# Generation of High-Brilliance Polarized $\gamma$-Rays via Vacuum
Birefringence

Chong Lv  Department of Nuclear Physics, China Institute of Atomic Energy, P.
O. Box 275(7), Beijing 102413, 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  Yousef I. Salamin  Department of Physics, American
University of Sharjah, Sharjah, POB 26666 Sharjah, United Arab Emirates  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  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  Ruirui Xu  Department of Nuclear Physics, China Institute
of Atomic Energy, P. O. Box 275(7), Beijing 102413, China  Jian-Xing Li
<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  Department of Nuclear Physics,
China Institute of Atomic Energy, P. O. Box 275(7), Beijing 102413, China

###### Abstract

High-brilliance circularly polarized $\gamma$-photon beams are of great
significance for a wide range of applications. However, their generation
through nonlinear Compton scattering must require a high-density
longitudinally-spin-polarized electron beam and consequently is still a great
challenge. Here, we put forward a novel method to generate such
$\gamma$-photon beams via the vacuum dichroism (VD)-assisted vacuum
birefringence (VB) effect, only utilizing a well-established unpolarized
electron beam. We split a linearly polarized laser pulse into two subpulses
with the first one colliding with a dense unpolarized electron beam to
generate a linearly polarized $\gamma$-photon beam (via nonlinear Compton
scattering), which then further collides with the second subpulse and is
transformed into a circularly polarized one via the VB effect. We find that by
manipulating the relative polarization of two subpulses, one can “purify” the
polarization of the $\gamma$-photon beam via the VD effect, thereby
significantly enhancing the circular polarization of the $\gamma$-photon beam.
Due to the VD assistance, the VB effect reaches optimal when the relative
polarization is nearly $30^{\circ}$, not the widely used $45^{\circ}$ in the
common VB detection methods. Our numerical results show that one can obtain a
circularly polarized $\gamma$-photon beam with average degree of about
$30\%~{}(43\%)$ for energies above $500~{}(1000)$ MeV and brilliance of about
$10^{24}~{}(10^{23})~{}\mathrm{photons/(s\cdot mm^{2}\cdot mrad^{2}\cdot
0.1\%BW)}$ at $500~{}(1000)$ MeV by using a currently feasible laser with a
peak intensity of about $10^{22}~{}\mathrm{W/cm^{2}}$. And, it can be further
improved to above $60\%~{}(75\%)$ by increasing the laser pulse duration.
Moreover, our method is shown to be robust with respect to the laser and
electron beam parameters, and can also be used to efficiently confirm the
well-known VB effect itself, which has been predicted a very long time ago but
has not been directly observed in experiments yet.

It is conjectured that ultraintense, high-brilliance, and highly polarized
$\gamma$-photons, generated via laser-based means, will play crucial roles in
advancing contemporary research in particle physics [1], nuclear physics [2],
astrophysics [3], as well as in many applications in materials science and
medicine [4]. For example, circularly polarized (CP) $\gamma$ photons can act
as powerful probes for investigating a wide range of fundamental processes
that involve spin angular momentum, including parity violation [5], vacuum
birefringence (VB), elastic photon-photon scattering [6], and photoproduction
of mesons [7]. Currently, high-energy $\gamma$-photon beams are mainly
generated in processes involving synchrotron radiation, bremsstrahlung, and
laser-Compton effect. Compton scattering of CP light [8, 9, 10] or
bremsstrahlung employing longitudinally spin-polarized (LSP) electrons [11,
12, 13, 14] can produce CP $\gamma$ photons. However, the generated beam
brilliance is limited due to the small scattering cross-section and low
brilliance of the LSP electron beam produced in conventional accelerators.

Rapid progress in ultraintense laser technology has pushed the peak intensity
available to laboratory experiments to the level of $I_{0}\simeq
10^{23}~{}\mathrm{W/cm^{2}}$ [15, 16, 17, 18, 19, 20]. Thus, high-brilliance
$\gamma$-photon beams can be generated via nonlinear Compton scattering (NCS)
of ultraintense laser off dense electron beams produced via laser-plasma
wakefield acceleration (LWFA) [21, 22, 23]. In NCS, the yield of $\gamma$
photons increases with increasing laser intensity, and the dominant factor of
the emission polarization gradually shifts from the polarization of the
scattering laser to the electron beam due to the multiphoton absorption effect
[24, 25]. Meanwhile, an LSP electron beam is mandatory for generating a highly
CP $\gamma$-photon beam via NCS [23, 26] or bremsstrahlung [27, 28]. Also, to
attain high-brilliance $\gamma$ photons via NCS, the dense LSP electrons can
only be generated via LWFA [29, 30, 31]. However, preparing the pre-polarized
plasma and maintaining the polarization degree of the accelerated electron
beam is still challenging for LWFA. At the same time, the circular
polarization degree of such $\gamma$ photons increases linearly with energy,
but the yield decreases exponentially; only very high-energy photons can
achieve a high circular polarization degree [26]. Thus, it is still a great
challenge to generate a high-brilliance and highly CP $\gamma$-photon beam by
a feasible unpolarized electron beam.

