††thanks: These authors have contributed equally to this work.††thanks: These
authors have contributed equally to this work.

# Brilliant circularly polarized $\gamma$-ray sources via single-shot laser
plasma interaction

Yu Wang  Ministry of Education Key Laboratory for Nonequilibrium Synthesis and
Modulation of Condensed Matter, Shaanxi Province Key Laboratory of Quantum
Information and Quantum Optoelectronic Devices, School of Physics, Xi’an
Jiaotong University, Xi’an 710049, China  Mamutjan Ababekri  Ministry of
Education Key Laboratory for Nonequilibrium Synthesis and Modulation of
Condensed Matter, Shaanxi Province Key Laboratory of Quantum Information and
Quantum Optoelectronic Devices, School of Physics, Xi’an Jiaotong University,
Xi’an 710049, China  Feng Wan<EMAIL_ADDRESS>Ministry of Education Key
Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,
Shaanxi Province Key Laboratory of Quantum Information and Quantum
Optoelectronic Devices, School of Physics, Xi’an Jiaotong University, Xi’an
710049, China  Qian Zhao  Ministry of Education Key Laboratory for
Nonequilibrium Synthesis and Modulation of Condensed Matter, Shaanxi Province
Key Laboratory of Quantum Information and Quantum Optoelectronic Devices,
School of Physics, Xi’an Jiaotong University, Xi’an 710049, China  Chong Lv
Department of Nuclear Physics, China Institute of Atomic Energy, P. O. Box
275(7), Beijing 102413, China  Xue-Guang Ren  Ministry of Education Key
Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,
Shaanxi Province Key Laboratory of Quantum Information and Quantum
Optoelectronic Devices, School of Physics, Xi’an Jiaotong University, Xi’an
710049, China  Zhong-Feng Xu  Ministry of Education Key Laboratory for
Nonequilibrium Synthesis and Modulation of Condensed Matter, Shaanxi Province
Key Laboratory of Quantum Information and Quantum Optoelectronic Devices,
School of Physics, Xi’an Jiaotong University, Xi’an 710049, China  Yong-Tao
Zhao  Ministry of Education Key Laboratory for Nonequilibrium Synthesis and
Modulation of Condensed Matter, Shaanxi Province Key Laboratory of Quantum
Information and Quantum Optoelectronic Devices, School of Physics, Xi’an
Jiaotong University, Xi’an 710049, China  Jian-Xing Li  Ministry of Education
Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed
Matter, Shaanxi Province Key Laboratory of Quantum Information and Quantum
Optoelectronic Devices, School of Physics, Xi’an Jiaotong University, Xi’an
710049, China

###### Abstract

Circularly polarized (CP) $\gamma$-ray sources are versatile for broad
applications in nuclear physics, high-energy physics and astrophysics. The
laser-plasma based particle accelerators provide accessibility for much higher
flux $\gamma$-ray sources than conventional approaches, in which, however, the
circular polarization properties of emitted $\gamma$-photons are used to be
neglected. In this letter, we show that brilliant CP $\gamma$-ray beams can be
generated via the combination of laser plasma wakefield acceleration and
plasma mirror techniques. In weakly nonlinear Compton scattering scheme with
moderate laser intensities, the helicity of the driving laser can be
transferred to the emitted $\gamma$-photons, and their average polarization
degree can reach about $\sim 37\%$ ($21\%$) with a peak brilliance of $\gtrsim
10^{21}~{}$photons/(s $\cdot$ mm${}^{2}\cdot$ mrad${}^{2}\cdot$ 0.1% BW)
around 1 MeV (100 MeV). Moreover, our proposed method is easily feasible and
robust with respect to the laser and plasma parameters.

