2 Simulation methods and setup

We perform simulations using the spin-resolved QED PIC code SLIPs^[ Reference Wan, Lv, Xue, Dou, Zhao, Ababekri, Wei, Li, Zhao and Li ^{71 ]}, which is coupled to a Monte Carlo algorithm for QED processes based on the local constant field approximation^[ Reference Ritus ^{45 ,} Reference Ilderton ^{72 –} Reference Di Piazza, Tamburini, Meuren and Keitel ^{74 ]}. As one of the primary QED processes during the beam–target interaction, the nonlinear Compton scattering process is characterized by the nonlinear QED parameter ${\chi}_{\mathrm{e}}\equiv \mid e\mid \mathrm{\hslash}/\left({m}_{\mathrm{e}}^3{c}^4\right)\sqrt{-{\left({F}_{\mu \nu}{p}^{\nu}\right)}^2}$ ^[ Reference Ritus ^{75 –} Reference Koga, Esirkepov and Bulanov ^{77 ]}. Here, $e$ and ${m}_{\mathrm{e}}$ are the charge and mass of the electron, respectively, $c$ is the speed of light in vacuum, $\mathrm{\hslash}$ is the reduced Planck constant, ${p}^{\nu }$ is the four-momentum of the electron and ${F}_{\mu \nu}$ is the electromagnetic field tensor. In nonlinear Compton scattering, the polarization of a $\gamma$ photon can be described by the Stokes parameters ( ${\xi}_1,{\xi}_2,{\xi}_3$ ). Since the polarization of an emitted $\gamma$ photon is associated with the directions of the velocity $\widehat{\mathbf{v}}$ and acceleration $\widehat{\mathbf{a}}$ of the parent electron, we define the polarization in an instantaneous frame ( ${\widehat{\mathbf{k}}}_{\gamma }$ , ${\widehat{\mathbf{e}}}_1$ , ${\widehat{\mathbf{e}}}_2$ )^[ Reference McMaster ^{78 ]} comoving with the electron, with ${\widehat{\mathbf{e}}}_1=\widehat{\mathbf{a}}-\widehat{\mathbf{v}}\left(\widehat{\mathbf{v}}\cdot \widehat{\mathbf{a}}\right)$ and ${\widehat{\mathbf{e}}}_2=\widehat{\mathbf{v}}\times \widehat{\mathbf{a}}$ . Here, the direction of photon propagation ${\widehat{\mathbf{k}}}_{\gamma }$ is considered to be aligned with $\widehat{\mathbf{v}}$ , since the emission angle is approximately $1/{\gamma}_{\mathrm{e}}$ $\ll$ 1, where ${\gamma}_{\mathrm{e}}$ is the Lorentz factor of the electron. On this basis, ${\xi}_3$ corresponds to the degree of linear polarization along ${\widehat{\mathbf{e}}}_1$ versus ${\widehat{\mathbf{e}}}_2$ ; ${\xi}_1$ represents linear polarization at an angle of $\pm 45{}^{\circ}$ to the ${\widehat{\mathbf{e}}}_1$ -axis; and ${\xi}_2$ denotes the degree of circular polarization^[ Reference Li, Shaisultanov, Hatsagortsyan, Wan, Keitel and Li ^{79 ]}. This instantaneous frame changes as the electron moves. To detect the mean polarization of photons propagating in a given direction, their Stokes parameters must be transformed to a common observation frame ( ${\widehat{\mathbf{k}}}_{\gamma }$ , ${\widehat{\mathbf{o}}}_1$ , ${\widehat{\mathbf{o}}}_2$ ) and then averaged.