The recent advances in laser technologies have also revived theoretical and
experimental studies of all-optical strong field quantum electrodynamics (QED)
effects [32, 33], such as radiation-reaction effects [34, 35, 36, 37, 38, 39,
40], NCS [21, 35, 22], nonlinear Breit-Wheeler process (NBW) [41, 42], and all
related polarization effects [43, 23, 26, 44, 45, 46, 47, 48]. Ultraintense
lasers also facilitate exploration of the polarization properties of the
quantum vacuum, such as VB [49] and vacuum dichroism (VD) [50, 51, 52].
Conventional detection methods of the VB effect mainly focus on the low-energy
probe photons traversing the magnetically-polarized vacuum. For instance,
Polarization of Vacuum with LASer (PVLAS) [53, 54], Observing VAcuum with
Laser (OVAL) [55], and Biréfringence Magnétique du Vide (BMV) [56] use low-
energy photons as probes, and other theoretical proposals also use high-energy
probes [57, 58]. However, the detection accuracy is still beyond the
attainable signal strength. Petawatt laser-based VB detection proposals,
involving X-ray free electron lasers (XFEL) [59, 60, 61, 62, 63, 64] or GeV-
scale $\gamma$-ray probes [50, 65, 51], also attract much attention. For
instance, VB can be studied by detecting the reduction of linear polarization
of the CP $\gamma$ photons, and vice versa, while VD can be investigated by
measuring the change of linear polarization due to $e^{+}e^{-}$ pair
production [51, 50, 66]. Recent proposals have also suggested alternative
methods for detecting VB, such as Coulomb-field assistance [67], flying
focusing [68], and plasma effects [69, 70]. Nevertheless, direct observation
of the VB effect in experiments with currently available laser systems is
still an open question.

Figure 1: (a) Conventional nonlinear Compton scattering and nonlinear Breit-
Wheeler pair production setup. (b) Generation of a CP $\gamma$-photon beam via
VB. The main pulse is split into two subpulses. In the first one, the
polarization is rotated to $\theta_{<1,2>}=30^{\circ}$ with respect to the
second subpulse (main pulse), to conduct the NCS, and the second one is used
to collide with the NCS $\gamma$-photons. $\ket{\parallel}$ and $\ket{\perp}$
denote photon polarization parallel and perpendicular to the first subpulse,
respectively. The pair production rate $W_{\pm}(\ket{\parallel,\perp})$ is
asymmetric (symmetric) for $\theta_{<1,2>}=30^{\circ}$ ($45^{\circ}$), and
thereby with (without) the polarization purification, respectively.

In this Letter, we put forward a novel method to generate high-brilliance CP
$\gamma$-photon beams via VD-assisted quantum VB effect, only utilizing a
well-established unpolarized electron beam; see Fig. 1 for a schematic. In the
conventional setup in Fig. 1(a), an intense linearly polarized (LP) laser
pulse colliding with a relativistic electron beam only generates brilliant LP
$\gamma$-photons via NCS. By contrast, in our proposed setup in Fig. 1(b), an
LP laser pulse is split into two subpulses: the first one is used to perform
NCS and generate a brilliant LP $\gamma$-photon beam; the second one is used
to polarize the vacuum, i.e., convert the LP $\gamma$-photon beam into a CP
one via the VB effect (to ensure sufficient photon production in the first
subpulse and significant VB effect in the second subpulse, the second subpulse
is chosen to be relatively stronger than the first one). Meanwhile, as shown
in the dash-dotted box in Fig. 1(b), by fine tuning the polarization angle
$\theta_{<1,2>}$ between two subpulses, we find that NBW-induced VD effect can
enhance the polarization of the $\gamma$-photon beam by changing the relative
numbers of photons in different modes $\ket{\parallel}$ and $\ket{\perp}$,
i.e., indirectly “purifying” the beam polarization. The circular polarization
degree can reach about 43% for photon energies above 1 GeV, with a peak
brilliance of $10^{23}~{}\mathrm{photons/(s\cdot mm^{2}\cdot mrad^{2}\cdot
0.1\%BW)}$ ; see details in Fig. 2. And, it can be further improved up to 75%
by increasing the laser pulse duration. Note that, the optimal
$\theta_{<1,2>}$ is about $30^{\circ}$, not the $45^{\circ}$ used in the
common VB detection methods [50, 51, 65]; see detailed explanations in Fig. 3.
Moreover, the circular polarization degree of the generated $\gamma$-photon
beam with respect to the relative polarization angle between two subpulses can
serve as a reliable indicator of the VB effect. In addition, our method is
shown to be robust with respect to the laser and electron beam parameters; see
Fig. 4.