The polarization property of the $\gamma$-rays is of great significance to
reveal the emission mechanisms in the pulsar [1, 2], magetars and other
galactic objects [3]. As an essential tool, $\gamma$-ray beams can be used in
researches of nuclear physics [4, 5], high-energy physics [6], industrial
applications of medical imaging [7] and object tomography, etc. In particular,
circularly polarized (CP) ones have broad applications in, for instances, the
generation of longitudinally-polarized positron beams [8, 9], polarization-
dependent photo-fission of nucleus in the giant dipole resonance [10] and
photo-production of mesons [11]. Conventionally, highly polarized photons can
be generated via free electron laser (FEL), Thomson and Compton scattering,
bremsstrahlung and synchrotron radiation [12]. However, FEL can only deliver
X-ray photons of tens of keV, and the peak brilliance and flux of other high-
energy $\gamma$-ray sources are normally limited by the scattering
probabilities and flux of the driving electron beam. For instance, the peak
flux of state-of-the-art High Intensity Gamma-ray Source (HI$\gamma$S) [5] is
about $10^{10}$ photons$\cdot\mathrm{s}^{-1}$ around 10 MeV with beam diameter
of $D\simeq 12$ mm. Recently, rapid developments of ultra-short ultra-intense
laser technologies [13, 14] have promoted high-brilliance laser-plasma based
particle [15, 16] and radiation sources [17, 18, 19]. When multi-PW lasers
interacting with plasmas, MeV-GeV $\gamma$-photons can be produced with
ultrahigh brilliance of $10^{25}$-$10^{27}~{}\mathrm{photons/(s\cdot
mm^{2}\cdot mrad^{2}\cdot 0.1\%BW)}$ [20, 21, 22]. However, these produced
$\gamma$-photons are either unpolarized or only linearly polarized. And all-
optical CP $\gamma$-ray beams can be generated via multi-PW laser pulses
colliding with high-energy longitudinally spin-polarized electron beams in
strongly nonlinear Compton scattering [11, 23], which faces potential
difficulties of spatial and temporal synchronization. Thus, efficient
generation of brilliant CP $\gamma$-ray beam is still a great challenge.

Figure 1: Interaction scenario for all-optical generation of brilliant CP
$\gamma$-ray beam via single-shot laser plasma interaction.

In this Letter, we put forward an efficient all-optical method for generating
brilliant CP $\gamma$-ray beams via single-shot laser plasma interaction. See
the interaction scenario in Fig. 1, which is similar to the setup in Ref.
[24], however, here we take into account the helicity transfer in the laser
plasma interaction. A moderately intense CP laser incidents into a gas plasma
and drives the laser plasma wakefield acceleration (LWFA) to generate high-
energy electrons. The CP driving laser pulse is then reflected by the plasma
mirror and further collides with accelerated electrons to emit CP
$\gamma$-photons via weakly nonlinear Compton scattering
($e+n\omega_{L}\rightarrow e^{\prime}+\omega_{\gamma}$). Here we find that the
helicities of emitted $\gamma$-photons in the low-energy regime is mainly
associated with those of the driving laser photons. While, for high-energy
$\gamma$-photons, due to multi-photon absorption, the average helicity will
saturate to a constant for unpolarized eletrons or linearly depend on the
energies of emitted photons for polarized electrons; see more details in Fig.
3. (By contrast, in strongly nonlinear Compton scattering the circular
polarization of emitted $\gamma$-photon is determined by the electron helicity
[11].) With a moderately intense CP laser, CP $\gamma$-photon beams with
brilliance of $\gtrsim 10^{21}~{}\mathrm{photons/(s\cdot mm^{2}\cdot
mrad^{2}\cdot 0.1\%BW)}$ can be generated with polarization degree of $\simeq
37\%$; see more details in Fig. 2. Moreover, the proposed method is robust
with respect to the laser and plasma parameters; see more details in Fig. 4.

