High-brilliance high-polarization *γ* rays based on Compton scattering are of great significance in broad areas, such as nuclear physics, high-energy physics, astrophysics, etc. However, the transfer mechanism of spin angular momentum in the transition from linear through weakly into strongly nonlinear processes is still unclear, which severely limits the simultaneous control of brilliance and polarization of high-energy *γ* rays. In this work, we clarify the transfer mechanism in the transition regions and put forward a clear way to efficiently manipulate the polarization of emitted photons. We find that to simultaneously generate high-energy, high-brilliance, and high-polarization *γ* rays, it is better to increase the laser intensity for the initially spin-polarized electron beam. However, for the case of employing the initially spin-nonpolarized electron beam, in addition to increasing laser intensity, it is also necessary to increase the energy of the electron beam. Because the *γ* photon polarization emitted through the single-photon absorption channel is mainly attributed to the spin transfer of laser photons, while in multi-photon absorption channels, the electron spin plays a major role. Moreover, we confirm that the signature of *γ*-ray polarization can be applied to observing the nonlinear effects (multi-photon absorption) of Compton scattering with moderate-intensity laser facilities.

## I. INTRODUCTION

Polarized *γ* rays are powerful probes for basic research and applications.^{1–5} For instances, polarized photons below 1 MeV enable the crystallographic probing and the biomedical imaging with femtosecond time resolution.^{4} In the energy range of several MeV to tens of MeV, polarized photo-induced nuclear reactions are highly effective in studying nuclear physics, transmutation, and astrophysics.^{6–9} Furthermore, polarized *γ* rays can be readily deployed for testing the interaction between two real photons leading to the linear Breit–Wheeler (BW) electron–positron pair production^{10–13} and photon–photon elastic scattering.^{14,15} For energy scales ranging from hundreds of MeV to GeV, polarized *γ* rays are significant for probing vacuum birefringence.^{16–19}

Traditionally, polarized *γ*-ray sources are mainly obtained via synchrotron radiation,^{4,20} bremsstrahlung,^{21,22} and linear Compton scattering (LCS).^{23–26} For synchrotron radiation facilities, higher-energy electrons are required due to insertion devices such as undulators or wigglers with a wavelength λ of a few centimeters. This requirement arises because the energy of *γ* rays, denoted as ε_{γ}, is proportional to $ \epsilon e 2 \lambda $,^{27,28} where $ \epsilon e$ is the electron energy. Compared with bremsstrahlung, LCS is characterized by good directionality, collimation, and high polarization.^{29} However, the low scattering probability sets a practical limit for the maximum flux no larger than 10^{10} photons/s.^{8}

Recently, a rapidly advanced high-power laser technique^{30–33} promoted the research of high-energy and high-brilliance polarized *γ* rays,^{34–36} where the interaction mechanism transitions from the linear to the nonlinear regime.^{37–40} At an intermediate laser intensity, polarized *γ* rays can be generated by spin-nonpolarized (SNP) electron beams via weakly nonlinear CS (NLCS).^{41–44} Furthermore, the stronger laser field scattered with initially spin-polarized (SP) electron beams enables the generation of more brilliant high-polarization *γ* rays in strong NLCS.^{45} Importantly, in the near future, experiments, such as LUXE at DESY^{46} and E320 at FACET-II,^{47} will be performed using the conventionally accelerated $ \epsilon e \u223c$ 10 GeV electron beam in collision with dozens up to hundreds of TW (corresponding to the laser invariant intensity $ a 0 \u223c 10$) laser pulses to probe the transition from a linear to a strongly nonlinear QED regime. Moreover, all-optical PW up to 10 PW laser facilities have also been commissioned or will soon be online.^{48,49} However, there is no charted transfer mechanism of spin angular momentum (SAM) in the transition from linear through weakly into strongly nonlinear CS. Therefore, the simultaneous control of brilliance and polarization of high-energy *γ* rays is still a great challenge.

In this paper, the manipulation of *γ*-ray polarization in CS employing the S(N)P ultrarelativistic electron beam is investigated. We analyze the transfer mechanism of SAM in the transition regions from linear through weakly nonlinear to strongly nonlinear Compton scattering. We find that the polarization is predominantly contributed by the electron (laser photon) spin via the multi-photon (single-photon) absorption channel (see Figs. 1 and 2). And, SNP electrons in a high-intensity laser pulse can also generate high-brilliance high-polarization *γ* photons (see Fig. 3). The transfer mechanism of SAM is also valid in the generation of vortex *γ* photons due to the same radiation dynamics as the plane wave *γ* photons.^{50} Moreover, the polarization of *γ* photons radiated by the electrons with different initial spin states proceeds in different ways in LCS and strong NLCS, respectively, which can give a clear signature of the nonlinear effects in CS with moderate-intensity laser facilities (see Fig. 5).

The paper is organized as follows. Section II gives a brief description of the radiation probability of a photon as well as the polarization in CS. Section III presents the theoretical analysis of the manipulation of *γ*-ray polarization. This is followed by the numerical analysis of the polarized *γ*-ray manipulation considering the radiation reaction during the propagation of the electrons through the laser field. The diagnosis of nonlinear effects in CS is also discussed. Section IV summarizes our results.

## II. METHODS

In this work, we consider the collision system of an ultrarelativistic electron beam colliding head-on with an intense laser field, of which the intensity is depicted by an invariant parameter $ a 0 = | e | E rms / ( m e \omega L )$,^{51,52} with the charge *e* and rest mass *m _{e}* of the electron, $ E rms = \u27e8 E 2 \u27e9 1 / 2$,

**E,**and

*ω*are the root-mean-squared (rms) electric field, electric field, and frequency of the laser field, respectively. In CS, the invariant parameter characterizing the quantum effects is $ \chi e \u2261 | e | \u2212 ( F \mu \nu p e \nu ) 2 / m e 3 \u2248 2 \omega L a 0 \epsilon e / m e 2$,

_{L}^{51,52}with the field tensor $ F \mu \nu $ and four-vector momentum $ p e = ( \epsilon e , p e )$ of the electron. Relativistic units with $ c = \u210f = 1$ are used throughout.

*γ*photon via CS. The rate for an electron emitting a single photon in a monochromatic field is given as

^{38}

^{,}

*W*

_{0}= $ \alpha m e 2 a 0 2 / ( 16 \epsilon eff ) , \u2009 \epsilon eff = \epsilon e + a 0 2 \omega L / \Lambda $ is the effective energy of the initial electron in the laser field, $ \delta = ( k L \xb7 k \gamma ) / ( k L \xb7 p e ) \u2248 \epsilon \gamma / \epsilon e$ is the energy ratio parameter, $ \Lambda = 2 ( k L \xb7 p e ) / m e 2$ is an invariant variable, $\phi $ is the azimuthal scattering angle,

*α*is the fine structure constant,

*k*and $ k \gamma $ are the four-momenta of the laser photon and emitted photon, and

_{L}*n*is the number of absorbed laser photons (

*n*th harmonics), respectively. According to the energy-momentum conservation $ q e + n k L = q e \u2032 + k \gamma $, the cutoff energy fraction (

*n*th Compton edge) is derived as $ \delta n = 2 n ( k L \xb7 p e ) m e 2 ( 1 + a 0 2 ) + 2 n ( k L \xb7 p e )$ when the emitted photon is along the propagation direction of the initial electron.

