_{y}(\psi)E_{y}(\psi)$. This
means that the energy loss at $\psi^{\prime}<0$ cancels out the energy gain at
$\psi^{\prime}>0$. As a result, the elongation caused by the frequency
modulation produces no net energy gain in the considered regime.

The obtained result is not specific to the choice of the phase offset
$\psi_{0}$. In fact, there is no net energy gain at
$\langle\omega^{\prime}\rangle=2\omega_{\beta}$ for any $\xi_{0}$. To show
this, we rewrite the expression for $\Delta\gamma$ given by Eq. 53 as

$\displaystyle\Delta\gamma$ $\displaystyle=$
$\displaystyle-\frac{\Delta_{\max}}{\pi}\int_{-\pi}^{\pi}\cos(\xi+\xi_{0})\cos(\psi)d\psi$
(67) $\displaystyle=$
$\displaystyle\frac{\Delta_{\max}}{\pi}\sin(\xi_{0})\int_{-\pi}^{\pi}\sin(\xi)\cos(\psi)d\psi$
$\displaystyle-$
$\displaystyle\frac{\Delta_{\max}}{\pi}\cos(\xi_{0})\int_{-\pi}^{\pi}\cos(\xi)\cos(\psi)d\psi.$

It follows from the expression for $\xi$ given by Eq. 58 that, in general,

$\xi(-\psi)=-\xi(\psi).$ (68)

Therefore, the integral that involves the product $\sin(\xi)\cos(\psi)$ is
equal to zero and we are left only with the integral that contains
$\cos(\xi)\cos(\psi)$:

$\Delta\gamma=-\frac{\Delta_{\max}}{\pi}\cos(\xi_{0})\int_{-\pi}^{\pi}\cos(\xi)\cos(\psi)d\psi.$
(69)

It is important to point out that the value of the integral in Eq. 69 is
independent of $\xi_{0}$. To provide a specific example, we have already
considered the case with $\xi_{0}=0$ and showed that the integral is equal to
zero. By changing $\xi_{0}$, we can only change the multiplier, whereas the
integral will remain equal to zero. Therefore, $\Delta\gamma$ remains equal to
zero for all possible values of $\xi_{0}$.

The period of the frequency modulation is the primary reason why there is no
net energy gain in the considered regime. As seen in Eq. 49, $\omega^{\prime}$
has two minima during each betatron oscillation. These minima correspond to
two transverse turning points. If the number of laser field oscillations is
three, then these minima can be matched with the locations where $E_{y}=0$. In
this case, the electron gains energy near the turning points and the frequency
modulation emphasizes the energy gain while reducing energy losses. The same
logic applies to the regimes with $\ell>3$ where $\ell$ is odd. If the number
of laser field oscillations is two, then it is not possible to match up the
turning points with zeroes of $E_{y}$, which is necessary for taking advantage
of the frequency modulation. As a result, there is no net energy gain despite
the frequency modulation. The same logic applies to the regimes with $\ell>2$
where $\ell$ is even.

Figure 8: Impact of the frequency modulation on the evolution of the laser
electric field for $\langle\omega^{\prime}\rangle=2\omega_{\beta}$ and
$\xi_{0}=0$. The solid curve in the upper panel shows $E_{y}/E_{0}$ for
$C_{2}=0.5$. The dashed curve is the field profile without the frequency
modulation ($C_{2}=0$). The lower panel shows the normalized value of
$-E_{y}v_{y}$ versus $\psi$ for the same field profiles. The empty circle
marks a transverse turning point ($v_{y}=0$).

##  VIII Summary

In this paper, we examined the role of the laser frequency modulation caused
by the betatron oscillation of the electron in quasi-static plasma electric
and magnetic fields. It is shown that the frequency modulation experienced by
the electron reduces the net energy gain per betatron oscillation during the
betatron resonance. However, there is a regime where the presence of the
modulation is beneficial. In this regime, the laser performs three
oscillations per betatron period. There is no energy gain without the
modulation, because the energy gain is fully compensated by the energy loss.
The experienced frequency modulation slows down the laser field oscillation
near transverse stopping points, which then prolongs the time interval during
which the electron experiences energy gain. As a result, a net energy gain
becomes possible. The same mechanism applies to other regimes where the number
of laser oscillations per betatron period is odd and greater than one. If the
number of oscillations is even, then the energy gain remains fully compensated
by the energy loss despite the frequency modulation.

The results for an odd number of laser oscillations $\ell$ over a single
betatron oscillation can be summarized with a single expression. For a given
odd $\ell$, the change of the relativistic $\gamma$ factor over one betatron
oscillation is

$\frac{\left(\Delta\gamma\right)_{\ell}}{\Delta_{\max}}=(-1)^{k+1}\cos(\xi_{0})\left[J_{k}(C_{\ell})-J_{k+1}(C_{\ell})\right],$
(70)

where

$k=(\ell-1)/2.$ (71)

This expression can be verified by performing the integral in Eq. 69 with
$\xi$ given by Eq. 58. Equation 70 provides a general expression that is valid
for any odd $\ell$ rather than just for $\ell=1$ and $\ell=3$. For even values
of $\ell$, we have $\left(\Delta\gamma\right)_{\ell}=0$, i.e. there is no net
energy gain.

## Acknowledgments

This material is based upon work supported by the Department of Energy
National Nuclear Security Administration under Award Number DE-NA0004030.
Simulations were performed with EPOCH (developed under UK EPSRC Grants No.
EP/G054940/1, No. EP/G055165/1, and No. EP/G056803/1) using HPC resources
provided by TACC at the University of Texas at Austin.

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