In our simulations, we use three-dimensional (3D) particle-in-cell (PIC) code
EPOCH [25] to simulate the LWFA process, and the Monte Carlo code CAIN [26,
11] to simulate the weakly nonlinear Compton scattering process. As an
example, we employ a right-hand CP laser pulse (helicity $h_{L}=-1$)
propagating along $z$ direction with wavelength
$\lambda_{0}=0.8~{}\mu\mathrm{m}$, normalized intensity $a_{0}\equiv
eE_{\mathrm{rms}}/m_{e}c\omega_{L}=2.8$ [corresponding peak intensity
$I_{0}\approx 2.74\times
10^{18}a_{0}^{2}\left(1~{}\mu\mathrm{m}/\lambda_{0}\right)^{2}~{}\mathrm{W/cm^{2}}\approx
3.46\times 10^{19}~{}\mathrm{W/cm^{2}}$], and transverse Gaussian profile with
focal radius $w_{0}=11.5~{}\mu\mathrm{m}$, where $e$ and $m_{e}$ are the
charge and mass of the electron, respectively,
$E_{\mathrm{rms}}\equiv\langle(E^{2})^{1/2}\rangle$, $E$ and $\omega_{L}$ the
root mean square (rms) electric field, electric field and frequency of the
laser field, respectively, and $c$ the light speed in vacuum. The temporal
profile is composed by a flat-top part of $\tau_{\mathrm{flat}}=6T_{0}$ and
rising (falling) part of $\tau_{\mathrm{rising~{}(falling)}}=12T_{0}$ with
Gaussian-like 5th order symmetric polynomial $10t^{\prime 3}-15t^{\prime
4}+6t^{\prime 5}$ (here, $t^{\prime}\equiv t/T_{0}$ and $T_{0}$ is the laser
period) [27]. The number density of the gas plasma is linearly rising from
$n_{e}=0$ at $z=0$ to $n_{e}=1.5\times 10^{18}~{}\mathrm{cm^{-3}}$ at
$z=100~{}\mu\mathrm{m}$ and then distributed uniformly to $z=5.515~{}$mm. The
aluminum plasma mirror is placed at $z=5.515~{}$mm with a thickness of
$l=2~{}\mu\mathrm{m}$, number density $n_{e}=451n_{c}$ and scale length
$L=\lambda_{0}$ for the preplasma [28], where the critical plasma density is
$n_{c}=m_{e}\omega_{L}^{2}/4\pi e^{2}$. The spatial sizes of the simulation
box are $z\times x\times y=127\lambda_{0}\times 80\lambda_{0}\times
80\lambda_{0}$ with cell sizes of $4000\times 256\times 256$. The numbers of
macro-particle per cell are assigned as 2 and 1 for electrons and Helium ions,
respectively. Note that the above mentioned spatial sizes are not fine enough
to resolve the plasma frequency of the aluminum target, thus the reflection of
the laser pulse is recalculated with finer grid sizes of $\Delta z\times\Delta
x=\frac{\lambda_{0}}{200}\times\frac{\lambda_{0}}{50}$ via the two dimensional
PIC to efficiently reduce the massive 3D computation time, and, the numbers of
macro-particle per cell for electrons and ions are set to 100 and 30,
respectively.

Figure 2:  (a) Number density of emitted photons
d${}^{2}N_{\gamma}/$d$\theta_{x}$d$\varepsilon_{\gamma}$ (mrad-1 MeV-1) vs
deflection angle $\theta_{x}=p_{\gamma,x}/p_{\gamma,z}$ and $\gamma$-photon
energy $\varepsilon_{\gamma}$. (b)-(d) Angle-resolved average helicity
$\overline{h}_{\gamma}$ of emitted photons with unpolarized electrons
(electron helicity $h_{e}=0$), and, polarized electrons with $h_{e}=-1$ and
$+1$, respectively. In (c) and (d), only photons with
$\varepsilon_{\gamma}\geq 100~{}$MeV are taken into account. Solid lines
indicate the lineout along the deflection angle
$\theta_{y}=p_{\gamma,y}/p_{\gamma,z}=0$ with $\Delta\theta_{y}=0.12~{}$mrad.
(e) Blue lines indicate the emission spectra
d$N_{\gamma}/$d$\varepsilon_{\gamma}~{}$(MeV-1) vs $\varepsilon_{\gamma}$, and
black solid, dashed and dotted lines indicate $\overline{h}_{\gamma}$ vs
$\varepsilon_{\gamma}$ for the cases of $h_{e}=0$, $-1$ and $+1$,
respectively. (f) Relative deviation of energy spectra
$\Delta_{\gamma}=2\frac{\mathcal{N}_{\mathrm{pol}}-\mathcal{N}_{\mathrm{unpol}}}{\overline{\mathcal{N}_{\mathrm{pol}}+\mathcal{N}_{\mathrm{unpol}}}}$
with $\mathcal{N}\equiv~{}$d$N_{\gamma}/$d$\varepsilon_{\gamma}$, where blue
and red lines indicate $h_{e}=+1$ and $-1$, respectively; solid lines are from
the simulation results, and dash-dotted lines are fitting curves.