^{38,53}Here, $ q e = p e + a 0 2 m e 2 2 ( k L \xb7 p e ) k L$ and $ q e \u2032 = p e \u2032 + a 0 2 m e 2 2 ( k L \xb7 p e \u2032 ) k L$ are the four-quasimomenta of the initial and final electrons, and $ p e \u2032 = ( \epsilon e \u2032 , p e \u2032 )$ is the four-momenta of the final electron. The sums in Eq. (1) running from 1 to 3 are over the terms related to the polarization properties of final particles, which can be specified by three real numbers ( $ \xi j \u2032$ for the emitted photon and $ \zeta j \u2032$ for the final electron, with

*j*= 1, 2, and 3). Here, $ \xi j \u2032$ and $ \zeta j \u2032$ are defined with respect to the unit-basis vectors ( $ e \u0302 1 , \u2009 e \u0302 2 , \u2009 e \u0302 3$) and ( $ g \u0302 1 , \u2009 g \u0302 2 , \u2009 g \u0302 3$) for the final photon and electron, respectively. For the emitted photon, $ e \u0302 3$ is along the emitted photon momentum $ n \u0302 = k \gamma | k \gamma |$ with the emitted photon momentum $ k \gamma , \u2009 e \u0302 1 = E \u0302 \u2212 n \u0302 ( n \u0302 E \u0302 )$ and $ e \u0302 2 = e \u0302 3 \xd7 e \u0302 1$ with the unit vector $ E \u0302 = E | E |$ along the electric field

**E**. For the final electron, in the collision system, $ g \u0302 j$ (

*j*= 1, 2, 3) are $ g \u0302 1 = p e \xd7 p e \u2032 | p e \xd7 p e \u2032 | , \u2009 g \u0302 3 = p e \u2032 | p e \u2032 |$, and $ g \u0302 2 = g \u0302 3 \xd7 g \u0302 1$, respectively. The terms of $ F k n , \u2009 H k n$, and $ G k n$ ( $ k = 0 , 1 , 2 , 3$) for cases of the circularly polarized (CP) laser and the linearly polarized (LP) laser are detailed in Ref. 38. For the limit of $ a 0 2 \u2192 0$, the radiation rate coincides with the result known for the LCS case.

^{38,54,55}

^{38,56}According to the usual rules,

^{56}after summing over the polarization states of the final electron, one can obtain the Stokes parameters of the final photon as follows:

*k*= 0, 1, 2, and 3) are the average values of $ F k n$ over $\phi $.

*γ*photon helicity by merging and reorganizing formulas $ \u27e8 F k n \u27e9$ (

*k*= 0, 1, 2, and 3) in Ref. 38. For the case of the CP laser, i.e., $ \xi 1 i = \xi 3 i = 0$ and $ \xi 2 i = h L = \xb1 1 , \u2009 \u27e8 F 1 n \u27e9 = \u27e8 F 3 n \u27e9 = 0$, therefore, the emitted photons have no degree of linear polarization. Also, $ \u27e8 F 0 n \u27e9 = C 0 n + h L h e C 1 n$ and $ \u27e8 F 2 n \u27e9 = h L C 2 n + h e C 3 n$. Here, $ \xi j i$ (

*j*= 1, 2, and 3) are polarization states of the laser photon and are defined with unit-basis vectors ( $ e \u0302 1 , L , \u2009 e \u0302 2 , L , \u2009 e \u0302 3 , L$), where, $ e \u0302 3 , L$ is along the direction of laser propagation, $ e \u0302 1 , L$ is perpendicular to $ e \u0302 3 , L$, and $ e \u0302 2 , L = e \u0302 3 , L \xd7 e \u0302 1 , L$. Here,

*h*is the laser photon helicity, $ h e = 2 \zeta p e | p e |$ is the helicity of the electron with initial spin $\zeta $, and $ C k n$ (k = 0, 1, 2, and 3) are given as

_{L}^{57}

^{,}

*z*defined as

_{n}*v*for a given n is $ v n = n \Lambda / ( 1 + a 0 2 )$. Therefore, the average helicity of the emitted

*γ*photon is determined by

^{38,57}

## III. RESULTS AND DISCUSSION

### A. Theoretical analysis of manipulation of *γ*-ray polarization

Let us illustrate the transfer mechanism of SAM in CS from linear to strongly nonlinear processes by employing the CP laser (see Fig. 1). For $ a 0 \u2272 O ( 0.1 )$, there is a distinct edge at the end of the first harmonic followed by smaller probabilities of higher harmonics [see Figs. 1(a) and 1(b)]. For instance, for $ a 0 = 0.1 , \u2009 d 2 W rad d t d \delta \u2248 10 \u2212 5.31$ at $ \delta \u2248 0.192$ is two orders of magnitude smaller than $ d 2 W rad d t d \delta \u2248 10 \u2212 3.01$ at $ \delta = \delta 1 \u2248 0.190$, and for larger *δ*, $ d 2 W rad d t d \delta $ is smaller, where *W _{rad}* is the radiation probability after summing over the polarization states of the final particles and averaging over the initial electron spin with $ d 2 W rad d t d \delta = 4 W 0 C 0$, and

*δ*

_{1}is the first Compton edge of the emitted photon. Therefore, for $ a 0 \u2272 O ( 0.1 )$, the electron absorbs almost only one laser photon and radiates a

*γ*photon with $ \delta \u2264 \delta 1$, i.e., the scattering process is LCS. For $ a 0 \u223c O ( 1 )$, the process of absorbing dozens of laser photons appears with $ d 2 W rad d t d \delta \u2248 10 \u2212 2.99$ and the average number of absorbed laser photons is $ n \xaf = \u2211 n n \xb7 C 0 n \u2211 n C 0 n \u2248 10$ (corresponding to $ \delta \u2248 0.4$), termed as weak NLCS. As

*a*

_{0}increases to

*O*(10), due to $ W rad \u221d a 0 2$, the electron will have a greater probability of absorbing thousands of laser photons to emit a higher energy

*γ*photon with $ d 2 W rad d t d \delta \u2248 10 \u2212 1.22$ and $ n \xaf \u2248 1000$ (corresponding to $ \delta \u2248 0.55$), described as strong NLCS. Therefore, electrons will radiate more brilliant

*γ*rays in a higher intensity laser field. Note that the electron quivers in the transverse direction under the high-intensity laser field with $ a 0 \u2273 1$. Therefore, the electron obtains an effective mass $ m e * = m e 1 + a 0 2$ while propagating in the laser field.