To simulate the interactions in Fig. 1(b) self-consistently, the spin-resolved
NCS and NBW are investigated by our Monte-Carlo methods [26, 44, 45, 47],
incorporating with the VB effect by evolution of the Stokes parameters [50,
70, 47]. Importance of the NCS, NBW, and VB effects is signified by the
electron/photon nonlinear quantum parameter
$\chi_{e,\gamma}=e\hbar/(m^{2}_{e}c^{3})\sqrt{(F^{\mu\nu}p_{\mu})^{2}}\simeq
2\mathcal{E}_{e,\gamma}/(m_{e}c^{2})a_{0}(\hbar\omega_{r}/(m_{e}c^{2})$, where
$\hbar$, $-e$, $m_{e}$, and $c$ are the reduced Planck’s constant, charge and
mass of the electron, and the light speed in vacuum, respectively,
$\mathcal{E}$ and $p_{\mu}$ are energy and four-momentum vector of the
particle. $F^{\mu\nu}$, $a_{0}\equiv eE/(m_{e}c\omega_{r})$, and $E$ are the
field tensor, peak intensity, and electric field strength of the laser,
respectively, and $\omega_{r}$ the reference frequency used for normalization.
To describe the VB effect in a wide regime of $\chi_{\gamma}$, we use the
accurate refractive indices described below.

When an electron (positron) interacts with the laser field and generates a
$\gamma$ photon via NCS, a set of Stokes parameters
$\bm{\xi}=(\xi_{1},\xi_{2},\xi_{3})$ with respect to polarization basis
$\hat{e}_{1}$ and $\hat{e}_{2}$ are assigned to the photon [71, 72]. Here,
$\hat{e}_{1}\parallel\mathbf{E}_{\mathrm{red.,\perp}}$,
$\hat{e}_{2}\parallel\hat{k}\times\hat{e}_{1}$, $\hat{k}$ is the direction of
the photon momentum and
$\mathbf{E}_{\mathrm{red.}}\equiv(\mathbf{E}+\hat{k}\times\mathbf{B})$ is the
reduced electric field [73]. $\xi_{3}$ indicates the linear polarization
parallel to $\hat{e}_{1}$, $\xi_{1}$ linear polarization parallel to the
direction with $\pi/4$ relative to $\hat{e}_{1}$, and $\xi_{2}$ the circular
polarization [74]. In the subsequent evolution and pair production, the
eigenbasis should be obtained with the local field. When the eigenbasis
rotates by $\delta\psi$, the Stokes parameters transform via a rotation matrix
$\left(\begin{array}[]{l}\xi^{\prime}_{1}\\\
\xi^{\prime}_{3}\end{array}\right)=\left(\begin{array}[]{ccc}\cos
2\delta\psi&-\sin 2\delta\psi\\\ \sin 2\delta\psi&\cos
2\delta\psi\end{array}\right)\left(\begin{array}[]{l}\xi_{1}\\\
\xi_{3}\end{array}\right)$ [58, 50], where primed and unprimed Stokes
parameters are evaluated in the updated and original frames, respectively.
Note that primed ones will denote Stokes parameters in the second subpulse in
the later. Before the photon decays into an $e^{+}e^{-}$ pair, its Stokes
parameters evolve due to the VB effect. Within a single time step, the
evolution can be represented by a phase rotation
$\bigg{(}\begin{array}[]{c}\xi_{1}^{f}\\\ \xi_{2}^{f}\\\
\end{array}\bigg{)}=\bigg{(}\begin{array}[]{cc}\cos\delta\phi&-\sin\delta\phi\\\
\sin\delta\phi&\cos\delta\phi\end{array}\bigg{)}\bigg{(}\begin{array}[]{c}\xi_{1}^{i}\\\
\xi_{2}^{i}\end{array}\bigg{)}$ [58], where $i$ and $f$ indicate the initial
and final Stokes parameters of the $\gamma$ photon. Also,
$\delta\phi=\int_{0}^{L}{\rm d}l(2\pi/\lambda_{\gamma})\Delta
n(\omega_{\gamma})$ is the phase delay between two eigenstates for a photon
(wavelength $\lambda_{\gamma}$ and frequency $\omega_{\gamma}$) traversing a
polarized vacuum of length $L$, and $\Delta n=n_{\perp}-n_{\parallel}$, with
$\parallel$ and $\perp$ denoting the polarization modes parallel to
$\hat{e}_{1}$ and $\hat{e}_{2}$, respectively. $n_{\parallel,\perp}$ follow
from the real and imaginary parts
$\mathrm{Re}(n)=1-\frac{45}{4}\mathcal{D}\int_{0}^{1}d\upsilon
f(\upsilon)\left[\pi x^{4/3}\mathrm{Gi}^{\prime}(x^{2/3})\right]$ and
$\mathrm{Im}(n)=\frac{45}{4}\mathcal{D}\int_{0}^{1}d\upsilon
f(\upsilon)\left[\frac{x^{2}}{\sqrt{3}}\mathrm{K}_{2/3}\left(\frac{2}{3}x\right)\right]$
of the refractive index $n$ [75, 50, 70]. Here,
$\mathcal{D}=\frac{\alpha}{90\pi}\left(\frac{e|{\bf E}_{\rm
red,\perp}|}{m_{e}^{2}}\right)^{2}\equiv\frac{\alpha}{90\pi}\frac{\chi_{\gamma}^{2}}{\omega^{2}/m_{e}^{2}}$,
$f(\upsilon)=(1-\upsilon^{2})\left\\{\begin{matrix}\frac{1}{2}(1+\frac{1}{3}\upsilon^{2})\\\
1-\frac{1}{3}\upsilon^{2}\end{matrix}\right\\}$, with
$\mathrm{Gi}^{\prime}(x)$ being the derivative of the Scorer function,
$\mathrm{K}_{\nu}(x)$ the modified Bessel function of the second kind, and
$x=4/(1-\upsilon^{2})\chi_{\gamma}$. The first and second rows of
$f(\upsilon)$ denote the polarization states parallel to $\hat{e}_{1}$ and
$\hat{e}_{2}$, respectively, i.e., labeled with $\parallel$ and $\perp$.
Re($n$) accounts for the photon propagation, and Im($n$) accounts for the
$e^{+}e^{-}$ pair production, which is resolved by the NBW process [73, 75,
66]. In each time step for a $\gamma$-photon propagation, the Stokes
parameters will first be mapped to the instantaneous frame of $\hat{e}_{1}$,
$\hat{e}_{2}$, and the NBW is checked. If no pairs are created, the photon
will continue to propagate including the VB-induced phase delay, i.e.,
rotating the Stokes parameters.