The average Stokes parameters of a photon emitted by an electron with initial spin ${\mathbf{S}}_\mathrm{i}$ are given by the following^[ Reference Cao, Xue, Liu, Li, Hu, Liu, Dou, Wan, Zhao, Yu and Li ^{63 ]}:

(1a) $$\begin{align}{\overline{\xi}}_1&=\frac{u/\left(1-u\right){K}_{\frac{1}{3}}\left(\rho \right)\left({\mathbf{S}}_\mathrm{i}\cdot \widehat{\mathbf{a}}\right)}{w-{uK}_{\frac{1}{3}}\left(\rho \right)\widehat{\mathbf{b}}\cdot {\mathbf{S}}_\mathrm{i}},\end{align}$$

(1b) $$\begin{align}{\overline{\xi}}_2&=\frac{-\left[u\mathrm{Int}{K}_{\frac{1}{3}}\left(\rho \right)-\left(2u-{u}^2\right)/\left(1-u\right){K}_{\frac{2}{3}}\left(\rho \right)\right]\left({\mathbf{S}}_\mathrm{i}\cdot \widehat{\mathbf{v}}\right)}{w-{uK}_{\frac{1}{3}}\left(\rho \right)\widehat{\mathbf{b}}\cdot {\mathbf{S}}_\mathrm{i}},\end{align}$$

(1c) $$\begin{align}{\overline{\xi}}_3&=\frac{K_{\frac{2}{3}}\left(\rho \right)-u\left(1-u\right){K}_{\frac{1}{3}}\left(\rho \right)\left({\mathbf{S}}_\mathrm{i}\cdot \widehat{\mathbf{b}}\right)}{w-{uK}_{\frac{1}{3}}\left(\rho \right)\widehat{\mathbf{b}}\cdot {\mathbf{S}}_\mathrm{i}}, \end{align}$$

where $w=-\mathrm{Int}{K}_{\frac{1}{3}}\left(\rho \right)+\frac{u^2-2u+2}{1-u}{K}_{\frac{2}{3}}\left(\rho \right)$ , $\widehat{\mathbf{b}}=\widehat{\mathbf{v}}\times \widehat{\mathbf{a}}/\mid \widehat{\mathbf{v}}\times \widehat{\mathbf{a}}\mid$ , $\rho =2u/\left[\left(1-u\right)3{\chi}_{\mathrm{e}}\right]$ , $u={\omega}_{\gamma }/{\varepsilon}_\mathrm{i}$ is the ratio of the photon energy ${\omega}_{\gamma }$ to the initial electron energy ${\varepsilon}_\mathrm{i}$ , and $\mathrm{Int}{K}_{\frac{1}{3}}\left(\rho \right)\equiv {\int}_{\rho}^{\infty}\mathrm{d}{zK}_{\frac{1}{3}}(z)$ , with ${K}_n$ being the $n$ th-order modified Bessel function of the second kind. The degree of polarization for each propagation direction ${\widehat{\mathbf{k}}}_{\gamma }$ is given by ${P}_{\gamma}\equiv \sqrt{{\overline{\xi_1}}^2+{\overline{\xi_3}}^2}$ .