^{53}Hence, the

*n*th Compton edge

*δ*can also be expressed as $ \delta n = 2 n ( k L \xb7 p e ) m e * 2 + 2 n ( k L \xb7 p e )$ and their gaps of harmonic cutoffs are obtained as $ \Delta \delta n = \delta n \u2212 \delta n \u2212 1 \u221d 1 / m e * 2$. Then, as the laser intensity

_{n}*a*

_{0}increases, the electron effective mass $ m e *$ increases, and the edge locations

*δ*of the emission spectrum and their gaps $ \Delta \delta n$ are both decreasing. As a result, in a higher intensity laser field, the energy cutoffs of the emitted photon shift toward the lower energy region for absorbing a certain number of laser photons, and the harmonic structures become more closely spaced. Therefore, the distinct harmonic structures become smoother due to the increased effective mass of the electron in stronger NLCS. Note that the trends of harmonic cutoffs as well as the laser intensity are crucial for manipulating high-quality polarized

_{n}*γ*photons and testing the nonlinear effects in strong NLCS (see Figs. 2 and 5, which will be discussed later). In addition, the impact of electron spin on the radiation rate weakens as the laser intensity increases. For instances, for $ a 0 = 0.1$, the radiation probability of the longitudinally spin-polarized (LSP) electrons (satisfying $ h L h e = \u2212 1$) at the first edge can be increased by $ \Delta W i = ( W i LSP \u2212 W i SNP ) / W i SNP = \u2212 C 1 C 0 \u2248 20 %$, yet, for $ a 0 = 10 , \u2009 \Delta W i \u223c 0$ [see Fig. 1(c)], where $ W i LSP$ ( $ W i SNP$) is the radiation probability of LSP (SNP) electrons with $ d 2 W i LSP d t d \delta = 4 W 0 ( C 0 + h L h e C 1 )$.

The electron spin plays an increasingly significant role in the transfer of SAM with stronger nonlinear effects. The entanglement term $ C 1 C 0$ of the laser helicity and electron spin in the *γ*-photon helicity Eq. (10) mainly operates at the end of the first few harmonics and decreases as *a*_{0} increases. While $ C 1 C 0$ can reach to −0.68 at $ \delta = 0.7$ for $ a 0 = 0.1$, it is irrelevant due to the relatively low radiation probability with $ d 2 W rad d t d \delta \u2248 10 \u2212 25$ [see Figs. 1(a) and 1(c)]. Therefore, the helicity of emitted *γ* photon is mainly related to the independent laser helicity term *C*_{2} and the electron spin term *C*_{3}. For emitted photons with $ \delta \u2272 \delta 1$ at any *a*_{0}, the laser helicity contribution $ | C 2 | | C 2 | + C 3 \u2248 1$ and correspondingly the electron spin contribution $ C 3 | C 2 | + C 3 = 1 \u2212 | C 2 | | C 2 | + C 3 \u2248 0$, i.e., regardless of whether the scattering process is linear or not, the average helicity of *γ* photons via the single-photon absorption channel is almost completely determined by the laser [see Fig. 1(d)]. When electrons simultaneously absorb dozens of laser photons, higher energy *γ* photons are emitted, where the contribution of laser helicity gradually decreases and the electron spin comes into play, i.e., the SAM transfer enters a competitive stage of the laser and electron for controlling the *γ*-photon polarization [see the black-dotted line in Fig. 1(d), which almost corresponds to $ n \xaf \u2248 10$ in Fig. 1(b)]. As the number of absorbed laser photons continues to increase, the electron spin plays a dominant role in the SAM transfer, for instance, as $ n \xaf$ increases up to 1000, $ | C 2 | | C 2 | + C 3 \u2248 0.05$ and $ C 3 | C 2 | + C 3 \u2248 0.95$ at $ \delta = 0.55$ for $ a 0 = 10$.

The transfer mechanism of SAM indicates a way to manipulate the polarization of γ rays. For LSP ( $ h e = \u2212 1$) electrons scattered with the CP (*h _{L}* = 1) laser, in LCS ( $ a 0 \u2272 0.1$), the emitted photons are almost completely radiated via the single-photon absorption channel and helicities are mainly contributed by the laser [see Fig. 2(a)]. In the low-energy parts of the first harmonic,

*γ*photons are radiated in the forward scattering with average helicity $ h \xaf \gamma \u2243 h L = 1$, while near the edge $ h \xaf \gamma \u2243 \u2212 h L = \u2212 1$ via the backward scattering. This behavior, obtained as the limit of a small number of absorbed photons during an NLCS event, also exists in the LCS.

^{58}As

*a*

_{0}increases, the interaction process transitions into the NLCS regime, due to $ \delta 1 \u221d 1 / a 0 2$, the energy regions of $ h \xaf \gamma \u223c 1$ of radiated photons via the single-photon absorption channel tend toward lower energy areas with smaller

*δ*. For a certain $ a 0 \u2273 1$,

*γ*photons with high energy ( $ \delta \u2273 \delta 100$) obtain increasingly high $ h \xaf \gamma $ transferred by the electron spin. For the case of

*h*= 1, the helicity behavior is similar to that of $ h e = \u2212 1$ but slightly different especially at the end of the first few harmonics with $ a 0 \u2272 1$ due to the entanglement term of the laser helicity and electron spin [see Fig. 6(a) in Appendix A]. If electrons are SNP (

_{e}*h*= 0), $ h \xaf \gamma $ of

_{e}*γ*photons is only provided by the laser helicity. Therefore, $ h \xaf \gamma $ of high-energy

*γ*photons via multi-photon absorption channels ( $ n \xaf \u2273 10$) gradually decreases as

*a*

_{0}increases, for instance, at $ \delta = 0.3 , \u2009 h \xaf \gamma \u2248 \u2212 0.40$ and −0.03 for $ a 0 = 1$ and 10, respectively [see Fig. 2(b)]. Note that in the NLCS, the polarization of emitted photons in the linear region ( $ \delta \u2272 \delta 1$) is averaged over all possible multi-photon absorption channels (see Fig. 7 in Appendix B). For instance, $ h \xaf \gamma $ shifts up to −0.76 from −1 near the first harmonic edge ( $ \delta \u2248 \delta 1$) for $ a 0 = 1$, while, $ h \xaf \gamma \u2248 \u2212 0.53$ for $ a 0 = 10$ [see Figs. 7(d) and 7(f) in Appendix B].

The above-mentioned analytical discussions are based on the single-photon emission of electrons with $ \epsilon e = 10$ GeV. However, the transfer mechanism also holds true for other electron energies. The slight difference lies in the positions of harmonic edges and the average number of absorbed laser photons (see Fig. 8 in Appendix C). Moreover, for LP *γ* photons generated in the LP laser, the competition between the laser and electron for controlling the *γ* photon polarization also exists (see Fig. 10 in Appendix E).