Figure 2: (a)-(c) Photon number density
$\log_{10}($d${}^{2}(N/N_{e})/$d$\theta_{x}$d$\theta_{y})$ (rad-2), average
circular polarization $P_{\rm CP}\equiv\overline{\xi}_{2}$ and linear
polarization $P_{\rm
LP}\equiv\sqrt{\overline{\xi}^{2}_{1}+\overline{\xi}_{3}^{2}}$ with respect to
angles $\theta_{x}$ and $\theta_{y}$. (d) Circular polarization distribution
with respect to the angle $\theta_{x}$ and energy $\mathcal{E}_{\gamma}$. (e)
Photon energy spectra from primary electrons and $e^{+}e^{-}$ pairs, denoted
as $A$ and $A_{\pm}$, respectively. (f) and (g) Energy-dependent $P_{\rm LP}$
and $P_{\rm CP}$ within $|\theta_{x}|\lesssim 5$ mrad and
$|\theta_{y}|\lesssim 0.3$ mrad, respectively. $A,B$ and $C$ labels result
from $\theta_{<1,2>}=30^{\circ},45^{\circ}$, and $30^{\circ}$ (without pair
production), respectively. In (g) the thin-red, gray, and green lines indicate
the cases of only considering the second-generation $\gamma$-photons created
by pairs, depleting primary electrons before they collide the second subpulse,
and using an initial LSP electron beam, respectively.

