Compact Compton γ-ray source from a spatiotemporal-modulated pulse scattering a high-energy electron beam

Abstract

A novel plasma mirror is proposed for realizing all-optical Compton scattering, and its performance is compared with that of planar and concave plasma mirrors. Compared to a planar mirror, a concave mirror augments the radiation energy, but it decreases the collimation of the emitted photon beam. With the aid of the increased pulse length of the reflected laser, our proposed plasma mirror boosts the radiation energy and simultaneously improving the collimation of the emitted radiation. The pulse length and radius of the reflected laser can be controlled by adjusting the parameters of the proposed plasma mirror. The dependences of the pulse length and radius on the mirror parameters have been demonstrated. The impact of non-ideal conditions encountered in real experiments on the proposed mechanism has been discussed, which precisely demonstrates the robustness of the proposed mechanism. Additionally, the required gas density for a wakefield accelerator is derived to achieve optimal scattering under the given plasma mirror configurations.

Full text

Compact Compton c-ray source
from a spatiotemporal-modulated pulse
scattering a high-energy electron beam
Cite as: Phys. Plasmas 31, 083101 (2024); doi: 10.1063/5.0211695
Submitted: 1 April 2024 .Accepted: 6 July 2024 .
Published Online: 1 August 2024
Q. Yu,
1,a)
Y. J. Gu,
2,3
Y. Zhang,
1,a)
Q. Kong,
4,a)
and S. Kawata
5
AFFILIATIONS
1
School of Mechanical Engineering and Rail Transit, Changzhou University, Changzhou 213164, China
2
Institute of Laser Engineering, Osaka University, Suita, Osaka 565-0871, Japan
3
Institute of Physics of the ASCR, ELI-Beamlines, Na Slovance 2, 18221 Prague, Czech Republic
4
Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of Modern Physics,
Department of Nuclear Science and Technology, Fudan University, Shanghai 200433, China
5
Graduate School of Engineering, Utsunomiya University, Utsunomiya 321-8585, Japan
a)
Authors to whom correspondence should be addressed: qqyu@cczu.edu.cn;zhangyi1976@126.com; and qkong@fudan.edu.cn
ABSTRACT
A novel plasma mirror is proposed for realizing all-optical Compton scattering, and its performance is compared with that of planar and con-
cave plasma mirrors. Compared to a planar mirror, a concave mirror augments the radiation energy, but it decreases the collimation of the
emitted photon beam. With the aid of the increased pulse length of the reflected laser, our proposed plasma mirror boosts the radiation
energy and simultaneously improving the collimation of the emitted radiation. The pulse length and radius of the reflected laser can be con-
trolled by adjusting the parameters of the proposed plasma mirror. The dependences of the pulse length and radius on the mirror parameters
have been demonstrated. The impact of non-ideal conditions encountered in real experiments on the proposed mechanism has been dis-
cussed, which precisely demonstrates the robustness of the proposed mechanism. Additionally, the required gas density for a wakefield accel-
erator is derived to achieve optimal scattering under the given plasma mirror configurations.
V
C2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (https://
creativecommons.org/licenses/by/4.0/).https://doi.org/10.1063/5.0211695
I. INTRODUCTION
High-quality electron beams have been realized by exploiting
high-power lasers and advanced plasma-based accelerator technolo-
gies.
1–9
They can be regarded as compact particle accelerators that can
be used to drive ultra-compact light sources, yielding a broad spectrum
of electromagnetic radiation ranging from THz to x-ray or c-ray
regions. These light sources have excellent applications in numerous
fields, including phase contrast imaging,
10–12
radiosurgery,
13
lithogra-
phy,
14
nuclear resonance fluorescence,
15–17
transmutation of nuclear
wastes,
18,19
and generation of medical isotopes.
20–23
Based on the laser
and plasma parameters, different regimes can be realized, contributing
to the generation of high-quality multi-MeV photon beams. These
beams include the betatron radiation produced in a laser wakefield
accelerator (LWFA) with under-dense plasmas,
24
synchrotron radia-
tion in laser–solid interactions,
25
bremsstrahlung emission by laser-
accelerated fast electrons interacting with high-Z atoms,
26
Compton
scattering (CS), or Thomson scattering,
27–30
and in-flight positron
annihilation.
31
Recently, all-optical controllable CS or Thomson scat-
tering sources have been widely explored,
29,32,33
aided by routine gen-
eration of laser-driven GeV electron beams
5,34–37
in laboratories, thus
facilitating the realization of all-optical Compton/Thomson sources.
A CS source bombards a bunch of relativistic electrons with an
intense laser pulse. The electrons traveling in the electromagnetic field
oscillate and emit synchrotron-like radiation, which is commonly
referred to as CS when the quantum effect is considered. The all-
optical CS concept has been widely discussed theoretically
38–43
and has
been demonstrated in proof-of-principle experiments.
44–46
The
entirely optical realization of CS typically requires two counterpropa-
gating relativistic laser pulses:
33,43,47,48
one to evoke an LWFA and the
other to scatter the wakefield-accelerated electrons. This leads to two
stringent requirements for the experiments: the two counterpropagat-
ing laser pulses need to be spatially aligned and temporally
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synchronized. To meet these requirements, an alternative
approach
29,40,49–53
has been proposed, wherein a planar plasma mirror
(PPM) is placed behind the plasma-based accelerator. After the accel-
eration of the electrons, the PPM reflects the incident laser, which
causes it to automatically overlap with the subsequent accelerated
beam, thereby achieving all-optical CS. Since this single-pulse CS
scheme employs only one laser pulse, the temporal synchronization
and spatial alignment of two counterpropagating ultra-short pulses
can be disregarded. Thus, this approach can easily be experimentally
implemented. Through this single-pulse CS scheme, numerous
researchers have successfully generated highly energetic x-ray/c-ray
beamsinbothexperiments
50,51,54,55
and simulations.
29,53,56
Harvey
et al.
30
comprehensively investigated the effect of the laser profile on
scattering and provided a detailed explanation about the dependence
of scattering on various laser parameters.
In the aforementioned PPM-based CS mechanism, the emitted
photon energy
57,58
is limited by the LWFA driving laser intensity,
which is generally small owing to the focusing condition limitations,
59–62
namely, 17 nc
ne