To summarize, if electrons are SP, one can increase the laser intensity to generate high-energy, high-brilliance, and high-polarization *γ* rays. However, for SNP electrons, due to the energy regions $ \delta \u2272 \delta 1 \u221d \epsilon e$ for a certain *a*_{0}, apart from the high-intensity laser, higher energy electrons are also required to produce the similar high-quality *γ* rays, where the polarization is mainly contributed by the laser via the single-photon absorption channel. Importantly, when the scattering process satisfies the rotational symmetry around the collision axis, e.g., in the case of CP laser, the angular momentum conservation holds and one could expect the generation of vortex *γ* photons with intrinsic orbital angular momentum (OAM) via multi-photon absorption channels. In the monochromatic CP laser, *γ* photons corresponding to the *n*th harmonic carry OAM $ l \gamma = n h L + h e \u2212 h e \u2032 \u2212 h \gamma $ with the final electron helicity $ h e \u2032$. Since the radiation dynamics of the vortex *γ* photons are the same as those of the plane wave *γ* photons,^{50} the transfer mechanism of SAM discussed above is still valid. However, the electrons are required to possess coherence in the transverse plane with axial symmetry in order to attain vortex *γ* photons. In theory, such a transverse coherence for the final electron can be achieved using the generalized measurement. In this scenario, the momentum azimuthal angle of the final electron is not measured, such that the final *γ* photon assumes a vortex state carrying OAM.^{59,60}

The SAM transfer mechanisms of LCS and weak NLCS presented above are consistent with the existing experimental data. However, the SAM transfer mechanism of strong NLCS will be verified as the fast-paced development of high-intensity laser facilities. For LCS, polarized *γ* photons have been generated with low-intensity laser devices,^{61} such as BNL-ATF,^{23} SPring-8,^{24} NewSUBARU,^{25} UVSOR-III,^{26} etc. In these LCS experiments mentioned above, the polarization degree of the emitted photons near the peak of the spectra exceeds 99%. The experimental results coincide with the SAM transfer mechanism in LCS, indicating that the polarization of the emitted photons is mainly transferred by the laser spin. Near the edge of the first harmonic with $ \delta \u2248 \delta 1$, the average polarization states of *γ* photons are $ | \xi \xaf 2 f | = \xi 2 i = 100 %$ and $ \xi \xaf 1 f = \xi \xaf 3 f = 0$ for the CP laser, and $ \xi \xaf 1 f \u2248 \xi \xaf 2 f = 0$ and $ \xi \xaf 3 f \u2248 \xi 3 i = 100 %$ for the LP laser [see Figs. 2(b) and 7(b) in Appendix B and Fig. 10(b) in Appendix E]. For weak NLCS, the CP- or LP-emitted photons have been obtained experimentally in an all-optical moderate-intensity laser setup ( $ a 0 \u2248 0.86 \u2272 1$) by tuning the polarization state of the driving laser,^{62} where the average linear polarization degree of the emitted photons is 75 (±3)% within 18.2 mrad. The experimental results are consistent with the SAM transfer mechanism in weak NLCS, wherein the polarization of emitted photons is still mainly transferred by the laser spin, albeit with a slightly decreased contribution [see Figs. 2(b) and 7(d) in Appendix B]. For strong NLCS, the transfer mechanism of the *γ*-photon polarization from the electron spin has not been confirmed. Subsequently, the verification plan is proposed based on current and upcoming high-intensity ( $ a 0 \u2273 10$) laser setups, utilizing either conventional accelerator setups, such as LUXE at DESY^{46} and E320 at FACET-II,^{47} or all-optical configurations in the colliding-beams geometry, such as ELI-NP,^{63} SILEX-II,^{64} Apollon,^{65} SULF,^{66} etc. We allow the high-intensity laser to collide head-on with the LSP and SNP electron beams, using each setup, respectively, and then measure the polarization degree of the high-energy *γ* photons. If the polarization degree of high-energy *γ* photons emitted by the LSP electrons increases with energy (from the middle to the rear segments of the spectra), while in the case of the SNP electrons, the high-energy *γ* photons are nearly nonpolarized [see Figs. 2 and 3(h)], then the transfer mechanism of SAM in strong NLCS will be verified. The high-energy LSP electrons can be generated in a controllable laser wakefield acceleration scheme,^{67–73} in which a pre-polarized gas target can be prepared experimentally via ultraviolet photodissociation of a hydrogen-halogen molecule.^{74–77}

### B. Numerical analysis of manipulation of *γ*-ray polarization

*γ*rays via different CS mechanisms in a realistic laser pulse is detailed in Fig. 3. In the region of $ a 0 \u2272 10$, we employ the method of locally monochromatic approximation (LMA) to simulate the scattering process

^{57}[For the strong NLCS ( $ a 0 \u226b 1$), the computational cost of evaluating very high-order harmonics restricts the implementation of the LMA and the locally constant field approximation (LCFA) method is appropriate].

^{49,51,52,78}The LMA method, which has been benchmarked against exact calculations in pulses and has been extended to include chirped pulses under a “slowly varying chirp” approximation,

^{79,80}treats the fast dynamics related to the carrier frequency of the plane wave exactly, but uses a local expansion to describe the slow dynamics associated with the pulse envelope.

^{81}The focused Gaussian left-hand CP laser pulse (

*h*= 1) propagating along the –

_{L}*z*direction is employed with peak intensity $ a 0 , peak = 1$, wavelength $ \lambda 0 = 0.8 \u2009 \mu $m, pulse duration $ \tau = 10 \u2009 T 0$ (full width at half maximum) with the period

*T*

_{0}, and beam waist (the radius at which the intensity falls to 1/

*e*

^{2}of its central value) $ w 0 = 5 \u2009 \mu $m. The normalized amplitude is

^{79}

^{,}

*t*is the pulse shift in the coordinates to make sure that the electrons are initially outside of the laser pulse, and we neglect the wavefront curvature and Gouy phase. On the longitudinal propagation axis, let $ X = ( 0 , r \u22a5 , 0 )$ be the origin of the laser, i.e., the initial time and position, thus, one gets $ z \u2248 t shift / 2$ and $ f ( t + z ) \u2248$ 1 at $ t = t shift / 2$. The head-on collision (along $ + z$ direction) cylindrical electron beam is uniformly distributed between $ z = \u2212 l e$ and $ z = 0 \u2009 \mu $m in the longitudinal direction with the beam length $ l e = 3 \u2009 \mu $m, initial kinetic energy $ \epsilon \xaf e = 10$ GeV, energy spread $ \Delta \epsilon e / \epsilon \xaf e = 12 %$, and electron number $ N e = 5 \xd7 10 6$. The distribution in the transverse direction has a Gaussian profile with the radius $ r e = 2 \u2009 \mu $m. The collision polar angle with respect to the laser propagation direction is $ \theta e = 180 \xb0$ with angular divergence $ \Delta \theta e = 0.2$ mrad. Such electron bunches can be obtained via laser wakefield acceleration.

_{shift}^{82–85}To ensure that the electron beam is initially outside of the laser pulse, we have $ t shift / 2 = 4 \sigma \tau = 4 \tau / 2 \u2009 ln \u2009 2 \u2248 34 \u2009 T 0$.