The simulation box has dimensions of $x\times y\times z=7\kern0.22em \unicode{x3bc} \mathrm{m}\times 10\kern0.22em \unicode{x3bc} \mathrm{m}\times 10\kern0.22em \unicode{x3bc} \mathrm{m}$ , with grid cells of $560\times 400\times 400$ . A rotating electron beam with an energy of 1 GeV, a charge of 1.73 nC, a relative energy spread of 5% and an angle spread of $0.5{}^{\circ}$ propagates along the + $x$ -axis. Here, the ‘rotation’ refers to electrons possessing an initial azimuthal momentum that leads to helical trajectories. This is distinct from electrons in a quantum vortex state^[ Reference Ivanov ^{80 ]}. Given that an electron beam with azimuthal momentum possesses a hollow density profile as a result of centrifugal forces, we specify the initial beam density in our simulations as ${{n}_{\mathrm{b}}={n}_{\mathrm{b}0}\exp \left[-{x}^2/{\sigma}_x^2-{\left(r-{r}_{\mathrm{c}}\right)}^2/{\sigma}_{\perp}^2\right]}$ . Here, $r=\sqrt{y^2+{z}^2}$ , ${\sigma}_x=0.5\;\unicode{x3bc} \mathrm{m}$ , ${r}_{\mathrm{c}}={\sigma}_{\perp }=1\;\unicode{x3bc} \mathrm{m}$ and ${n}_{\mathrm{b}0}$ is the maximum density of the beam. Here, ${n}_{\mathrm{b}0}=1.1\times {10}^{21}\;{\mathrm{cm}}^{-3}$ has been employed to generate a strong enough CTR field for nonlinear Compton scattering. This density must remain below the foil density to prevent target penetration, which would suppress the CTR field^[ Reference Sampath, Davoine, Corde, Gremillet, Gilljohann, Sangal, Keitel, Ariniello, Cary, Ekerfelt, Emma, Fiuza, Fujii, Hogan, Joshi, Knetsch, Kononenko, Lee, Litos, Marsh, Nie, O'Shea, Peterson, San Miguel Claveria, Storey, Wu, Xu, Zhang and Tamburini ^{81 ]}. Electron beams with comparable parameters can be produced using advanced conventional accelerators or laser-driven accelerators^[ Reference Babjak, Willingale, Arefiev and Vranic ^{82 –} Reference Yakimenko, Meuren, Gaudio, Baumann, Fedotov, Fiuza, Grismayer, Hogan, Pukhov, Silva and White ^{86 ]}, as detailed in the Supplementary Material. A solid aluminum foil with density ${n}_{\mathrm{Al}}=5.4\times {10}^{22}\;{\mathrm{cm}}^{-3}$ and thickness $d=0.5\kern0.22em \unicode{x3bc} \mathrm{m}$ is employed with its front surface at $x=5\kern0.22em \unicode{x3bc} \mathrm{m}$ . In this scheme, thin, low-Z targets are preferred to suppress unwanted energy loss via bremsstrahlung and Bethe–Heitler processes, since both cross-sections scale approximately as ${Z}^2$ ^[ Reference Heitler ^{87 ]}. In our setup, the contributions of these processes are simulated using the EPOCH code^[ Reference Arber, Bennett, Brady, Lawrence-Douglas, Ramsay, Sircombe, Gillies, Evans, Schmitz, Bell and Ridgers ^{88 ]}, as detailed in the Supplementary Material. A moving window is used, starting at $t=8.25\;\mathrm{fs}$ and moving along the $+x$ -axis at a speed of $c$ . In the simulations, each cell contains 10 macroparticles for beam electrons and aluminum atoms. In addition, field ionization is modeled using a hybrid approach that incorporates both tunnel ionization^[ Reference Ammosov, Delone and Krainov ^{89 ]} and barrier-suppression ionization^[ Reference Posthumus, Thompson, Frasinski and Codling ^{90 ]}.

3 Generation and properties of hybrid cylindrical vector $\gamma$ rays

The main results for the generated $\gamma$ rays are summarized in Figure 2, which compares three cases with different initial azimuthal momenta of the electron: ${p}_{\varphi }=10{m}_{\mathrm{e}}c$ (Figures 2(a1) and 2(b1)), $20{m}_{\mathrm{e}}c$ (Figures 2(a2) and 2(b2)) and $40{m}_{\mathrm{e}}c$ (Figures 2(a3) and 2(b3)). Here, only photons with energies above 1 MeV are analyzed to focus on the high-energy $\gamma$ rays. As shown in Figure 2(a), the emitted $\gamma$ rays exhibit hybrid CV polarization. With increasing $\mid {p}_{\varphi}\mid$ , the radial polarization component gradually decreases, whereas the azimuthal polarization component gradually increases. Since the angle-resolved distribution and polarization degree of the emitted $\gamma$ rays exhibit cylindrical symmetry, we analyze their dependence on the polar angle $\theta$ , as shown in Figure 2(b). As $\mid {p}_{\varphi}\mid$ increases, the polar angle $\theta$ at the peak of $\gamma$ -ray number rises approximately linearly with $\mid {p}_{\varphi}\mid$ , reaching $0.3{}^{\circ}$ , $0.6{}^{\circ}$ and $1.2{}^{\circ}$ for ${p}_{\varphi }=10{m}_{\mathrm{e}}c$ , $20{m}_{\mathrm{e}}c$ and $40{m}_{\mathrm{e}}c$ , respectively. In all cases, most of the $\gamma$ rays distribute within a $\theta$ width of $0.8{}^{\circ}$ . Furthermore, the degree of polarization increases with $\mid {p}_{\varphi}\mid$ , with average polarization degrees of 0.44, 0.57 and 0.65 for the cases shown in Figures 2(b1), 2(b2) and 2(b3), respectively. This enhancement occurs because a larger ${p}_{\varphi }$ more effectively suppresses the polarization cancellation, which will be discussed in detail in Figure 3. The result of average polarization degrees is comparable to that achieved in laser–electron collision schemes ( $\sim$ 60%)^[ Reference Li, Shaisultanov, Chen, Wan, Hatsagortsyan, Keitel and Li ^{58 ,} Reference Xue, Dou, Wan, Yu, Wang, Ren, Zhao, Zhao, Xu and Li ^{60 ]}.