GW <P<neR4
nck4. Here, neand ncare the plasma and
critical densities, respectively; kis the laser wavelength; and P is the
laser power. Furthermore, the radius of the scattering laser is usually
considerably larger than that of the scattered electron beam, denot-
ing that only the near-axis portion of the laser energy partakes in
CS, which significantly reduces the CS efficiency and utilization of
the laser. To address these two issues, researchers
29,63–65
have pro-
posed the use of a concave focusing plasma mirror instead of a PPM.
For instance, Feng et al.
29
successfully fabricated intense c-rays by
employing a focusing plasma mirror to refocus the reflected laser
pulse on the LWFA electrons. However, owing to the increase in the
transverse field strength of the scattering pulse, the collimation and
emittance of the radiation deteriorated. The concave focusing
plasma mirror also significantly shortened the effective pulse length
of the reflected laser. Nevertheless, as stated in Ref. 30, this parame-
ter is crucial in CS as it affects the scattering time and consequently
affects the energy of the radiated photons and laser utilization.
Moreover, the negative impact of the reduction in the effective pulse
length could neutralize or even overcome the positive effects of the
laser intensity enhancement, discussed in Ref. 30.
This study employs an exponential convex focusing plasma mir-
ror (E-FPM) instead of a PPM [or a parabolic concave focusing plasma
mirror (P-FPM)] in a PPM-based (or P-FPM-based) CS mechanism
(Fig. 1). This particular shape is employed for the mirror because the
optical path analysis showed that this shape can longitudinally extend
the pulse length of the reflected laser while transversely focusing the
reflected laser toincrease the laser intensity. This design has the advan-
tages of a P-FPM without its disadvantages. The base of E-FPM is
designed to be planar to effectively modulate the reflected laser radius
and ensure that it can fully cover the scattered electron beam trans-
versely at the focal point. This is another advantage of E-FPM over P-
FPM, as ultra-tight focusing, when using a P-FPM to reflect the laser,
can sometimes yield too small a radius of the reflected laser to fully
cover the colliding electron beam transversely at the focal spot.
Section II presents a comparison of the effects of PPM, P-FPM,
and E-FPM on the CS process and quality of the radiated rays. Section
III verifies the promoting effect of the extended pulse length afforded
by E-FPM on the quality improvement of the emitted photon beam.
Section IV discusses the impacts of various adverse experimental
factors on the three scattering schemes. Section Vinvestigates the
impact of E-FPM on the reflected laser profile and the scattering effect.
Section VI theoretically presents the LWFA density required for realiz-
ing high-efficiency laser–electron scattering, with the relevant
technical details presented in Appendix.SectionVII illustrates the
conclusions.
II. COMPARISON OF SINGLE-PULSE CS BASED ON PPM,
P-FPM, AND E-FPM
A. Simulation setup
Figure 2 illustrates the schematic of the single-pulse CS scheme
based on an LWFA and plasma mirrors. A relativistic laser propagates
through an under-dense plasma, evoking a wakefield. The wakefield
captures and accelerates the background electrons, and the acceleration
ends when the electrons reach the dephasing point. Upon reaching the
plasma mirror placed behind the gas target, the driving laser is
reflected by the plasmamirror, and the reflected laser reverses its prop-
agation direction and naturally overlaps with the wakefield-accelerated
forward-propagating electron beam, thus realizing CS and affording a
high-quality light source. Three single-pulse CS processes based on
PPM, P-FPM, and E-FPM were investigated via theoretical analyses
and numerical simulations. To save simulation resources, the wakefield
acceleration process was ignored, which has been already thoroughly
researched in numerous studies.
2–4,66–68
Simulations were performed based on the Particle-in-Cell
code VLPL, which represents Virtual Laser Plasma Lab.
69
Unless
stated otherwise, the same simulation parameters were employed
in all the three cases. The driving pulse was a linearly polarized
Gaussian laser pulse with a wavelength (kÞof 1 lm, radius (RÞof
10 lm, intensity (a0) of 1, and pulse duration (LÞof 30 k.The
dimensions of the simulation box along and perpendicular to the
laser propagation direction were 101k70k, with a corresponding
resolution of 0:03k0:1k. The wakefield-accelerated energetic
electron beam was a preformed 0:01ncelectron beam measuring
7k2kalong and perpendicular to the laser propagation direc-
tion. The electron beam propagated from the left to the right with
a peak energy of 0.5 GeV and an energy spread of 5%. Four macro-
particles were present in each cell for the electron beam. With
regard to the plasma mirrors, each cell comprised eight macro-
particles and the mass ratio of electrons to ions was 1/1836. The
radius and thickness of the plasma mirrors were 35kand 5k,
respectively, with a density of 5nc.TheleftpanelofFig. 3 depicts
the inner surface outlines of the plasma mirrors, which are
described as
FIG. 1. Schematic of the single-pulse CS based on an exponential focusing reflec-
tor and plasma wakefield.
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y
jj
¼
x¼x0;for PPM
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k1x1x
ðÞ
p;for P-FPM
ek2c0x
ðÞ
y0;x<x1for E-FPM
8
>
>
>
>
<
>
>
>
>
:
(1)
with x0¼50k,x1¼86k,y0¼0, k1¼12:5, k20:06;and c0¼101k.
Specifically, k1is a coefficient that determines the focal length (f)
and curvature change of a parabola through f¼k1
4. The larger k1is,
the greater is the change in the parabolic function; conversely, the
smaller k1is, the more gradual is the change. In other words, k1
determines the focal length, aperture size, and curvature change of
the parabolic mirror, directly influencing the geometric shape and
optical properties of the mirror. k2is another coefficient whose mag-
nitude affects the rate of curvature change in an exponential func-
tion. The larger k2is, the more dramatic is the change in the
exponential function; conversely, the smaller k2is, the more gradual
is the change. k2governs the geometric structure and optical proper-
ties of an exponential mirror by influencing its aperture size and cur-
vature change. The longitudinal distances between the electron
beam mass center and laser center were 26:5k,6k, and 32:5kfor the
PPM, P-FPM, and E-FPM cases, respectively. The employed delay
distances between the electron beam mass center and laser center in
all the three cases guaranteed that the electrons scattered the lasers
at the focal points.
B. Simulation results and discussion
The optical path diagrams in the left panel of Fig. 3 show that
PPM barely influences the reflected laser pulse length, P-FPM signifi-
cantly decreases the reflected laser pulse length, and E-FPM exhibits
an extension effect for the reflected laser pulse length. The simulated
laser profiles reflected by PPM, P-FPM, and E-FPM (right panel of
Fig. 3) verify the theoretical predictions. Based on the theoretical analy-
ses, the pulse lengths of the reflected lasers (Lref )are
Lref 
L;for PPM
0;for P FP
lnR
k2
þRtan 2h2p
2