In the laser pulse, the intensity sensed by electrons changes in real time and is strongest at $ t \u2248 35 \u2009 T 0$, which is larger than $ t shift / 2 \u2248 34 \u2009 T 0$ due to the electron beam with a certain length, and copious *γ* rays are radiated at $ t \u2248 35 \u2009 T 0$ [see Fig. 3(a)]. According to Eq. (11), the normalized amplitude $ a 0 , gauss$ of the focused Gaussian laser not only depends on the time *t* but also on the space positions ( $ r \u22a5 , z$) in longitudinal and transverse directions. Therefore, we calculate the average intensity $ a \xaf 0$ sensed by the spatially distributed electron beam in real time. For SNP electrons, the helicities of *γ* photons come from the laser. In the front of the laser pulse with $ t \u2272 23 \u2009 T 0$ and the average intensity $ a \xaf 0 \u2272 0.1$, electrons only radiate a small amount of high polarized *γ* photons due to the low-radiation probability of the LCS regime [see Figs. 3(a) and 3(b)]. The polarization directions of *γ* photons with $ \epsilon \gamma \u2273 1$ GeV ( $ \delta \u2273 0.1$) and $ \epsilon \gamma < 1$ GeV ( $ \delta < 0.1$) are opposite [see Figs. 2(b) and 3(b)]. For $ 23 \u2009 T 0 \u2272 \u2009 t \u2272 35 \u2009 T 0 , \u2009 a \xaf 0$ gradually increases to 1 and since $ \delta 1 \u221d 1 / a 0 2$ the energy regions of $ h \xaf \gamma \u2243 \u2212 h L = \u2212 1$ decrease from $ \epsilon \gamma \u2272 2$ GeV to $ \u2272 1$ GeV. There are high-energy *γ* photons radiated by electrons absorbing dozens of laser photons and $ h \xaf \gamma $ of GeV *γ* photons decreases from $ \u223c \u2212 1$ to $ \u223c \u2212 0.43$ due to the decline of the transfer efficiency of the driving laser. As *t* continues to increase, $ a \xaf 0$ begins to decrease and the trends of radiated spectra and helicities of *γ* photons are basically symmetrical with those of $ t \u2272 35 \u2009 T 0$. Therefore, the final $ h \xaf \gamma $ of *γ* photons is the average result of the laser pulse effect, and the first few harmonics are smoothed out and the $ h \xaf \gamma $ of *γ* photons with $ \epsilon \gamma \u2273 2$ GeV is about −0.53 which is higher than $ h \xaf \gamma \u2248 \u2212 0.43$ of the plane wave with $ a 0 = 1$ [see Fig. 2(b) and the green-solid line in Fig. 3(h)]. For LSP electrons,^{67–73} the helicities of *γ* photons contributed by the CP laser are almost identical to the case of SNP electrons for all energy regions radiated at both ends of the laser pulse with $ t \u2009 \u2272 23 \u2009 T 0$ and $ t \u2273 49 \u2009 T 0$ and $ \epsilon \gamma \u2272 1$ GeV radiated at the middle of the laser pulse with $ 23 \u2009 T 0 \u2272 t \u2272 49 \u2009 T 0$. However, at the middle of the laser pulse, the *γ* photons with $ \epsilon \gamma \u2273 1$ GeV receive helicities from the scattering electrons and $ h \xaf \gamma $ linearly falls (rises) as $ \epsilon \gamma $ increases for $ h e = \u2212 1$ (*h _{e}* = 1) [see Fig. 3(c), the blue-dash-dotted line in Fig. 3(h), and Figs. 6(b) and 6(c) in Appendix A]. In addition, the relative deviation of energy spectra $ \Delta \gamma $ can reach 30% (−30%) for $ h e = \u2212 1$ (

*h*= 1) [see the red dash-dotted line in Fig. 3(g)].

_{e}Usually, one can increase the laser intensity to obtain high-brilliance *γ* rays. For instance, the *γ*-photon yields in CS with $ a 0 , peak = 10$ are two to three orders of magnitude higher than that of the $ a 0 , peak = 1$ case [see Figs. 3(d) and 3(g)]. The yield and helicity distributions are not completely symmetric due to the radiation reaction. Importantly, SNP electrons can also generate high-brilliance high-polarization *γ* rays with the energy scale of keV to MeV [see Fig. 3(e) and the black-solid line in Fig. 3(h)]. Similar to the $ a 0 , peak = 1$ case, one gets $ | h \xaf \gamma | \u2248 h L = 1$ for *γ* photons via linear and weakly nonlinear CS at the front and rear of the laser pulse. During $ 23 \u2009 T 0 \u2272 t \u2272 49 \u2009 T 0$, the scattering sequentially undergoes weakly nonlinear, strongly nonlinear, and again weakly nonlinear processes. At about 35*T*_{0}, due to $ \delta 1 \u221d 1 / a 0 2$, the energy regions of $ h \xaf \gamma \u2243 h L = 1$ drop below 100 keV, which is much smaller than that in the plane wave case ( $ \epsilon \gamma = \delta \epsilon e \u2272$ 10 MeV) in Fig. 2, due to radiation reaction. Meanwhile, electrons experience the strongest field and have larger probabilities of absorbing thousands of laser photons to radiate higher energy *γ* photons [see Fig. 3(g)]. For $ \epsilon \gamma \u2273 1$ GeV, $ h \xaf \gamma $ decreases from −0.28 at $ t \u2248 23 \u2009 T 0$ to −0.05 at $ t \u2248 35 \u2009 T 0$ since the transfer efficiency of laser helicity decreases as *a*_{0} increases. The corresponding brilliances $B$ (average helicities $ h \xaf \gamma $) are $ 1.49 \xd7 10 20$ (0.98), $ 1.14 \xd7 10 21$ (0.59), and $ 5.23 \xd7 10 21$ (0.16) photons/(s $\xb7$ mm^{2} $\xb7$ mrad^{2} $\xb7$ 0.1% bandwidth) for $ \epsilon \gamma = 100$ keV, 1, and 10 MeV, respectively. Such polarized beams can be readily deployed to the near-future photon–photon colliders, e.g., the Beijing electron–positron collider (BEPC) in IHEP,^{86} for studying linear Breit–Wheeler pair production^{13} and elastic photon–photon scattering.^{15} Note that the difference in the average helicity of the emitted photons in the linear region ( $ \epsilon \gamma \u2272 \epsilon \xaf e \delta 1$) between LCS, weak NLCS, and strong NLCS can be attributed to the combined effects of multi-photon absorption channels and radiation reaction. The effects of multi-photon absorption channels lead to the average helicity of emitted photons in the linear region shifting upward compared to that of the LCS (see Fig. 7 in Appendix B). The helicity of emitted photons is averaged over those photons radiated by the electrons with a wide energy spread caused by the radiation reaction (see Fig. 9 in Appendix D). For LSP electrons, the *γ* photons spectra are almost identical to the case of *h _{e}* = 0 with $ \Delta \gamma \u2248 5 %$ at $ \epsilon \gamma = 9$ GeV [see the red-solid line in Fig. 3(g)]. However, electrons transfer SAM to high-energy

*γ*photons with $ \epsilon \gamma \u2273 1$ GeV via strong NLCS and $ h \xaf \gamma $ is linearly falling (rising) as $ \epsilon \gamma $ in the case of $ h e = \u2212 1$ (

*h*= 1) and reaches $ \u2243 \u2212 1$ (1) at $ \epsilon \gamma = 9$ GeV [see Fig. 3(f), the red-dash-dotted line in Fig. 3(h), and Figs. 6(d) and 6(e) in Appendix A]. Note that with $ a 0 , peak = 10$, the LCFA method commonly used to simulate strong NLCS performs poorly not only in radiation spectra but also in

_{e}*γ*-photon polarization (see Fig. 11 in Appendix F).