Simulation results of emitted CP $\gamma$-ray beam are shown in Fig. 2. The
peak intensity of emitted $\gamma$-rays is in the order of
$10^{7}~{}\mathrm{mrad^{-1}}$ for photon energies $\varepsilon_{\gamma}\geq
5~{}\mathrm{keV}$. The corresponding brilliances (average helicities
$\overline{h}_{\gamma}$) are $4.35\times 10^{21}~{}(0.37)$, $1.37\times
10^{22}~{}(-0.11)$, $5.89\times 10^{21}~{}(-0.20)$ and $1.46\times
10^{21}~{}(-0.21)~{}\mathrm{photons/(s\cdot mm^{2}\cdot mrad^{2}\cdot
0.1\%BW)}$ for $\varepsilon_{\gamma}=~{}$1 MeV, 10 MeV, 100 MeV and 200 MeV,
respectively. Here, the angular spread of the $\gamma$-ray beam
$\Delta\theta_{\gamma}$ originates from the angular spread of the electron
beam $\Delta\theta_{e}$ and the laser induced transverse momentum
$p_{e,\perp}=\sqrt{p_{e,x}^{2}+p_{e,y}^{2}}\sim a_{0}$, i.e.,
$\Delta\theta_{\gamma}\simeq\Delta\theta_{e}+2p_{e,\perp}/p_{e,z}\simeq\Delta\theta_{e}+2a_{0}/\overline{\gamma}_{e}$,
where $\overline{\gamma}_{e}$ is the average Lorentz factor of electrons. Note
that here for ultra-relativistic electrons $\gamma_{e}\gg 1$ the
$\gamma$-photons are assumed to be emitted along the electron momentum (due to
the emission solid angle $\sim 1/\gamma_{e}$). However, since high-energy
electrons (with electron energies $\varepsilon_{e}\approx 900~{}$MeV and 1400
MeV) are concentrated in a narrow angle of less than 1 mrad,
$\Delta\theta_{\gamma}\approx 2a_{0}/p_{e,z}\approx 3$-4 mrad [see Figs. 2(a)
and 3(b)]. The average helicity $\overline{h}_{\gamma}$ of $\gamma$-photons
within a narrow cone of $\Delta\theta\lesssim 1~{}$mrad can reach $-0.38$, but
for other photons of $\Delta\theta\gtrsim 3~{}$mrad, $\overline{h}_{\gamma}$
is in the range of ($-0.01$, $0.01$) [see Fig. 2(b)]. The energy-resolved
$\overline{h}_{\gamma}$ presents quite different feature, and can reach nearly
1.0 at low energies of $\sim$10 keV. $\overline{h}_{\gamma}$ quickly drops to
$\sim-0.59$ in the vicinity of 20 keV and then rises to $\sim 0.37$ around 1
MeV [see Fig. 2(e)]. In the high-energy regime $\varepsilon_{\gamma}\gtrsim
20$ MeV, $\overline{h}_{\gamma}$ saturates at $\simeq-0.2$. When employing
polarized electron bunches, the radiation spectra and angular distribution are
almost identical to the unpolarized case with an relative deviation of 0.9%,
1.7% and 3.3% around $\varepsilon_{\gamma}\approx 50,100$ and 200 MeV,
respectively [see Fig. 2(f)]. Angle-resolved $\overline{h}_{\gamma}$ of
polarized cases also show similar patterns to that in Fig. 2(b) and
consequently are excluded. For $\varepsilon_{\gamma}\gtrsim 100$ MeV, angle-
resolved $\overline{h}_{\gamma}$ is in the order of $-0.31~{}(-0.08)$ for
$h_{e}=-1~{}(+1)$ [see Figs. 3(c) and (d)]. Especially, in a narrow cone of
$\Delta\theta\leq 1~{}$mrad, $\overline{h}_{\gamma}\simeq-0.42~{}(-0.21)$ for
$h_{e}=-1(+1)$ which means that large numbers of $\gamma$-photons with
$\varepsilon_{\gamma}\simeq(20$-$40)~{}$keV are generated with small angular
spread [see Figs. 2(c)-(e)]. Besides, for $\gamma$-photons with energies of
$\varepsilon_{\gamma}\lesssim 20$ MeV, $\overline{h}_{\gamma}$ is identical to
the unpolarized case. However, in the high-energy part of
$\varepsilon_{\gamma}\gtrsim 20$ MeV, $\overline{h}_{\gamma}$ is linearly
rising (falling) as $\varepsilon_{\gamma}$ in the case of $h_{e}=+1~{}(-1$)
and reaches $\simeq 0.25~{}(-0.61)$ at the energy cutoff of
$\varepsilon_{\gamma}\simeq 500$ MeV [see Fig. 2(e)]. Such brilliant CP
$\gamma$-ray beam can be used for polarized lepton creation with
$\varepsilon_{\gamma}\gtrsim 1.022~{}\mathrm{MeV}~{}(2m_{e}c^{2})$ [9] and
photo-nuclear physics with $\varepsilon_{\gamma}\simeq 10$-$100$ MeV [10].