;for E FPM;
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
(2)
where h2¼arctanðk2RÞ. The simulation results approximately
matched the theoretical predictions for all the three plasma mirrors,
especially in the E-FPM case, where the simulated Lref of approxi-
mately 30kclosely agreed with the theoretical result of 33k.
FIG. 2. Schematic of a single-pulse CS based on an LWFA and (a) PPM, (b)
P-FPM, and (c) E-FPM. The arrows indicate the propagation directions of the laser
photons before and after the reflection.
FIG. 3. Laser beam path diagrams after reflection by (a) PPM, (b) P-FPM, and (c)
E-FPM. (d)–(f) Corresponding profiles of the laser beams reflected by the three
plasma mirrors and corresponding transverse electric field distributions of the
reflected lasers along y¼0 before and after reflection.
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Furthermore, P-FPM and E-FPM were capable of refocusing the
reflected pulse. Depending on the refocusing effect, P-FPM and
E-FPM increased the reflected a0to approximately 2.9 and 2:0, respec-
tively, from the original value of 1.0. The reflected a0value in the
E-FPM case was lower than that in the P-FPM case. This was because
compared to the latter, the Lref extension effect in the former weakened
the focusing effect.
Compared to P-FPM, which caused the reflected pulse to rapidly
diverge after the focal spot, E-FPM maintained the focusing profile of
the reflected pulse for more than 25T
0
(T
0
being the laser period), as
shown in Fig. 4. The CS time (tCS)isrelatedtoLref via tCS ¼Lre f =2c,
where Lref ¼30kand cis the light speed. The tCS value was 15T
0
in
the E-FPM case, which denotes that E-FPM kept the reflected pulse
focused during the entire CS process, thus guaranteeing highly efficient
CS.
Figure 5 illustrates the temporal evolutions of the electron energy
and the radiated photon beam quality during the scattering process
under the three cases. The left column depicts that as the collision pro-
cess progressed, the electrons transferred energy to the radiated pho-
tons, which decreased the electron energy (Ee) and increased the
photon energy (Ep) until they both stabilized, marking the end of the
CS process. Furthermore, the right column shows that as the collision
process advanced, increasing number of photons were generated, lead-
ing to a sharp increase in the photon yield (Np) until it stabilized at the
end of the CS process. During the scattering process, due to the trans-
verse force exerted by the laser on the electrons, the radiated photons
acquired a certain emittance. Consequently, the divergence angle of
the photons (hp) increased until the end of the CS process, whereupon
the divergence angle stabilized.
Moreover, approximately 7, 9, and 13lJ electron energies were
transported to the emitted radiations through CS in the PPM, P-FPM,
and E-FPM cases, respectively, as shown in the left column figures in
Fig. 5. This implies that the laser utilization efficiency of E-FPM was
higher by 86% and 44% compared to that of PPM and P-FPM, respec-
tively. This can be attributed to the boosted Lref and a0of E-FPM.
Compared to PPM and P-FPM, since the radiation energy of E-FPM
was boosted, as shown in Figs. 5(d)–5(f), the photon number and colli-
mation of the emitted radiation of E-FPM were also improved.
Specifically, E-FPM increased the photon number of the emitted ray to
approximately 12 107, an increase in 50% and 70% compared to
PPM and P-FPM, respectively. Additionally, E-FPM decreased the
average emission angle of the emitted photons to 1:8102,which
was 3/5 and 3/25 of that afforded by PPM and P-FPM, respectively.
While P-FPM improved the radiation energy compared to PPM, its
boosted transverse electric field of the reflected pulse stemming from
the ultra-tight focusing effect degraded the collimation of the radiated
photons.
The enhanced a0by P-FPM or E-FPM arising from the refocus-
ing effect could improve the energy and photon number in the radia-
tion beam. This is attributed to the enhanced radiation reaction force
and photon number density of the reflected laser resulting from the
increase in the reflected a0. First, the radiation reaction force
70
is scaled
by a0as fRR 4pahxLc2
0a2
0=ð3kÞ, indicating that the radiation recoil
force increases with the a0value. Thus, the enhanced a0after reflection
by P-FPM or E-FPM augmented the radiation recoil force, thereby
improving the radiation energy. Second, a2