Figure 2 (a1)–(a3) Angle-resolved distribution ${\log}_{10}\left(\mathrm{d}{N}_{\gamma }/\mathrm{d}\Omega \right)$ (background heatmap) and average polarization ${P}_{\gamma }$ of the emitted $\gamma$ photons with respect to the polar angle $\theta$ and the azimuth angle $\varphi$ . Here, $\mathrm{d}\Omega =\sin \theta \mathrm{d}\theta \mathrm{d}\varphi$ , where $\theta$ is the angle between the photon momentum and the $+x$ -axis, and $\varphi$ is the angle between the projection of the momentum onto the $yz$ -plane and the $+y$ -axis. The superimposed double-headed arrows indicate the average polarization direction, while their color represents the degree of polarization ${P}_{\gamma }$ . (b1)–(b3) Angle-resolved polarization degree ${P}_{\gamma }$ (blue) and distribution $\mathrm{d}{N}_{\gamma }/\mathrm{d}\theta$ (red) of all emitted $\gamma$ photons versus $\theta$ . Here, panels ((a1), (b1)), ((a2), (b2)) and ((a3), (b3)) correspond to the case with an initial electron azimuthal momentum of ${p}_{\varphi }=10{m}_{\mathrm{e}}c$ , $20{m}_{\mathrm{e}}c$ and $40{m}_{\mathrm{e}}c$ , respectively. (c) The angle $\delta$ as a function of the electron initial azimuthal momentum ${p}_{\varphi }$ . (d) Energy-resolved polarization degree ${P}_{\gamma }$ (blue) and distribution $\mathrm{d}{N}_{\gamma }/\mathrm{d}{\varepsilon}_{\gamma }$ (red) of $\gamma$ photons within $0.15{}^{\circ}<\theta <0.65{}^{\circ}$ versus the photon energy ${\varepsilon}_{\gamma }$ for ${p}_{\varphi }=20{m}_{\mathrm{e}}c$ . (e) The brilliance (photons $/\left(\mathrm{s}\ {\mathrm{mm}}^2\ {\mathrm{mrad}}^2\times 0.1\%\ \mathrm{bandwidth}\right)$ ) of the $\gamma$ rays as a function of the photon energy ${\varepsilon}_{\gamma }$ .

Figure 3 (a) Distribution of the effective electric field ${E}^{\prime }$ in the $xy$ -plane at $z=0$ . (b) Time-dependent photon generation rate, where the blue line represents the rate for first-photon emission and the red line represents the rate for multiple-photon emission. (c) Average polarization degree ${P}_{\gamma }$ as a function of energy ratio ${\varepsilon}_{\gamma }/{\varepsilon}_{\mathrm{e}}$ and QED parameter ${\chi}_{\mathrm{e}}$ . (d) Radial displacement as a function of longitudinal position for selected electrons during interaction with the CTR field. (e) Distribution of radial forces ${F}_{\mathrm{r}}$ acting on electrons at different times, with color representing the number of electrons. The red solid line shows the evolution of the average radial momentum ${p}_{\mathrm{r}}$ of the beam. (f) Distribution of photon polarization directions (black double arrows) and momentum directions (red arrows) in the $yz$ -plane, with the color scale representing the effective electric field ${E}^{\prime }$ .