The generation of the second-generation *γ* photons radiated by positron–electron pairs, which are produced by the first-generation *γ* photons during the propagation in laser pulse via nonlinear BW (NLBW), is shown in Fig. 4. The decay probability of *γ* photons into positron–electron pairs in a laser field is characterized by an invariant parameter $ \chi \gamma \u2261 | e | \u2212 ( F \mu \nu k \gamma \nu ) 2 / m e 3 \u2248 2 a 0 \epsilon \gamma \omega L / m e 2$ with the four-vector momentum $ k \gamma = ( \epsilon \gamma , k \gamma )$ of the *γ* photon. For $ a 0 \u2272 10$ and $ \epsilon \gamma \u2272 10$ GeV, $ \chi \gamma \u2272 1.18$, therefore, only a very small fraction of the first-generation *γ* photons generated by high-energy electrons via NLCS decay into positron–electron pairs.^{51} For instance, the ratio of the number of positrons to one of the first-generation *γ* photons is $ N p / N \gamma 1 st \u2248 3.3 \xd7 10 \u2212 4$, where *N _{p}* and $ N \gamma 1 st$ are the total numbers of the positrons via NLBW and the first-generation

*γ*photons via NLCS, respectively [spectra of the positrons $ d N p / d \epsilon p$ and the first-generation

*γ*photons $ d N \gamma 1 st / d \epsilon \gamma $ are shown in Figs. 4(a) and 4(b), respectively]. As a result, the proportion $ P = d N \gamma 2 nd / d \epsilon \gamma d N \gamma 1 st / d \epsilon \gamma + d N \gamma 2 nd / d \epsilon \gamma \u2272 0.03$ of the second-generation

*γ*photons radiated by positron–electron pairs is very small, where $ d N \gamma 2 nd / d \epsilon \gamma $ is the number of the second-generation

*γ*photons [see Figs. 4(b) and 4(c)]. Therefore, the final yields and polarization of

*γ*photons are almost the same as those of the first-generation

*γ*photons.

In addition, the vacuum birefringence (VB) effect on the polarization of *γ* photons can also be neglected. For CP *γ* photons radiated in CP laser pulse, the unit-basis vector ( $ e \u0302 1 , \u2009 e \u0302 2 , \u2009 e \u0302 3$) rotates due to the rotation of the polarization direction of the CP laser field, and then the VB effect cancels out. For LP *γ* photons, since the polarization of the emitted photons is mainly along the polarization direction of the LP laser field, $ \xi 1 f \u2248 0$ is very small. The VB effect is related to $ \xi 1 f$ and $ \xi 2 f$ but not to $ \xi 3 f$.^{17} Therefore, the VB effect is not included in the calculation code.

### C. Diagnostic of nonlinear effects in CS

As mentioned above, the nonlinear effects increase with the laser intensity. In experiments to test the nonlinear effects of CS, the main methods involve measuring the energy spectra of scattered electrons or emitted photons. However, visible harmonic structures of the final particles cannot be detected in strong NLCS as seen in experiments conducted for weak NLCS at SLAC.^{37} The reason is that the gaps of harmonic cutoffs decrease as *a*_{0} increases and the harmonic structures become smoother in stronger NLCS. In addition, the method of measuring *γ*-photon energy spectra requires the use of the highest possible laser intensity to generate high-energy *γ* photons exceeding the energy cutoffs evaluated by the LCS.^{34,36} Therefore, detecting the nonlinear effects of CS in the experiment remains a crucial issue. The relevant theoretical studies predict different behaviors of the nonlinear effects in CS compared with the linear case,^{87,88} also primarily focusing on the energy spectra of emitted photons. However, the polarization of the emitted photons introduces another dimension to the investigation of nonlinear effects in CS.

An alternative method is proposed for testing the nonlinear effects in CS by detecting the polarization degree of *γ* photons (see Fig. 5). The polarization behavior of *γ* photons is studied in such a setup, where a moderate-intensity laser pulse collides head-on with an electron beam, considering the energy and angular spreads and the radiation reaction effect. In LCS, the first cutoff is $ \delta LCS = \Lambda / ( 1 + \Lambda ) \u2248 4 \epsilon e \epsilon L / ( m e 2 + 4 \epsilon e \epsilon L )$ and higher harmonics (*n* > 1) disappear. Therefore, there is a distinct edge at $ \epsilon \gamma , edge = \delta LCS \epsilon e = 1.9$ GeV, which is smaller than 2.5 GeV of the numerical result due to the initial electron energy spread [see the red-solid line in Fig. 5(c)]. In addition, the helicities of the emitted photons are almost completely derived from the laser, and due to the opposite scattering direction, $ h \xaf \gamma $ at both ends of the spectra are anti-parallel. Because of the radiation reaction, $ \epsilon \gamma , edge$ and the turning points from $ h \xaf \gamma \u2248 1$ to −1 decrease as *t* increases [see Figs. 5(a) and 5(b)]. As a result, for 10 MeV $ \u2272 \epsilon \gamma \u2272 1$ GeV, $ h \xaf \gamma $ is averaged to a smaller polarization degree [see the green-solid line in Fig. 5(d)]. Therefore, the strongly nonlinear effects not only broaden the spectra but also change the helicity distribution of radiated *γ* photons compared with the LCS process [see Figs. 5(c) and 5(d)]. Importantly, the polarization of *γ* photons exhibits notably distinct behaviors even in the overlapped energy regions predicted by the LCS and strong NLCS, respectively. Therefore, the method of detecting the polarization degree of *γ* photons reduces the need for higher energy *γ* photons beyond the LCS cutoff. As a result, the polarization degree of *γ* photons can serve as a diagnostic tool to test the nonlinear effects in CS with the use of currently available laser facilities. Moreover, if electrons are LSP, the nonlinear signals will be more sensitive. For instance, the electron spin hardly affects the *γ*-photon helicity in the LCS process, while, in strong NLCS, LSP electrons will absorb abundant laser photons and transfer SAM to high-energy *γ* photons [see Fig. 5(d)].

Therefore, the transfer mechanism of SAM can offer an alternative approach for testing the nonlinear effects in CS using presently accessible laser facilities. For instance, the collision of SNP high-energy electrons with a moderate-intensity CP laser pulse ( $ a 0 \u223c 10$) will enable the experimental study of nonlinear effects in CS by detecting the average circular polarization degree $ \xi \xaf 2 f$ of *γ* photons. If $ \xi \xaf 2 f$ transitions gradually from 100% to −100% as the energy of the emitted photons increases, the scattering processes of electrons will exhibit LCS. Conversely, if $ \xi \xaf 2 f$ decreases from 100% to approximately 0, the nonlinear effects in CS will be tested [see Fig. 5(d)].