0is proportional to the den-
sity of the laser photon number ðnc): a2
0¼he2
m2
ec2xLnc¼4pa2k3nc,
where hxL=mec2is the normalized laser frequency; k3ncis the
number of laser photons in a cube with side k;his the Dirac constant;
eis the unit charge; meis the electron mass; and xLis the laser fre-
quency. The probability of multiphoton scattering eþncL!eþc
(where nis the number of laser photons participating the in multipho-
ton scattering) is proportional to the laser photon density, that is, pro-
portional to the scattering laser intensity: nn
ca2n
0. Thus, the enlarged
scattering laser intensity improved the probability of multiphoton scat-
tering, thereby boosting the number of CS photons.
After CS, 0.5, 0.7, and 0.7MeV photon beams were obtained in
PPM, P-FPM, and E-FPM cases, respectively, as shown in Fig. 6.In
the E-FPM case, the energy spread of the emitted photon bunch was
20%, which was 1/2 that of the P-FPM case, and the angular spread
waslessthan0.02
, which was only 1/28 that of the P-FPM case and
FIG. 4. Electric field distributions of the laser reflected by E-FPM at (a) 55T
0
, (b)
65T
0
, (c) 75T
0
, and (d) 85T
0
.
FIG. 5. Time evolutions of the electron energy (Ee) and radiated photon energy
(Ep) in the (a) PPM, (b) P-FPM, and (c) E-FPM cases. (d)–(f) Corresponding time
evolutions of the average emission angle (hp) and emitted photon number (Np)in
the three cases.
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1/3 that of the PPM case. In terms of the energy–angle distribution,
the photon distribution afforded in the E-FPM case was more compact
than that afforded by the other two cases.
During the realization of CS with laser–electron beam colli-
sions, the electron beam energy distribution plays a crucial role in
determining the spectral distribution of the resulting photon beam.
A non-monoenergetic electron beam yields a broad spectrum of
the produced photon beam. However, the study simulations
employed an electron beam generated by an LWFA, which is nota-
bly characterized by its monoenergetic properties. The use of such
electronbeamsforCSresultsinphotonbeamsthatexhibithigh
monochromaticity. This has been effectively demonstrated by Yu
et al.
65
In Yu et al., the head-on collision of an LWFA-produced a
monoenergetic electron beam with a laser yielded c-raysinthe
MeV range with excellent monochromaticity—approximately 30%,
with energy typically ranging from 20% to 40% in the structured
plasma mirror cases. The simulation and experimental results of
Yu et al. confirmed the monochromatic nature of Compton pho-
ton beams produced through LWFAs. Conversely, electron beams
from other types of acceleration mechanisms tend to exhibit
reduced monoenergetic qualities, which decreases the monochro-
maticity of the resultant c-rays. Hence, LWFA-produced electron
beams were employed in the CS analyses herein.
III. VERIFICATION OF IMPACT OF EXTENDING
REFLECTED PULSE LENGTH ON CS BY PARAMETER
SCANNING
The above discussion showed that the CS efficiency can be
improved by increasing the reflected a0.BothP-FPMandE-FPMcan
boost the reflected a0, but P-FPM yields slightly higher reflected a0than
E-FPM. However, compared to P-FPM, E-FPM boosts the CS effi-
ciency, which is attributed to the boosted L
ref
in the latter case. This
study verified this inference through multiple sets of numerical simula-
tions. In each simulation set, a crucial parameter, including the reflected
a0, focal length of reflected laser, or delay between the electrons and
driving laser, was maintained constant. Table I illustrates the simulation
results, showing that when any one of the aforementioned parameters
was kept constant, E-FPM still substantially enhanced the CS efficiency
compared to P-FPM. This eliminated the possibility of scattering
improvement stemming from the aforementioned parameters. When
theincreaseinL
ref
was weak, E-FPM did not exhibit an advantage in
terms of CS efficiency compared to P-FPM. These simulation results
further demonstrate that the CS efficiency increases with L
ref
.
IV. DISCUSSION ON THE IMPACT OF VARIOUS
ADVERSE EXPERIMENTAL FACTORS ON THESE THREE
SCATTERING SCHEMES
Considering the complexities of real-world experiments, the
impact of non-ideal conditions on various scattering mechanisms need
to be assessed. The following non-ideal conditions were considered:
(1) misalignment between the beams and structured target axis, (2)