The hybrid CV polarization state is characterized by a polarization angle $\delta$ , defined as the angle between the polarization direction ${\widehat{\mathbf{e}}}_{\mathrm{P}}$ and the radial unit vector ${\widehat{\mathbf{e}}}_{\mathrm{r}}$ . Here, $\delta =0{}^{\circ}$ corresponds to purely radial polarization, while $\mid \delta \mid =90{}^{\circ}$ corresponds to purely azimuthal polarization. Our simulations show that $\mid \delta \mid$ increases with $\mid {p}_{\varphi}\mid$ , reaching $55{}^{\circ}$ , $69{}^{\circ}$ and $79{}^{\circ}$ at ${p}_{\varphi }=10{m}_{\mathrm{e}}c$ , $20{m}_{\mathrm{e}}c$ and $40{m}_{\mathrm{e}}c$ , respectively (Figure 2(c)), with a limit of $90{}^{\circ}$ . This dependence makes it possible to diagnose the azimuthal momentum of short-pulse electron beams, which remains a great challenge in current laboratories^[ Reference Thaury, Guillaume, Corde, Lehe, Le Bouteiller, Phuoc, Davoine, Rax, Rousse and Malka ^{70 ]}. Since the sign of $\delta$ is determined by the sign of ${p}_{\varphi }$ , the angle $\delta$ can cover the range $\left(-90{}^{\circ},90{}^{\circ}\right)$ .

We now discuss the energy dependence of the $\gamma$ -ray properties, using the case of ${p}_{\varphi }=20{m}_{\mathrm{e}}c$ as an example. Figure 2(d) shows the energy-resolved polarization ${P}_{\gamma }$ and energy spectrum $\mathrm{d}{N}_{\gamma }/\mathrm{d}{\varepsilon}_{\gamma }$ of photons within $0.15{}^{\circ}<\theta <0.65{}^{\circ}$ for the case of ${p}_{\varphi }=20{m}_{\mathrm{e}}c$ . These $\gamma$ rays exhibit an exponential energy spectrum with a cutoff at approximately 200 MeV. The average polarization increases with photon energy ${\varepsilon}_{\gamma }$ from approximately 60% at the MeV range to nearly 100% above 25 MeV. This trend is consistent with the theoretical prediction, as shown in Figure 3(c), that ${P}_{\gamma }$ increases with ${\varepsilon}_{\gamma }/{\varepsilon}_{\mathrm{e}}$ over our parameter range ( ${\varepsilon}_{\gamma }/{\varepsilon}_{\mathrm{e}}<0.2,{\varepsilon}_{\mathrm{e}}\sim 1\;\mathrm{GeV}$ ). These photons have a root-mean-square angular divergence of 0.79 $\mathrm{mrad}$ $\times$ 0.79 $\mathrm{mrad}$ , and spatial dimensions of $0.22$ μm (longitudinal) and $1.20\kern0.22em \unicode{x3bc} \mathrm{m}\times 1.20\kern0.22em \unicode{x3bc} \mathrm{m}$ (transverse). The corresponding brilliances at ${{\varepsilon}_{\gamma }=1}$ , 10 and 50 MeV are $1.37\times {10}^{24}$ , $7.86\times {10}^{23}$ and $1.43\times {10}^{23}$ photons $/\left(\mathrm{s}\ {\mathrm{mm}}^2\ {\mathrm{mrad}}^2\times 0.1\%\ \mathrm{bandwidth}\right)$ , respectively, as shown in Figure 2(e).