As an example, we use a driving laser with a total intensity of $I_{0}\simeq
5.48\times 10^{22}~{}\mathrm{W/cm^{2}}$ (corresponding to $a_{0}\simeq 200$),
pulse duration $\tau=10T_{0}$ ($T_{0}=\lambda/c\simeq 3.33~{}\mathrm{fs}$),
focal radius $w=3~{}\mathrm{\mu m}$ and wavelength $\lambda=1~{}\mathrm{\mu
m}$. As shown in Fig. 1(b), the main pulse is split into the first subpulse
with $a_{1}=30$ and the second one with
$a_{2}=\sqrt{a_{0}^{2}-a_{1}^{2}}\simeq 197.7$. Polarization of the first
subpulse is then rotated to $30^{\circ}$ with respect to the original
polarization (the second one). Since these two subpulses are derived from the
same source and reflected at the same angle by mirrors, the synchronization is
automatically achieved. We use a feasible electron beam with a peak energy of
5 GeV, energy spread of $3\%$, angular spread of $0.1~{}\mathrm{mrad}$,
longitudinal size of $5~{}\mathrm{\mu m}$, transverse radius of
$0.4~{}\mathrm{\mu m}$, and number of $6.25\times 10^{7}$ (total charge of 10
pC) [76, 77, 78]. In the absence of the dense LSP electron beams from LWFA in
experiments, here we employ the well-generated unpolarized electron beams.

In all simulations, the first subpulse is assumed to be polarized along the
$x$-axis. The maximum photon yield per electron can reach
$10^{8}/\mathrm{rad}^{2}$; see Fig. 2(a). The peak degrees of circular and
linear polarization can reach about $45\%$ and $73\%$, respectively; see Figs.
2 (b)-(d). In certain applications (e.g., photoproduction of $\pi^{0}$ pairs
from nucleons [79] and deuteron photodisintegration [80]), one can obtain
about $8\times 10^{6}$ ($4\times 10^{7}$) $\gamma$ photons per shot with
$\mathcal{E}_{\gamma}\gtrsim 1000$ MeV (500 MeV) and a circular polarization
degree of approximately $43\%$ ($30\%$) [Figs. 2(e), (g) and Fig. 4]. Note
that, for the case of even using an initial LSP electron beam scattering with
the laser [26], shown as a green line in Fig. 2(g), the circular polarization
degree in the energy band of $500-1000$ MeV is about $10\%-20\%$, which is
about half of the results of our method, and only for
$\mathcal{E}_{\gamma}\gtrsim$ 3 GeV, it can surpass our method. In addition,
the circular polarization degree is mainly contributed by photons generated by
the primary electrons in the first subpulse; see Fig. 2(g). Secondary photons
from NBW $e^{+}e^{-}$ pairs generated in the second subpulse, can barely be CP
(with degree of about $1\%-2\%$) due to much smaller $\xi^{\prime}_{1}$.

As shown schematically in Fig. 1 (b), the generation of high-energy CP
$\gamma$ photons is accomplished in two successive stages: NCS in the first
subpulse, followed by VB (accompanied by NBW and NCS cascade) in the second
subpulse. In the first stage, only primary electrons emit photons, and the NBW
pair production is negligible due to low intensity of the first subpulse; see
the evolution of photon spectra in Fig. 3(a) and normalized positron number
$N_{+}$ in Fig. 3(b). Since radiation loss by an electron is proportional to
the laser energy via $\delta\gamma_{e}\propto
a^{2}\gamma_{e}^{2}\omega^{2}\tau_{L}$ [34], with a stronger scattering laser
more photons are emitted by the primary electrons in the second stage; see
Fig. 3(a), but with much lower energies than in the first stage. In the second
stage, high-energy $\gamma$ photons can create $e^{+}e^{-}$ pairs via the NBW
process; see Fig. 3(b). These pairs, in turn, can emit abundant photons which
is negligible for the final photon energy spectrum and polarization in the
energy range of $\mathcal{E}_{\gamma}>500$ MeV; see Figs. 2(e) and (g).
Besides, further cascade is suppressed due to the decrease of $\chi$.

Figure 3:  (a) and (b) Photon number density
$\log_{10}($d${}^{2}N/$d$\phi$d$\mathcal{E})$ and angle-averaged
$\overline{\xi}_{2}$ generated by the primaries. Solid lines indicate the
relative temporal profile and $\eta\equiv\omega t-kz$ is the phase,
respectively, of the laser. (c) Evolution of Stokes parameters with respect to
the instantaneous eigenbasis of a sampled photon (generated by primary
electron) with red-solid, blue-solid, light blue-dotted, and black-dashed
lines indicating $\xi_{1}$, $\xi_{3}$, $P_{\mathrm{LP}}$, and $\xi_{2}$,
respectively. The inset shows rapid switching of $\xi_{1}$ and $\xi_{3}$ due
to the change of polarization basis. (d) Evolution of relative numbers in a
sampled simulation for photons with $\mathcal{E}_{\gamma}>1$ GeV, where the
blue and red lines show results for $\theta_{<1,2>}=30^{\circ}$ and
$45^{\circ}$, and solid and dash-dotted lines represent photons generated with
$\xi_{3}>0$ and $\xi_{3}<0$, respectively. (e) Evolution of
$\overline{\xi}_{3}$ corresponding to (d), with solid and dotted lines
indicating with and without considering the NBW process, respectively.