reduction in the driving laser pulse length from the laser depletion dur-
ing wakefield acceleration, and (3) presence of pre-plasma from the
reflection mirrors. Through a series of comparative simulations, this
study explored how the beam–target offset distance, driving laser pulse
length, and mirror pre-plasma size influence the scattering results.
Tables II–IV depict the effects of these parameters on the scattering
outcomes under the different schemes.
The tables show that the scattering efficiency diminished regard-
less of the type of mirror structure used under the three non-ideal con-
ditions. Furthermore, as these parameters increased, the degradation
in scattering efficiency became more pronounced. However, under
these non-ideal conditions, E-FPM still exhibited advantages in terms
of the CS efficiency relative to both P-FPM and PPM. These compara-
tive results highlight the robustness of E-FPM under less-than-ideal
experimental conditions and its potential superiority in enhancing the
CS efficiency in practical applications.
FIG. 6. (a)–(c) Energy–angular distribution, (d)–(f) energy spectra, and (g)–(i) angu-
lar spectra of the radiated photons in the PPM, P-FPM, and E-FPM cases,
respectively.
TABLE I. Photon number (NpÞ, total energy ðEpÞ, peak energy (EpeakÞ, and energy spread (DE=E) of the emitted beams for five comparative cases. Case I, as a reference
case, employed P-FPM with k1¼17. Cases II–V employed E-FPMs with the following parameters: Case II: k2¼0:045, c0¼x1¼86k, and y0¼3k; Case III: k2¼0:06,
c0¼85k,x1¼86k, and y0¼3k; Case IV: k2¼0:03, c0¼x1¼86k, and y0¼3k; and Case V: k2¼0:2, c0¼x1¼86k, and y0¼2k. The parameters were selected
to ensure that compared to case I, the reflected a
o
in case II, the focal length of the reflected pulse in case III, and the delay distance between electrons and driving laser in case
IV remained constant. Moreover, the increase in the effective L
ref
of the reflected laser was weak in case V.
N
p
(10
7
)E
p
(lJ) E
peak
(MeV) DE/E(%)
Case I Reference case 8 12 0.8 48
Case II Same laser intensity after reflection compared with case I 11.2 16 1.0 31
Case III Same focal length compared with case I 12.2 19 1.2 23
Case IV Same delay between the electron bunch and laser compared with case I 11.2 18 1.0 28
Case V Increase in effective duration is weak 5.74 4 0.45 48
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V. IMPACT OF THE E-FPM CONFIGURATION ON THE
REFLECTED LASER PROFILE AND SCATTERING
RESULTS
A. Scaling law of the pulse length and radius of the
reflected laser with E-FPM configuration
Equation (2) provides the dependence of L
ref
on the plasma mir-
ror parameters, indicating that in the E-FPM case, L
ref
is determined
by the parameter k2. To validate this dependence, two simulation cases
are investigated with k2¼0:06, and 0:1. Under these two cases, the
theoretical values of L
ref
were 33kand 23kand the simulation values
were 30kand 20k, respectively, as shown in Figs. 7(a) and 7(b).This
shows that the theoretical predictions and simulation results well
matched, demonstrating the reliability of the theoretical predictions
and proving that L
ref
varied with k2.
Analysis also demonstrated that the reflected laser radius (Rref Þ
was influenced by the E-FPM parameter rgap [as marked in Figs. 2(c)
and 3(c)]. Rref increased with rgap from 4:2kto 8:6k, as presented in
Figs. 8(a) and 8(b). Moreover, the dependence of Rref on rgap can be
represented as Rref ¼k0rgap. Through five more simulations and
the linear fitting of these numerical results, as shown in Fig. 8(c),
k0¼0:89 was obtained. Therefore, Rref is related to rgap as
Rref ¼0:89rgap:(3)
By exploiting Eqs. (2) and (3), the pulse length and radius of the
reflected laser can be adjusted to modulate its profile in the E-FPM case.
B. Dependence of the scattering effect on
parameter k2
The E-FPM configuration influences the reflected laser shape,
thereby impacting the scattering efficiency. Therefore, the dependence
of the scattering efficiency on the E-FPM configuration parameters
FIG. 7. Profiles of the reflected lasers in the (a) k2¼0:06 and (c) k2¼0:1 cases.
(b) and (d) Transverse electric field distributions of the reflected pulses along y ¼0
for the cases depicted in (a) and (c), respectively.
FIG. 8. Outlines of the reflection mirrors and reflected pulses with (a) rgap ¼4:2k
and (b) rgap ¼8:6k. (c) Dependence of the reflected laser spot radius on the