## IV. CONCLUSION

In conclusion, we have investigated the transfer mechanism of SAM in CS from linear through weakly into strongly nonlinear regimes. We have found that the SAM transfer is related to the number of absorbed laser photons. The polarization of emitted photons via the single-photon absorption channel is mainly contributed by the driving laser, and the contribution gradually decreases as the number of scattered laser photons increases, meanwhile, the electron spin plays an increasingly important role in the SAM transfer. The transfer mechanism gives us a clear direction for simultaneously manipulating the brilliance and polarization of high-energy *γ* rays, which can be applied to the study of nuclear physics, high-energy physics, astrophysics, etc. Moreover, we have shown that detecting the SAM of particles can help us observe the nonlinear effects of strong-field QED processes with currently feasible laser facilities.

## ACKNOWLEDGMENTS

The work was supported by the National Natural Science Foundation of China (Grants Nos. U2267204 and U2241281), and the Foundation of Science and Technology on Plasma Physics Laboratory (Grant No. JCKYS2021212008), and the Shaanxi Fundamental Science Research Project for Mathematics and Physics (Grant No. 22JSY014).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Yu Wang:** Conceptualization (equal); Data curation (equal); Investigation (equal); Writing – original draft (equal). **Mamutjan Ababekri:** Conceptualization (equal); Formal analysis (equal); Writing – review & editing (equal). **Feng Wan:** Conceptualization (equal); Formal analysis (equal); Writing – review & editing (equal). **Jia-Xing Wen:** Formal analysis (equal); Writing – review & editing (equal). **Wen-Qing Wei:** Formal analysis (equal); Writing – review & editing (equal). **Zhong-Peng Li:** Formal analysis (equal); Writing – review & editing (equal). **Hai-Tao Kang:** Formal analysis (equal); Writing – review & editing (equal). **Bo Zhang:** Conceptualization (equal); Funding acquisition (equal); Writing – review & editing (equal). **Yong-Tao Zhao:** Formal analysis (equal); Writing – review & editing (equal). **Wei-Min Zhou:** Conceptualization (equal); Funding acquisition (equal); Writing – review & editing (equal). **Jian-Xing Li:** Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Project administration (equal); Supervision (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 A: IMPACT OF ELECTRON SPIN (*h*_{e} = 1) ON THE AVERAGE HELICITY $ h \xaf \gamma $ OF *γ* PHOTONS

_{e}

Impact of electron spin (*h _{e}* = 1) on $ h \xaf \gamma $ of

*γ*photons in the transition regime is shown in Fig. 6. When the LSP electron beam scatters with the CP (

*h*= 1) laser field, $ h \xaf \gamma $ of emitted

_{L}*γ*photons via the single-photon absorption channel is almost unaffected by the electron spin, i.e., $ h \xaf \gamma \u2243 h L = 1$ in the low-energy parts of the first harmonic and $ h \xaf \gamma \u2243 \u2212 h L = \u2212 1$ near the end. As the number of absorbed photons increases, the electron spin begins to participate in the competition with the laser. As the average number of the absorbed laser photons increases up to $ n \xaf \u2273 100$, the electron spin mostly contributes to the high-energy

*γ*photon and $ h \xaf \gamma $ is anti-parallel to the case of $ h e = \u2212 1$ [see Figs. 2 and 6(a)]. In the realistic laser pulse, for both $ a 0 , peak = 1$ and $ a 0 , peak = 10 , \u2009 h \xaf \gamma $ of

*γ*photons rises as $ \epsilon \gamma $ increases in the medium- to high-energy regions. $ h \xaf \gamma $ with $ \epsilon \gamma \u2272 1$ GeV for $ a 0 , peak = 1$ is the same as the case of

*h*= 0, while due to $ \delta 1 \u221d 1 / a 0 2$, for $ a 0 , peak = 10$, in the range of $ \epsilon \gamma \u2272 0.36$ GeV, $ h \xaf \gamma $ is the same as in the case of

_{e}*h*= 0 [see Figs. 3 and 6(b)–6(e)].

_{e}### APPENDIX B: IMPACT OF NONLINEAR EFFECTS ON $ h \xaf \gamma $ OF EMITTED PHOTONS IN THE LINEAR REGION

In the NLCS, the polarization of the emitted photons in the linear region is averaged over all possible multi-photon absorption channels. The polarization of *γ* photons radiated by the single-photon absorption channel in NLCS is almost identical to that of the artificially squeezed LCS, which can be considered as the limit of NLCS as $ a 0 2 \u2192 0$ and *n* = 1.^{38} As the first harmonic edge of NLCS $ \delta 1 \u221d 1 / a 0 2$, *δ*_{1} decreases as *a*_{0} increases. For instance, *δ*_{1} decreases from 0.191 for $ a 0 = 0.1$ to $ 2.30 \xd7 10 \u2212 3$ for $ a 0 = 10$. Therefore, artificially squeezing the LCS helicity by a factor of $ S = \delta LCS / \delta 1$ along *δ* yields a result that is nearly identical to that of the single-photon absorption channel in the NLCS [see the black-dash-dotted and green-dotted lines in Figs. 7(b), 7(d), and 7(f)]. The squeezing factors *S* are about 1.0, 1.80, and 81.96 for $ a 0 = 0.1$, 1, and 10, respectively. Note that the polarization of *γ* photons radiated by the single-photon absorption channel is mainly determined by the laser spin. Due to the multi-photon absorption effect in the NLCS, in the following paragraphs, we analyze the photon polarization with respect to the laser intensity regime.

For $ a 0 \u2243 0.1$, the transition rate $ R multi . ( \delta ) \u2261 \u2211 n \u2265 2 R n ( \delta )$ of multi-photon absorption channels is about three orders of magnitude smaller than $ R n = 1 ( \delta )$ of the single-photon absorption channel [see Fig. 7(a)], which dominates the transition rate $ R ( \delta )$. Therefore, both the average helicity is almost identical to case of LCS [see Figs. 7(b)].

For $ a 0 \u2243 1 , \u2009 R multi . ( \delta )$ with $ \delta \u2264 \delta 1$ cannot be neglected [see Fig. 7(c)]. Therefore, the average helicity is significantly affected by the relative intensity of each channel and shifts upward (due to the contribution from multi-photon absorption channels) compared to the situation where LCS is artificially squeezed. For instance, the average helicity shifts up to $ h \xaf \gamma \u2248 \u2212 0.76$ from −1 at $ \delta \u2248 \delta 1$ [see Fig. 7(d)].

For $ a 0 \u2243 10 , \u2009 R multi . ( \delta )$ in the linear region ( $ \delta \u2272 \delta 1$) continues to increase [see Fig. 7(e)]. The deviation of $ h \xaf \gamma $ from the case of artificially squeezed LCS increases. For instance, $ h \xaf \gamma $ shifts up to −0.53 from −1 at $ \delta \u2248 \delta 1$, which is averaged over the multi-photon absorption channels [see Fig. 7(f)].