All these photons are initially LP along the external field (either in the
first or second subpulse) with an average Stokes parameter of $\xi_{3}\simeq
0.6$ ($\mathcal{E}_{\gamma}/\mathcal{E}_{e}\simeq 0$) and then decreasing
almost linearly to 0 for $\mathcal{E}_{\gamma}/\mathcal{E}_{e}\simeq 1$. In
the whole energy range, one has $\xi_{1}\simeq\xi_{2}\simeq 0$ [26]. For
photons emitted in the first stage, there is an angle
$\delta\psi=\theta_{<1,2>}$ between the instantaneous eigenbasis and the
polarization of the second subpulse. Subsequently, due to the change of the
instantaneous eigenbasis, their Stokes parameters change as well when they
enter the second subpulse; see the Stokes parameter evolution of a sampled
photon with an energy of $\mathcal{E}_{\gamma}\simeq 2.4$ GeV and
$\chi_{\gamma}\simeq 1.8$ in Fig. 3(c). In the range of $\eta\in(235,250)$,
intensities of the subpulses are comparable, so $\xi_{1}$ and $\xi_{3}$ change
frequently; see the inset of Fig. 3(c). When the intensity of the second
subpulse exceeds that of the first one, $\xi^{\prime}_{1}$ and
$\xi^{\prime}_{3}$ stabilize to $-\sin 2\theta_{<1,2>}\xi_{3}$ and $\cos
2\theta_{<1,2>}\xi_{3}$; see $\eta\gtrsim 250$ in Fig. 3(c). In principle,
$\xi^{\prime}_{1}$ can reach a maximum for $\theta_{<1,2>}=\pm\pi/4$
($45^{\circ}$), and $\xi^{\prime}_{3}\simeq 0$. When these photons propagate
in the second subpulse, some of them decay into $e^{+}e^{-}$ pairs, and others
experience the VB effect, i.e., $\xi_{1}^{\prime}$ transfers to $\xi_{2}$; see
Fig. 3(c) with $\eta\in(350,430)$. However, for photons emitted in the second
subpulse, the polarization basis at the instant of creation, and during
propagation or pair-production, is the same
($\hat{e}_{1}\parallel\mathbf{E}_{\mathrm{red.,\perp}}$ and $\delta\psi=0$).
Thus, for these photons, $\xi^{\prime}_{1}\simeq 0$, and no VB effect occurs
(VB only transfers polarization between $\xi^{\prime}_{1}$ and $\xi_{2}$).
Besides, since energy of the photons generated in the second subpulse is
relatively smaller than that of those generated in the first subpulse, they
affect the final circular polarization degree negligibly; see Fig. 3(a). This
is also confirmed by the gray line in Fig. 2(g), where depletion of the
primary electrons does not significantly affect the final circular
polarization degree. Meanwhile, since this generation scheme is based on the
VB effect, detecting the circular polarization degree also directly confirms
VB effect of the high-energy $\gamma$ photons.