Figure 3:  (a) Number density of accelerated electrons $n_{e}/n_{c}$ in
$z$-$y$ plane with $x=0$. The red band denotes the aluminum plasma target. (b)
Number distribution of electrons
$\log_{10}[$d${}^{2}N_{e}/$d$\theta_{x}$d$\varepsilon_{e}$] (mrad-1 MeV-1) vs
$\theta_{x}$ and $\varepsilon_{e}$. (c) Blue and red lines indicate the
envelop and helicity of the reflected laser pulse, respectively. (d) Black
solid line indicates the emission rate [normalized by $W_{0}$ in Eq. (1)] with
respect to $\varepsilon_{\gamma}/\varepsilon_{e}$. Blue solid, dash-dotted and
dashed lines indicate $h_{\gamma}$ as absorbing 1, 2 and 4 photons,
respectively. (e) and (f) $\overline{h}_{\gamma}$ vs
$\varepsilon_{\gamma}/\varepsilon_{e}$ for different laser intensities and
electron helicities, respectively.

The physical reasons of generating the CP $\gamma$-ray beam are analyzed in
Fig. 3. In LWFA process, two plasma bubbles are excited and trap electrons
[see Fig. 3(a)] to create two isolated quasi-monoenergtic electron bunches
with peak energy $\varepsilon_{e,\mathrm{peak}}\approx$ 900 MeV and 1.4 GeV,
respectively [see Fig. 3(b)] (similar to those in [27]). The total numbers of
accelerated electrons are $4.6\times 10^{8}$ (74 pC) with energy spread of
$5\%$ at $\varepsilon_{e,\mathrm{peak}}\approx$ 900 MeV and $1.8\times 10^{9}$
(288 pC) with energy spread of $10\%$ at
$\varepsilon_{e,\mathrm{peak}}\approx$ 1.4 GeV, respectively. Angular spread
of all electrons is $\Delta\theta_{e}\approx 12~{}\mathrm{mard}$, but for
high-energy electrons near $\varepsilon_{e,\mathrm{peak}}\simeq 900~{}$MeV
(1.4 GeV), $\Delta\theta_{e}\simeq 1~{}$mrad [see Fig. 3(b)]. Due to the
inhomogeneity of the electron density, the driving laser is chirped when
propagating in the front of the wakefield [see the $y$-component of the
reflected laser in Fig. 3(c)]. After reflection, due to the frequency
chirping, the helicity of the driving laser is flipped and changes from
negative $h_{L}=-1$ (right-hand rotation) to positive $\sim 0.98$ (left-hand
rotation) (the laser helicity is calculated via
$h_{L}=2\mathrm{Im}(E_{x}^{*}E_{y})/|E_{x}^{*}E_{y}|$ [29, 26], where
$E_{x,y}$ are the complex amplitudes of the electric field in $x$ and $y$
directions, respectively, and $E_{x}^{*}$ is the complex conjugate of
$E_{x}$). Note that the deviation due to the frequency chirping is evaluated
via semi-classical calculations [30] and the average relative errors in energy
spectra and helicity are both only about $1.3\%$.