plasma mirror parameter rgap, with the linear fitting results.
TABLE II. Total photon energy (Ep) radiated in the P-FPM and E-FPM cases under
various lateral beam-mirror offset conditions (D
offset
).
D
offset
¼0
Ep(lJ)
D
offset
¼1
kEp(lJ)
D
offset
¼5
kEp(lJ)
D
offset
¼8
kEp(lJ)
P-FPM 9.6 7.3 0.91 0.37
E-FPM 12.9 9.6 1.1 0.72
TABLE III. Photon energy output (Ep) in the PPM, P-FPM, and E-FPM cases under
various laser pulse lengths (L).
L¼15kEp(lJ) L¼12kEp(lJ) L¼8kEp(lJ)
PPM 6.9 5.5 3.5
P-FPM 9.6 8.5 5.6
E-FPM 12.9 10.3 6.7
TABLE IV. Photon energy output (Ep) in the PPM, P-FPM, and E-FPM cases under
various preplasma lengths (L
pre
).
L
pre
¼0Ep(lJ) L
pre
¼0.5kEp(lJ) L
pre
¼1kEp(lJ)
PPM 6.9 6.8 6.7
P-FPM 9.6 9 7.5
E-FPM 12.9 12.4 11.3
Physics of Plasmas ARTICLE pubs.aip.org/aip/pop
Phys. Plasmas 31, 083101 (2024); doi: 10.1063/5.0211695 31, 083101-6
V
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needs to be studied, primarily k2. By performing more simulations,
this section investigated the dependence of the scattering effect on k2,
and Fig. 9 presents the results. The figure shows that for the peak
energy of the radiated photons, under our simulation framework, the
optimal k2value is k2¼0:04;whereupon the peak energy of the radi-
ated photons (cppeak) is 2.24. This represents a 60% increase com-
pared to the cppeak ¼1:4 obtained with k2¼0:06 as used earlier.
Additionally, k2values ranging from 0.01 to 0.06 can yield radiated
photons with cppeak 1:4, demonstrating the robustness of the
E-FPM reflection mechanism.
VI. OPTIMAL LWFA PLASMA DENSITY FOR A GIVEN
E-FPM CONFIGURATION
For a given E-FPM configuration, an optimal spatial range exists
where the reflected laser exhibits excellent focus and high intensity,
which is referred to as the focal position range. Before the reflected
laser reaches or after it passes this focal position range, the laser is in a
divergent state with lower intensity. Thus, the electron and reflected
laser were designed to collide within this focal position range as the
reflected laser is well-focused and intense in this range, ensuring high
scattering efficiency and good quality of the radiated c-ray beam.
The collision position is determined by the pre-collision elec-
tron–laser delay distance del. The colliding electrons typically stem
from the LWFA mechanism. For the LWFA electrons, delis deter-
mined by the plasma wavelength, which is proportional to the plasma
density. Therefore, for the LWFA electrons, the collision position is
determined based on the plasma density. However, the focal position
is determined according to the E-FPM configuration. To align the col-
lision position with the focal position, the LWFA plasma density needs
to be matched with the E-FPM configuration. In other words, for a
given E-FPM configuration, an optimal LWFA plasma density exists.
Within this optimal density range, collisions occur at the focal
position of the reflected laser, leading to high scattering efficiency and
good quality of the photon beam. Outside this optimal density range,
electrons collide with the divergent weaker laser, resulting in poor col-
lision effects. Herein, through theoretical analysis, the optimal LWFA
gas density for a given set of E-FPM configuration parameters was
determined.
For single-pulse CS, delis a key factor affecting the scattering
location. When the collision occurs at the center of the focused
reflected laser, deland dlmshould fulfill the following condition:
del¼2dlm.Here,dlmis the distance between the center of the
focused reflected laser and the mirror and is determined by the mirror
configuration. In the E-FPM cases, the theoretical optimal value for
delwas 33k, which is consistent with the corresponding simulation
value of 32:5k. Single-pulse CS typically relies on the LWFA mecha-
nism, where delis typically half the plasma wavelength, i.e.,
del¼1
2kp,andkpdepends on the LWFA plasma density. Therefore,
to obtain the optimal del, the optimal LWFA plasma density needs to
be employed. Under this optimal condition, the scaling law del
¼1
2kp¼2dlmshould be satisfied. Upon substituting the expression
for kpinto this equation, where kpðlmÞ¼kðlmÞffiffiffiffiffiffiffiffiffiffiffi
nc=ne
pand
kðlmÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a0nc=ne
p=pin the linear and nonlinear regimes, respectively,
the optimal plasma density for LWFA was theoretically derived:
nLWFA1¼
nck2
16p2lnR
k2
þRtan 2h2p
2