An interesting finding that sounds counterintuitive is the optimal angle
between the two subpulses is not $45^{\circ}$ which is usually used in the
common VB detection methods [50, 51, 65]. After embedding the QED cascade of
NCS and NBW, the simulations show that the case of $\theta_{<1,2>}=30^{\circ}$
can yield a much higher circular polarization degree than that in the case of
$45^{\circ}$; see Fig. 2(g). This contradiction arises when the polarization-
dependent NBW acts as a purifier of the photon polarization, i.e. the VD
effect. The circular (linear) polarization degree of NCS photons could
increase by over 30% (20%); see Figs. 2(f), (g), and 3(e). This increment in
the circular polarization degree is not an expected result in the common VB or
VD detection proposals, where an exponential decay
$e^{-(\lambda_{1}+\lambda_{2})}$ is presented before all Stokes parameters due
to the VD effect ($\lambda_{1,2}>0$ are related to the pair production rate)
[50]. As indicated in Fig. 1(b), this VD-assisted purifying mechanism stems
from the NBW polarization-dependent pair-production rate, which has the form
of $W_{\pm}\propto W_{0}+\xi^{\prime}_{3}W_{3}$, with $W_{0}$ only depending
on the photon energy and $W_{3}$ being the polarization-dependent term [72].
For photons generated by NCS, the average polarization is $\bar{\xi_{3}}\simeq
60\%$ for the low energy range and $\simeq 0$ for the high energy range. This
means that for low-energy photons, about $20\%$ and $80\%$ of photons acquire
$\xi_{3}=-1$ and $1$, and for high-energy photons, about $50\%$ and $50\%$,
respectively. When these photons enter the second subpulse, the Stokes
parameters get rotated into the new frame with $\xi^{\prime}_{3}=\cos
2\theta_{<1,2>}\xi_{3}$ and $\xi^{\prime}_{1}=\sin 2\theta_{<1,2>}\xi_{3}$.
Therefore, for $\theta_{<1,2>}=45^{\circ}$ ($\bar{\xi^{\prime}_{3}}\simeq 0$),
the pair-production rate is symmetric for $\xi^{\prime}_{1}=\pm\sin
2\theta_{<1,2>}\simeq\pm 1$. However, for $\theta_{<1,2>}\neq 45^{\circ}$
($\bar{\xi^{\prime}_{3}}\neq 0$), it is no longer symmetric for
$\xi^{\prime}_{1}\simeq\pm\sin 2\theta_{<1,2>}$; see Fig. 3(d). More photons
with $\xi_{3}>0$ and fewer ones with $\xi_{3}<0$ are left for the $30^{\circ}$
than $45^{\circ}$. This asymmetric pair-production rate results in the
increase of $\bar{\xi^{\prime}_{1}}$ and, therefore, can yield higher
$\bar{\xi_{2}}$; see the final $\bar{\xi_{3}}$
($\simeq\bar{\xi^{\prime}_{1}}/\sin 2\theta_{<1,2>}$) in Fig. 3(e). In
principle, this angle can be optimized via the $n$-step pair-production rate
$\max(\bar{\xi_{1}^{\prime}})=\max\left(\xi_{1}^{\prime}\frac{\mathcal{N}_{+}-\mathcal{N}_{-}}{\mathcal{N}_{+}+\mathcal{N}_{-}}\right)=\max\left(\sin
2\theta_{<1,2>}\frac{\mathcal{N}_{+}-\mathcal{N}_{-}}{\mathcal{N}_{+}+\mathcal{N}_{-}}\right)$,
where $N_{\pm}$ and $\mathcal{N}_{\pm}\equiv
N_{\pm}[1-(W_{0}-\xi^{\prime}_{3,\pm}W_{3})]^{n}=N_{\pm}[1-(W_{0}\mp\cos
2\theta_{<1,2>}W_{3}])]$ denote the total and non-decay photon numbers with
$\xi_{3,\pm}=\pm 1$, respectively. With current laser and electron parameters,
we obtain the optimal $\theta_{<1,2>}\simeq 30^{\circ}$, not the commonly used
$45^{\circ}$.

Figure 4: Impact of the laser parameters on the VB effect and circular
polarization degree of the generated $\gamma$ photon with respect to [(a) and
(d)] second subpulse intensity $a_{2}$, [(b) and (e)] pulse duration $\tau$,
[(c) and (f)] relative polarization angle $\theta_{<1,2>}$. The first column
is photon energy-dependent and the light-blue area with “decay” indicates no
photons can be detected (due to pair production). The black circle in (c)
denotes the anomalous polarization region. In the second column, solid and
dash-dotted lines denote the results of photons with energies
$\mathcal{E}_{\gamma}>500$ MeV and $1$ GeV, respectively. All results are
collected within an angle of $|\theta_{x,y}|\leq 5$ mrad. The total numbers
are represented by $N_{>}$. The gray bands indicate parameters used in Figs. 2
and 3. All other parameters are the same as those in Fig. 2.