When electrons scatter with the reflected CP laser, they may absorb single or
multiple low-energy laser photons and then emit a high-energy $\gamma$-photon
via nonlinear Compton scattering. In the weakly nonlinear regime
($a_{0}\gtrsim 1$), the polarization-dependent cross section is given by [31,
26]

$W_{if}=W_{0}\sum_{n=1}^{\infty}\int_{0}^{\delta_{n}}d\delta[F_{1n}+h_{L}h_{e}F_{2n}+h_{\gamma}(h_{L}F_{3n}+h_{e}F_{4n})],$
(1)

with the photon helicity

$h_{\gamma}=\frac{h_{L}F_{3n}+h_{e}F_{4n}}{F_{1n}+h_{L}h_{e}F_{2n}},$ (2)

where $W_{0}$ = $\frac{\alpha
m_{e}^{2}a_{0}^{2}}{4\varepsilon_{\mathrm{eff}}}$, $\alpha=1/137$ is the fine
structure constant,
$\varepsilon_{\mathrm{eff}}=\varepsilon_{e}+a_{0}^{2}{\varepsilon}_{L}/\Lambda$
the effective energy of initial electron in the laser field,
$\delta=(k_{\gamma}\cdot k_{L})/(k_{L}\cdot p)$,
$\delta_{n}=n\Lambda/(1+a_{0}^{2}+n\Lambda)$ the cutoff energy fraction of
emitted photon absorbing $n$ laser photons [31], $\Lambda=2(k_{L}\cdot
p)/m_{e}^{2}$ the laser energy parameter, $p$, $k_{L}$ and $k_{\gamma}$ the
four-momenta of the initial electron, laser photon and emitted photon,
respectively, $\varepsilon_{L}$ the energy of the laser photon. $F_{kn}$
($k=1,2,3,4$) in Eq. (1) are given in detail in Refs. [31, 32]. In strongly
nonlinear Compton scattering with $a_{0}\gg 1$, the photon polarization
$h_{\gamma}$ is mainly determined by the electron helicity $h_{e}$ [33, 11],
however, in the weakly nonlinear regime, $h_{\gamma}$ not only depends on the
electron helicity $h_{e}$, but also on the scattering laser helicity $h_{L}$;
see Eq. (2).