2;
for the linear E-FPM regime
a0nck2
16p4lnR
k2
þRtan 2h2p
2

2;
for the non linear E-FPM regime:
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
(4)
For the E-FPM case, as the reflection pulse was able to maintain
the focusing shape for a long time, when the CS occurred at the leading
edge of the reflected laser, the scattering effect was considerable. In this
case, 2dlmþ1
2Lref ¼1
2kp.Therefore,thedensityforLWFAinthis
case was
nLWFA2¼
nck2
36p2lnR
k2
þRtan 2h2p
2

2;
for the linear E-FPM regime
a0nck2
36p4lnR
k2
þRtan 2h2p
2

2;
for the non linear E-FPM regime:
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
(5)
Thus, the density range of LWFAs for high-efficiency scattering in E-
FPM cases is nLWFA2<ne<nLWFA1.TheAppendix provides a
detailed discussion of the above process.
VII. CONCLUSION
To implement single-pulse CS, plasma mirrors of various shapes
are often employed. A planar mirror is often the first choice owing to
its less stringent experimental condition requirements and the robust-
ness of the scattering results. However, planar mirrors typically result
in low radiation energy and laser utilization. A common alternative is
the concave focusing mirror, which increases the reflected laser inten-
sity via focusing, thereby increasing the radiation energy. However,
disadvantageously, in some cases, the transverse and longitudinal sizes
FIG. 9. Dependence of the peak energy (cppeak ) of emitted photon beams on the
E-FPM configuration parameter k2.
Physics of Plasmas ARTICLE pubs.aip.org/aip/pop
Phys. Plasmas 31, 083101 (2024); doi: 10.1063/5.0211695 31, 083101-7
V
CAuthor(s) 2024

of the reflected laser might be very small, which reduces the collision
area with the electrons and the scattering time. Moreover, ultra-tight
focused reflected lasers weaken the collimation of the radiation photon
beam. The proposed convex plasma mirror can focus the reflected
laser, which increases its intensity and pulse length, consequently
enhancing the scattering time. Furthermore, the planar part at the bot-
tom can be used to adjust the reflected laser radius, allowing it to trans-
versely fully cover the colliding electron beam at the collision point
without reducing the scattering area. The impact of these three mirrors
on CS was compared through numerical simulations and theoretical
analyses. The simulation results confirmed the advantages and disad-
vantages of these three mirrors, highlighting the advantages of the pro-
posed mirror. The gas density range of the LWFA, required for
realizing high-efficiency CS, was also determined.
In this paper, we improve the scattering efficiency and the quality
of the radiation photon beam by shaping the colliding laser in
Compton scattering to match the configuration of the colliding electron
beam. In single-pulse Compton scattering, besides modulating the
reflected laser, the quality of the radiated photon beam can also be
improved by Yu et al.
62,71
enhancing the quality of the electron beam.
When the electron beam quality is sufficiently high, other novel mecha-
nisms, in addition to Compton scattering, can also produce high-
quality photon beams. For example, Zhu et al.
72
successfully obtained a
highly bright GeV c-ray beam through a two-stage laser-plasma interac-
tion. The first stage produces an electron beam, and the second stage
generates the photon beam. A crucial reason for achieving such high-
energy, high-brightness photon beams in the second stage is that the
wakefield acceleration in the first stage provides a high-charge (tens of
nC), high-energy (multi-GeV) electron beam. Such a high-quality elec-
tron beam sets the conditions for generating high-brightness, high-
energy photon beams. Therefore, in our next steps, we will focus on
how to apply improved electron acceleration mechanisms to our pro-
posed scheme. For instance, combine the electron acceleration scheme
from the work of Zhu et al.
72
with our proposed reflector.
ACKNOWLEDGMENTS
This work was supported by the Natural Science Foundation of
China under Contract No. 11804348.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Qin Yu: Conceptualization (equal); Investigation (equal); Software
(equal); Validation (equal); Visualization (equal); Writing –original
draft (equal); Writing –review & editing (equal). Yanjun Gu:
Conceptualization (equal); Investigation (equal); Software (equal);
Validation (equal); Visualization (equal). Yi Zhang: Conceptualization
(equal); Investigation (equal); Project administration (equal); Resources
(equal); Software (equal); Supervision (equal); Visualization (equal);
Writing –review & editing (equal). Qing Kong: Conceptualization
(equal); Resources (equal); Supervision (equal); Validation (equal);
Writing –review & editing (equal). Shigeo Kawata: Conceptualization
(equal); Writing –review & editing (equal).
DATA AVAILABILITY
The data that support the findings of this study are available from
the corresponding author upon reasonable request.
APPENDIX: LWFA PLASMA DENSITY WHEN THE
COLLISION OCCURS WITHIN THE OPTIMAL POSITION
RANGE
When the laser–electron collision occurs in the region from
the center to the leading edge of the focused reflected laser, the CS
efficiency is high and the emitted c-ray beam quality is optimal. In
this section, for E-FPM, the plasma density range for LWFA to
achieve efficient CS is theoretically derived, ensuring that the colli-
sion occurs in the region from the center to the leading edge of the
focused reflected laser.
Based on the E-FPM shape and using geometric optics, the
pulse length of the reflected laser is
Lref ¼lnR
k2
þRtan 2h2p
2