4 Mechanism of $\gamma$ -ray polarization

When a relativistic electron beam traverses a target, the abrupt change in dielectric constant generates an intense CTR field on both sides of the target^[ Reference Carron ^{91 ,} Reference Han, Yang, Cheng and Zhan ^{92 ]}, as shown in Figure 3(a). This leads to substantial $\gamma$ -ray emission from the beam electrons via the nonlinear Compton scattering process. In a cylindrical coordinate system aligned with the $x$ -axis ( ${\widehat{\mathbf{e}}}_x$ , ${\widehat{\mathbf{e}}}_{\mathrm{r}}$ , ${\widehat{\mathbf{e}}}_{\vartheta }$ ), this field takes the form ${\mathbf{E}}_{\mathrm{CTR}}={E}_{\mathrm{r}}{\widehat{\mathbf{e}}}_{\mathrm{r}}+{E}_x{\widehat{\mathbf{e}}}_x$ and ${\mathbf{B}}_{\mathrm{CTR}}=-{B}_{\vartheta }{\widehat{\mathbf{e}}}_{\vartheta }$ ^[ Reference Sampath, Davoine, Corde, Gremillet, Gilljohann, Sangal, Keitel, Ariniello, Cary, Ekerfelt, Emma, Fiuza, Fujii, Hogan, Joshi, Knetsch, Kononenko, Lee, Litos, Marsh, Nie, O'Shea, Peterson, San Miguel Claveria, Storey, Wu, Xu, Zhang and Tamburini ^{81 ]}. For relativistic electrons propagating along the $x$ -axis, the effective electric field can be written as ${\mathbf{E}}^{\prime }={\mathbf{E}}_{\perp }+\mathbf{v}\times \mathbf{B}\approx \left({E}_{\mathrm{r}}+{cB}_{\vartheta}\right){\widehat{\mathbf{e}}}_{\mathrm{r}}$ , where ${\mathbf{E}}_{\perp }$ is the transverse electric field. The intensity of the $\gamma$ -ray emission is governed by the nonlinear QED parameter ${\chi}_{\mathrm{e}}\approx {\gamma}_{\mathrm{e}}\mid {\mathbf{E}}^{\prime}\mid /{E}_{\mathrm{c}}$ ^[ Reference Ritus ^{45 ]}, where ${E}_{\mathrm{c}}={m}_{\mathrm{e}}^2{c}^3/\left(|e|\mathrm{\hslash}\right)\approx 1.3\times {10}^{18}\;\mathrm{V}/\mathrm{m}$ is the Schwinger critical field. The CTR field comprises components propagating along both the $-x$ and $+x$ directions. Electrons interacting with the field propagating along $-x$ experience a large ${\chi}_{\mathrm{e}}$ , thereby dominating the $\gamma$ -ray emission, while the contribution from the field propagating along $+x$ is negligible.

As depicted in Figure 3(b), high-energy photons primarily originate from single photon emissions from electrons, with multiple photon emissions accounting for only about 5.5% of the total photons. This is due to the significant reduction in electron energy and the corresponding decrease in ${\chi}_{\mathrm{e}}$ after the first emission. Consequently, we consider only the first photon emitted by an electron. For an initially unpolarized electron beam, the initial spin averages to zero ( ${\overline{\mathbf{S}}}_\mathrm{i}=0$ ). This leads to a simplification of Equation (1), yielding $\overline{\xi_1}=\overline{\xi_2}=0$ and a non-zero $\overline{\xi_3}={K}_{\frac{2}{3}}\left(\rho \right)/w$ . This result indicates that the emitted photon is linearly polarized along the axis ${\widehat{\mathbf{e}}}_1$ of the instantaneous frame. In this case, the degree of polarization is given by ${P}_{\gamma }=\overline{\xi_3}=\left(1-u\right){K}_{2/3}\left(\rho \right)/\left[-\left(1-u\right)\mathrm{Int}{K}_{1/3}\left(\rho \right)+\left({u}^2-2u+2\right)\cdot\right.\left. {K}_{2/3}\left(\rho \right)\right]$ . As illustrated in Figure 3(c), ${P}_{\gamma }$ decreases with increasing ${\chi}_{\mathrm{e}}$ at a fixed ${\varepsilon}_{\gamma }$ ; conversely, at a fixed ${\chi}_{\mathrm{e}}$ , ${P}_{\gamma }$ first increases with the photon energy ${\varepsilon}_{\gamma }$ , reaches a maximum, and then decreases to zero. In our scheme, since the emitted $\gamma$ photons have small polar angles and electrons exhibit negligible radial displacement [Figure 3(d)], we can approximate the velocity of the parent electron as $\widehat{\mathbf{v}}\approx {\widehat{\mathbf{e}}}_x$ and its transverse acceleration as ${\widehat{\mathbf{a}}}_{\perp}\approx {\mathbf{E}}^{\prime }/\mid {\mathbf{E}}^{\prime}\mid \approx -{\widehat{\mathbf{e}}}_{\mathrm{r}}$ . The ${\widehat{\mathbf{e}}}_1$ -axis of the instantaneous frame becomes ${\widehat{\mathbf{e}}}_1^0=\widehat{\mathbf{a}}-\widehat{\mathbf{v}}\left(\widehat{\mathbf{v}}\cdot \widehat{\mathbf{a}}\right)\approx -{\widehat{\mathbf{e}}}_{\mathrm{r}}\parallel -{\mathbf{E}}^{\prime }$ , which shows that the emitted $\gamma$ photon is polarized along the radial direction. The remaining axes of the instantaneous frame are ${\widehat{\mathbf{k}}}_{\gamma}\approx \widehat{\mathbf{v}}$ and ${{\widehat{\mathbf{e}}}_2^0=\widehat{\mathbf{v}}\times \widehat{\mathbf{a}}\approx -{\widehat{\mathbf{e}}}_{\vartheta }}$ . As all photons with a given wavevector ${\widehat{\mathbf{k}}}_{\gamma }$ share this common instantaneous frame ( ${\widehat{\mathbf{k}}}_{\gamma }$ , ${\widehat{\mathbf{e}}}_1^0$ , ${\widehat{\mathbf{e}}}_2^0$ ), we adopt it as our observation frame ( ${\widehat{\mathbf{k}}}_{\gamma }$ , ${\widehat{\mathbf{o}}}_1$ , ${\widehat{\mathbf{o}}}_2$ ).