For unpolarized electrons ($h_{e}=0$), the average helicity of emitted
$\gamma$-photons via the $n$-photon absorption channel is given by
$\overline{h}_{\gamma}=\frac{\sum_{n}F_{3n}}{\sum_{n}F_{1n}}h_{L}$. As
$\varepsilon_{\gamma}/\varepsilon_{e}\sim\delta\ll\delta_{1}\approx 0.0015$,
the emitted photons are mainly contributed by the one-photon absorption
channel ($n=1$) and $\overline{h}_{\gamma}\approx 1$, i.e., the helicities of
emitted $\gamma$-photons are solely determined by the laser helicity [see Fig.
3(e)]. For $10^{-3}\lesssim\delta\lesssim 7\times 10^{-3}$, corresponding to
multi-photon absorption with $1\leq n\leq 5$, $\overline{h}_{\gamma}$ is
rapidly oscillating due to the competition among different multi-photon
absorption channels with significant gaps of ($\delta_{n}-\delta_{n-1}$) for
small $n$ [see $h_{\gamma}$ for different channels in Fig. 3(d) and average
$\overline{h}_{\gamma}$ in Fig. 3(e)]. For $10^{-2}\lesssim\delta\lesssim
0.4$, $\overline{h}_{\gamma}$ saturates to $\sim-0.15$ [see Fig. 3(e)]. Above
theoretical analysis further confirms our simulation results in Fig. 2. For
instance, in the low-energy part ($\varepsilon_{\gamma}\lesssim 10~{}$keV)
with $\delta\simeq 1\times 10^{-5}\ll\delta_{1}$, one obtains
$\overline{h}_{\gamma}\approx 1$, while in the high-energy part
($\varepsilon_{\gamma}\gtrsim 20~{}$MeV) with $\delta\gtrsim
0.014$-$0.022\gtrsim\delta_{5}$, $\overline{h}_{\gamma}$ saturates to
$\sim-0.21$ [see Fig. 2(e)]. Moreover, since the first Compton edge (i.e., the
cutoff energy of one-photon absorption channel) occurs at
$\varepsilon_{\gamma}\simeq 1.3$-$2$ MeV, $\overline{h}_{\gamma}$ peaks around
$\varepsilon_{\gamma}\approx 1$ MeV, and, due to the mixture of
$\overline{h}_{\gamma}$ derived from $\varepsilon_{e}\approx 900$ MeV and 1.4
GeV, the valley zone of $\overline{h}_{\gamma}$ near
$10^{-4}\lesssim\delta\lesssim\delta_{1}$ (10 keV
$\lesssim\varepsilon_{\gamma}\lesssim$ 1 MeV) is broadened and the fast
oscillation near $10^{-3}\lesssim\delta\lesssim 10^{-2}$ (1.8 MeV
$\lesssim\varepsilon_{\gamma}\lesssim$ 15 MeV) is smoothed [see the comparison
between Fig. 2(e) and Fig. 3(e)]. Note that in the high-energy part
($\varepsilon_{\gamma}\gtrsim 20~{}$MeV) in Fig. 3(e), $\overline{h}_{\gamma}$
saturates to $\sim-0.21$ which lies between the analytical saturation values
of $-0.18$ and $-0.28$ for $a_{0}=3.2$ and 2.0 [see Fig. 2(e) and Fig. 3(e)],
and the slight deviation is derived from the finite-pulse effect in our
numerical simulation (by comparison, we employ a monochromatic plane wave in
the analytical estimation).

Furthermore, when electrons are polarized [34, 35], the impact of the electron
helicity $h_{e}$ on the emitted photon helicity $\overline{h}_{\gamma}$ is
negligible in the low-energy regime of
$\varepsilon_{\gamma}/\varepsilon_{e}\lesssim 10^{-2}$ [see Fig. 2(e) and Fig.
3(f)]. However, as $\varepsilon_{\gamma}/\varepsilon_{e}\gtrsim 10^{-2}$,
$\overline{h}_{\gamma}$ will linearly increase (decrease) for
$h_{e}=+1~{}(-1)$ [see Fig. 3(f)]. This analytical tendency completely
coincides with our numerical results in Fig. 2(e).

Figure 4: Impact of [(a) and (b)] $\varepsilon_{e,\mathrm{peak}}$, [(c) and
(d)] $a_{0}$, [(e) and (f)] laser pulse duration $\tau$ on
$\overline{h}_{\gamma}$. Other parameters are the follows: for the electrons
$\varepsilon_{e,\mathrm{peak}}=4$ GeV, energy spread $3\%$,
$\Delta\theta_{e}=0.2$ mrad, transverse beam size
$\sigma_{t}=2.1~{}\mu\mathrm{m}$, beam length $l_{e}=5~{}\mu\mathrm{m}$ and
total charge 300 pC; for the driving laser, the spatial profile is Gaussian
with $a_{0}=3.0$, $\tau=20~{}\mathrm{fs}$ and transverse focal radius
$w_{0}=11~{}\mu\mathrm{m}$.