### APPENDIX C: IMPACT OF ELECTRON ENERGY ON THE TRANSFER MECHANISM OF SAM

The above-mentioned analytical discussions are based on the single-photon emission of electrons with $ \epsilon e = 10$ GeV. However, the transfer mechanism also holds true for other electron energies (see Fig. 8). The slight difference is that the competed energy regions between laser and electron for controlling the *γ*-photon polarization are different. As $ \delta 1 \u221d \epsilon e$ for a certain *a*_{0}, the energy of *γ* photon radiated by the electron with smaller $ \epsilon e$, of which the helicity is mostly determined by the laser, moves to the lower energy region. To obtain high-energy *γ* photons, the lower energy electrons need to absorb more laser photons. For instances, for $ a 0 = 10$, electrons with $ \epsilon e = 10$ GeV radiate high-polarization *γ* rays with $ \delta \u2272 \delta 1 \u2248 2.35 \xd7 10 \u2212 3$ via single-photon channel, while, for $ \epsilon e = 1$ GeV, $ \delta \u2272 \delta 1 \u2248 2.35 \xd7 10 \u2212 4$. For high-energy *γ* photons, electrons with $ \epsilon e = 10$ GeV absorb laser photons with $ n \xaf \u2248 1000$ to radiate *γ* photons with $ \delta = 0.56$, while, electrons with $ \epsilon e = 1$ GeV require $ n \xaf \u2248 10000$ for the same *δ*.

### APPENDIX D: IMPACT OF RADIATION REACTION EFFECTS ON $ h \xaf \gamma $ OF EMITTED PHOTONS

The presence of radiation reaction leads to differences in the average polarization of emitted photons within the linear region between strong NLCS ( $ a 0 , peak = 10$) and weak NLCS ( $ a 0 , peak = 1$).

For $ a 0 , peak = 1$, the emitted photon number per electron is about 0.3 and the central energy of the radiating electrons is still around 10 GeV with the energy spread $ \Delta \epsilon e \u2248 2$ GeV [see the red-dash-dotted line in Fig. 9(a)]. Therefore, the polarization of the emitted photons is averaged over those photons radiated by electrons with different energies. For instance, $ h \xaf \gamma $ shifts up to −0.51 at $ \epsilon \gamma = \epsilon \xaf e \delta 1 \u2248 1.06$ GeV from $ h \xaf \gamma \u2248 \u2212 0.76$ of the case of $ \epsilon e = 10$ GeV, where $ \epsilon \xaf e = 10$ GeV is the central energy of the radiating electrons [see Fig. 9(b)].

For $ a 0 , peak = 10$, an electron emits approximately ten photons in the laser pulse. Consequently, multiple radiations lead to an expanded energy distribution of the radiating electrons, with an energy spread of $ \Delta \epsilon e \u2248 9.5$ GeV [see the green-dotted line in Fig. 9(a)]. Meanwhile, as $ \delta 1 \u221d \epsilon e$, the energies of the photons radiated by the single-photon absorption channel ( $ \epsilon \gamma \u2272 \delta 1 \epsilon e$) decrease as $ \epsilon e$ decreases. Therefore, $ h \xaf \gamma $ is averaged over those photons radiated by electrons with different energies and smoothly decreases from about 1 at $ \epsilon \gamma \u2248 0.1$ MeV to 0.08 at $ \epsilon \gamma \u2248 22.7$ MeV [see Fig. 9(c)].

### APPENDIX E: MANIPULATION OF *γ*-RAY LINEAR POLARIZATION

^{38}, $ \u27e8 F 1 n \u27e9 = 0$ and

*A*were introduced in Ref. 89 as follows:

_{k}For LP *γ* photons generated in the LP laser, the competition between the laser and electron for controlling the *γ*-photon polarization also exists (see Fig. 10). For instances, for SNP electrons in the LP laser field with $ a 0 \u223c O ( 0.1 )$, the radiation probability of the first harmonic is three orders of magnitude greater than that of the second and is greater than that of other harmonics. As a result, the emitted photons are almost completely radiated via the single-photon absorption channel and obtain an almost entirely linear polarization at the end of the first harmonic with $ \xi \xaf 3 f \u2248 1$ [see Figs. 10(a) and 10(b)]. As *a*_{0} increases to *O*(1), the absorption channels of dozens of laser photons occur and *γ* photons with higher energy are radiated, while $ \xi \xaf 3 f$ of *γ* photons at the first harmonic edge decreases to $ \u223c 0.88$. Since the electron spin plays an increasingly significant role in multi-photon absorption channels, $ \xi \xaf 3 f$ decreases as $ \delta \u2273 \delta 2$ [see Figs. 10(c) and 10(d)]. In addition, for $ a 0 \u223c O ( 10 )$, there are higher transition probabilities to radiate higher energy *γ* photons, and $ d 2 W rad d t d \delta \u2248 7.6$ at $ \delta \u2248 \delta 1 = 0.0023$ but with decreased $ \xi \xaf 3 f \u2248 0.64$. Note that when the average number of absorbed laser photons $ n \xaf$ increases to 100, the LCFA method can perfectly predict the radiation probability and linear polarization of *γ* photons [see Figs. 10(g) and 10(h)]. Expectedly, $ \xi \xaf 3 f$ is falling to 0 as *δ* increases to 1 because of more polarization contribution from the electron [see Fig. 10(f)]. Therefore, to generate high-brilliance high-energy LP *γ* rays, we not only need to increase the laser intensity but also accelerate the electrons to high energies due to $ \delta 1 \u221d \epsilon e$. Importantly, the spin effects of electrons on the liner polarization of radiated *γ* photons are balanced out in the LP laser pulse. However, if the *γ* photons can be separated, such as using an elliptically polarized laser pulse,^{45} the high polarization of *γ* photons at high-energy parts contributed by the transversely spin-polarized electrons via multi-photon absorption channels will be accessed.

### APPENDIX F: COMPARISON OF $ h \xaf \gamma $ BETWEEN THE METHODS OF LMA AND LCFA

Comparison of the *γ* photons, respectively, predicted by the methods of LMA^{57,79,81} and LCFA^{45} is shown in Fig. 11. LCFA requires $ a 0 \u226b 1$ and $ a 0 3 \u226b \chi e$,^{51,90–92} therefore, the radiation at the two tails of the laser pulse is overestimated and the spectra calculated by LCFA are broader than the case of LMA. Interestingly, even when the above conditions are fulfilled, the *γ*-photon spectra predicted by the methods of LMA and LCFA also have differences in the region of $ \epsilon \gamma \u2272 ( \chi e / a 0 3 ) \epsilon e$,^{93–96} where $ \chi e \u2248 1.18$ for the head-on collision scenario. As a result, the yields of low-energy photons $ \epsilon \gamma \u2272 11.87$ MeV are also overestimated near the peak intensity of the laser pulse [see Figs. 3(d), 11(a), and 11(d)]. Importantly, in the LCFA method, the helicity contribution from the CP laser is not included, therefore, one gets $ h \xaf \gamma = 0$ for spin-nonpolarized electrons (*h _{e}* = 0) [see Fig. 11(b) and the green-solid line in Fig. 11(e)]. As expected, high-energy

*γ*photons with $ \epsilon \gamma \u2273 1$ GeV radiated in the middle of the laser pulse obtain polarization from the LSP electrons [see Fig. 11(c) and the red-dash-dotted line in Fig. 11(e)], and the deviation $ \Delta h \xaf \gamma = h \xaf \gamma LCFA \u2212 h \xaf \gamma LMA$ decreases from 0.08 at $ \epsilon \gamma \u2248 1$ GeV to 0 at about 9 GeV.

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