To demonstrate the experimental feasibility of this method, we focus on the
dimensionless intensities $a_{1,2}$ of the two subpulses, pulse duration
$\tau$, and their relative polarization angle $\theta_{<1,2>}$. For a specific
laser facility, the total intensity is fixed, e.g., $a_{0}\simeq 200$. By
varying the relative intensity of both subpulses, the yield number is quite
stable, but the circular polarization degree decreases for $a_{1}\gtrsim 100$
[81]. For a fixed $a_{1}$ and a varied $a_{2}$ over a wide range, the circular
polarization degree continuously increases until $a_{2}\simeq 400$ and more
photons are converted into pairs. Increasing $a_{2}$ further induces the VB
phase retardation $\delta\phi\gtrsim\pi$ for high-energy photons, and one has
$\xi_{2}\simeq\sin\delta\phi\xi_{1}^{\prime}<0$; see Fig. 4(a). This pattern
also applies to the pulse duration $\tau$, as the circular polarization degree
or the phase retardation is proportional to the pulse duration $\tau$; see
Figs. 4(b) and (e), and high-energy photons acquire negative $\xi_{2}$ for
larger $\tau$. The optimal pulse duration of the currently available laser
facility is in the range of $10T_{0}-30T_{0}$. In the case of $\tau=25T_{0}$,
the circular polarization degree can reach $60\%$ and $75\%$ for photon energy
over 500 MeV and 1000 MeV, respectively. As expected, with increasing pulse
duration, more photons decay into pairs, and the decay ratio varies linearly
with $\tau$. Relative polarization angle $\theta_{<1,2>}$ of the two subpulses
is another key parameter for generating CP $\gamma$ photons via the VB effect.
As stated above, for photons emitted in the first stage, $\xi_{3}$ at creation
is first mapped into $\xi^{\prime}_{1}$ via $\theta_{<1,2>}$, and then rotated
to $\xi^{\prime}_{2}$ via the VB effect in the second stage. This mapping is
periodic with respect to $\theta_{<1,2>}$, and so is the induced circular
polarization degree; see Figs. 4(c) and (f). For photons with
$\mathcal{E}_{\gamma}<500$ MeV, the VD purifying mechanism is relatively weak,
and the circular polarization degree reaches 30% for
$\theta_{<1,2>}\in[30^{\circ},45^{\circ}]$ or $[135^{\circ},150^{\circ}]$. For
photons with $\mathcal{E}_{\gamma}\gtrsim 1$ GeV, the purifying mechanism
amplifies the circular polarization degree at $\theta_{<1,2>}\simeq
30^{\circ}$ or $150^{\circ}$, beyond the results corresponding to
$\theta_{<1,2>}\simeq 45^{\circ}$ or $135^{\circ}$. However, as shown in Fig.
4(c), an anomalous polarization regime (within the black circle) exhibits a
circular polarization degree opposite that of its neighboring region. This
anomalous CP reversal also originates from the NBW process. For
$\theta_{<1,2>}>45^{\circ}$, $\overline{\xi^{\prime}}_{3}\simeq\cos
2\theta_{<1,2>}\overline{\xi}_{3}\lesssim 0$, which indicates that photons
with $\xi_{3}=1$ exhibit larger pair-production rates than that with
$\xi_{3}=-1$. Although initially
$N_{\gamma}(\xi_{3}=1)>N_{\gamma}(\xi_{3}=-1)$, this inequality becomes weak
with the increase in photon energy and may reverse due to pair production. For
$N_{\gamma}(\xi_{3}=1)<N_{\gamma}(\xi_{3}=-1)$,
$\overline{\xi^{\prime}}_{1}\simeq\overline{\xi}_{3}\sin
2\theta_{<1,2>}\propto[N_{\gamma}(\xi_{3}=1)-N_{\gamma}(\xi_{3}=-1)]\sin
2\theta_{<1,2>}<0$, i.e., $\overline{\xi}_{2}$ will reverse sign; see the
region of $\mathcal{E}_{\gamma}\in[2,3.5]$ GeV and
$\theta_{<1,2>}\in[45^{\circ},90^{\circ}]$ in Fig. 4(c). The same analysis
applies to the region of $\mathcal{E}_{\gamma}\in[2,3.5]$ GeV and
$\theta_{<1,2>}\in[90^{\circ},135^{\circ}]$. Besides, since the total laser
intensity is the same, the number of generated photons is relatively stable.

In conclusion, we put forward a novel method for generating high-brilliance CP
$\gamma$-photon beams via the VD-assisted VB effect, which have significant
applications in material physics, nuclear physics, astrophysics, high-energy
particle physics, new physics beyond the Standard Model, etc. And, another
particularly intriguing outcome of our method is the potential confirmation of
the well-known VB effect itself, which was predicted more than eighty years
years ago but has not been directly observed in experiments yet.

Acknowledgments: This work is supported by the National Key Research and
Development (R&D) Program under Grant No. 2022YFA1602403, the National Natural
Science Foundation of China (Grants No. U2267204, No. 12022506, No. 12005305,
No. 12275209, and No. 12105217), the Foundation of Science and Technology on
Plasma Physics Laboratory (No. JCKYS2021212008), the Shaanxi Fundamental
Science Research Project for Mathematics and Physics (Grant No. 22JSY014), the
Fundamental Research Funds for Central Universities (No. xzy012023046), and
the Foundation under Grants No. FY222506000201 and No. FC232412000201.

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  * [81] See Supplemental Materials for the details on derivation of the refractive index, impact of the laser intensity and electron energy on the circular polarization of the generated $\gamma$-photon beam, impact of depletion of the primary electrons, and detection signal of the VB effect.