For the experimental feasibility, the impact of the laser and plasma
parameters on $\overline{h}_{\gamma}$ is analyzed in Fig. 4. In the LWFA
process, the peak energy of accelerated electrons
$\varepsilon_{e,\mathrm{peak}}$ scales with the laser power $P$ and plasma
density $n_{e}$ via $\varepsilon_{e}\propto
P^{1/3}n_{e}^{-2/3}\lambda_{0}^{-4/3}$ [27]. Thus, in Figs. 4(a) and (b), the
impact of $n_{e}$ is simulated via $\varepsilon_{e}\propto n_{e}^{-2/3}$ with
$\varepsilon_{e,\mathrm{peak}}=2,4$, and 8 GeV corresponding to
$n_{e}=8.8\times 10^{17}$, $3.1\times 10^{17}$ and $1.1\times
10^{17}~{}\mathrm{cm^{-3}}$, respectively. As the emission rate $W_{0}\propto
1/\varepsilon_{e,\mathrm{eff.}}\propto 1/\varepsilon_{e}$ [see Eq. (1)], lower
(higher) energies of incident electrons will induce higher (lower) yields [see
Fig. 4(a)]. Since the $n$-photon absorption cutoff
$\varepsilon_{\gamma,\mathrm{cutoff}}=\delta_{n}\varepsilon_{e}$ will shift
due to the variation of electron energy (but the shift in $\delta_{n}$ is
quite small), the energy spectra in Fig. 4(a) and $\overline{h}_{\gamma}$ in
Fig. 4(b) will shift to left (right) as increasing (decreasing) the electron
energy. But, the final saturation $\overline{h}_{\gamma}\simeq-0.2$ remains
unchanged [see Fig. 4(b)]. The laser intensity $a_{0}$ will affect both the
emission probability and the nonlinearity of scattering process. As
$W_{0}\propto a_{0}^{2}$, the radiation intensity will increase (decrease) for
smaller (larger) $a_{0}$ [see Fig. 4(c)]. For larger $a_{0}$, $\delta_{n}$
will be smaller, thus, $\overline{h}_{\gamma}$ will shift to left, and vice
versa [see Fig. 4(d)]. In addition, increasing (decreasing) the laser pulse
$\tau$ will yield more (fewer) photons per electron [see Fig. 4(e)]. However,
due to short pulse effect (average $a_{0}$ will be lower for shorter pulse),
$|\overline{h}_{\gamma}|$ is slight higher for $\tau=10$ fs than $\tau=20$ fs
and 30 fs [see Fig. 4(f)].

In conclusion, we put forward an efficient brilliant CP $\gamma$-rays
generation method via single-shot laser plasma interaction. We find that in
the weakly nonlinear regime of Compton scattering, the helicity of emitted
photon is subjected to the interplay of the laser and electron helicities. Our
proposed method can generate $\gamma$-photons with peak brilliance of
$10^{21}$-$10^{22}~{}\mathrm{photons/(s\cdot mm^{2}\cdot mrad^{2}\cdot
0.1\%BW)}$ and average polarization degree of $\sim 21\%$-$37\%$ with moderate
laser pulses. Moreover, our method is quite stable in a wide range of laser
and plasma parameters. With proper selection of $\gamma$-photon energy and
(polarized) plasma target, highly CP brilliant $\gamma$-ray sources can be
obtained for many applications, such as, photo-nuclear researches, generation
of polarized lepton sources, etc.

## Acknowledgements

This work is supported by the National Natural Science Foundation of China
(Grants Nos. 11905169, 12022506, 11874295), and the China Postdoctoral Science
Foundation (Grant No. 2020M683447).

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