;(A1)
where h2¼arctanðk2RÞ. Based on the shape evolution of the
reflected laser in simulations, the distance from the center point of
the focused reflected laser to the plasma mirror dlmis as follows:
dlm¼1
2Lref ¼lnR
2k2
þ1
2Rtan 2h2p
2

:(A2)
For the electrons to collide with the center of the reflected laser, the
distance delbetween the electrons and driving laser before the col-
lision should be
del¼2dlm¼lnR
k2
þRtan 2h2p
2

:(A3)
In the LWFA mechanism, at the end of the acceleration, the acceler-
ated electrons are positioned at the center of the plasma wave. Here,
the distance between the accelerated electrons and driving laser is
del¼1
2kp;(A4)
where kpis the plasma wavelength. For electrons from LWFA to
collide with the focused reflected laser at its center, delmust satisfy
both Eqs. (A3) and (A4). Thus,
kp¼2del¼4dlm¼2lnR
k2
þ2R tan 2h2p
2

:(A5)
If the driving laser of the plasma wave has non-relativistic intensity,
the plasma wavelength is directly determined by the plasma density:
kplm
ðÞ
¼klm
ðÞ
ffiffiffiffiffiffiffiffiffiffiffi
nc=ne
p:(A6)
If the driving laser reaches relativistic intensities, the plasma wave-
length needs to be relativistically corrected:
kplm
ðÞ
¼klm
ðÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a0nc=ne
p=p:(A7)
Substitution of Eqs. (A6) and (A7) into Eq. (A5) yields the LWFA
plasma density required for the collision of electrons with the
focused reflected laser at the laser’s center:
Physics of Plasmas ARTICLE pubs.aip.org/aip/pop
Phys. Plasmas 31, 083101 (2024); doi: 10.1063/5.0211695 31, 083101-8
V
CAuthor(s) 2024

nLWFA1¼
nck2
16p2lnR
k2
þRtan 2h2p
2

2;
for the linear E-FPM regime
a0nck2
16p4lnR
k2
þRtan 2h2p
2

2;
for the non linear E-FPM regime:
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
(A8)
Since the laser reflected by E-FPM can maintain a focused state
for a long time, collisions occurring at the leading edge of the
reflected laser can also ensure good scattering effects. When the col-
lision occurs at the leading edge of the reflected laser, Eq. (A3)
should be revised to
del¼3dlm¼lnR
k2
þRtan 2h2p
2

:(A9)
Substitution of Eq. (A9) into Eq. (A4) yields
kp¼2del¼6dlm¼3lnR
k2
þ3R tan 2h2p
2

:(A10)
Substituting Eqs. (A6) and (A7) into Eq. (A10), the LWFA plasma
density when the collision occurs at the leading edge of the focused
reflected laser is obtained:
nLWFA2¼
nck2
36p2lnR
k2
þRtan 2h2p
2

2;
for the linear E-FPM regime
a0nck2
36p4lnR
k2
þRtan 2h2p
2

2;
for the non linear E-FPM regime:
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
(A11)
When the collision occurs at any point from the leading edge to the
center of the focused reflected laser, the scattering exhibits relatively
high efficiency. Therefore, for high-efficiency CS, the LWFA plasma
density nLWFA must satisfy
nLWFA2<nLWFA<nLWFA1:(A12)
Equation (A12) presents the LWFA plasma density range for
achieving efficient CS.
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Physics of Plasmas ARTICLE pubs.aip.org/aip/pop
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