## 1 Introduction  
Laser technology has significantly evolved in recent years, with intensities having reached \(I\sim 1\times 10^{23}\) Wcm\({}^{-2}\)[1]. Intensities are further expected to reach \(I\sim 1\times 10^{24}-10^{26}\) Wcm\({}^{-2}\) by the next-generation ultra-intense laser facilities [2], such as extreme light infrastructure [3], EP-OPAL, and SEL [4]. Future advancements in laser intensities may provide opportunities and platforms for nonlinear quantum electrodynamics (QED) physics [5, 6, 7, 8], laser-plasma interaction physics [9, 10], and other related domains such as high-energy-density physics and laboratory astrophysics [11, 12, 13]. Although the required strong field intensity for vacuum decay (termed the Schwinger limit \(E_{\mathrm{cr}}\sim 10^{16}\) V/cm, where the corresponding laser intensity is \(I_{\mathrm{cr}}\sim 10^{29}\) W/cm\({}^{2}\)) is beyond the current and near-future capability of contemporary laser facilities [14], electron-positron pair production via laser collision using a high-energy electron beam was realized 27 years ago at SLAC [15, 16], and several promising QED experiments are currently being conducted based on the current laser technology and GeV-level electron beam [17, 18, 19] plan to measure the transition of nonlinear Compton scattering (NCS) to the nonperturbative regime. Polarization is an important intrinsic property of a \(\gamma\) photon and has crucial applications in QED experiments and other widespread research fields. For instance, polarized \(\gamma\)-rays can excite the polarization-dependent photofission and photoproduction of muon [20, 21]. Highly polarized high-energy photons are essential in generating polarized electron-positron pairs [22], probing radiation mechanisms, and the properties of dark matter and black holes [23; 24]. Another important application is the detection of the vacuum birefringence effect using strong laser and polarized high-energy photon collision [4]. Understanding the process of the \(\gamma\) photon polarization is essential in the high-precision detection of the high-field QED process and basis of the practical application of the polarization-dependent QED process. Currently, polarized \(\gamma\)-rays are mostly generated in conventional particle accelerators, which are typically huge and expensive. In the bremsstrahlung process, polarized \(\gamma\) photons produced by a pre-polarized seed electron and the spin polarization of the seed electron determine the photon polarization [25; 26]. Moreover, a high-energy pre-polarized seed electron beam is mostly produced by the Sokolov-Terno effect in a conventional storage ring [27], which is huge and has a long electron polarization time and low polarization degree. In linear Compton scattering, an unpolarized seed electron can produce polarized \(\gamma\) photons and the polarization of the photons is determined by an incident laser polarization [28]. However, for coherent bremsstrahlung, because of the threshold of crystal materials, the seed electron current density and the energy of the produced photons are limited [29]. The incoherent bremsstrahlung cannot generate linearly polarized photons, and the photon divergence angle and directivity are not very good [30]. In addition, polarized \(\gamma\) photons in bremsstrahlung and linear Compton scattering are not applicable in most leading studies. For example, the detection of the vacuum birefringence requires highly polarized high-flux high-energy photons [31]. With the rapid advancement in laser technologies, the laser plasma-based scheme is one of the preferred alternatives for high-energy electron acceleration and high-energy photon generation. Therefore, considerable theoretical and numerical studies on new physical mechanisms for producing highly polarized high energy \(\gamma\) photons have been conducted [32; 33; 30; 34; 35].