The hybrid $\gamma$ -ray polarization originates from the misalignment between their transverse momentum and polarization directions. During the interaction with the CTR field, an electron experiences a radial Lorentz force ${\mathbf{F}}_{\mathrm{r}}$ , and acquires a radial momentum component ( ${\sim}5{m}_{\mathrm{e}}c$ in Figure 3(e)). The azimuthal and radial momentum components of the parent electron ( ${p}_{\varphi }$ and ${p}_{\mathrm{r}}$ ) are subsequently transferred to the emitted photon. Therefore, the direction of the photon transverse wave vector ${\mathbf{k}}_{\perp }$ deviates from the radial direction by an angle $\arctan \left({p}_{\varphi }/{p}_{\mathrm{r}}\right)$ (Figure 3(f)). As a result, for a given observation angle $\phi$ , the detected photons originate from electrons at a specific position $\vartheta =\phi -\arctan \left({p}_{\varphi }/{p}_{\mathrm{r}}\right)$ , and their observed polarization is oriented at an angle $-\arctan \left({p}_{\varphi }/{p}_{\mathrm{r}}\right)$ relative to ${\widehat{\mathbf{e}}}_{\mathrm{r}}\left(\phi \right)$ . Consequently, the polarization angle $\delta$ can be approximated as $\delta \approx \arctan \left({p}_{\varphi }/{p}_{\mathrm{r}}\right)$ . By varying the initial azimuthal momentum ${p}_{\varphi }$ , $\delta$ can be continuously tuned, yielding the different polarization states shown in Figure 2(a). Furthermore, since the intensity of the CTR field is proportional to the density of the driving electron beam^[ Reference Sampath, Davoine, Corde, Gremillet, Gilljohann, Sangal, Keitel, Ariniello, Cary, Ekerfelt, Emma, Fiuza, Fujii, Hogan, Joshi, Knetsch, Kononenko, Lee, Litos, Marsh, Nie, O'Shea, Peterson, San Miguel Claveria, Storey, Wu, Xu, Zhang and Tamburini ^{81 ]}, adjusting the beam density changes ${\mathbf{F}}_{\mathrm{r}}$ and ${p}_{\mathrm{r}}$ , thereby providing an additional choice for manipulating the polarization states.

5 Influence of parameters

To demonstrate experimental feasibility, the impact of key parameters of the electron beam and the foil on the $\gamma$ -ray polarization degree ${P}_{\gamma }$ , number of the emitted photons ${N}_{\gamma }$ and cutoff energy ${\varepsilon}_{\mathrm{m}}$ is summarized in Figure 4. Here, only one parameter is changed in each case, while all other parameters are kept fixed. As the electron azimuthal momentum ${p}_{\varphi }$ increases, the polarization angle $\delta$ increases from approximately –90° to 90° (Figure 4(a)). This arises from the increase in $\arctan \left({p}_{\varphi }/{p}_{\mathrm{r}}\right)$ , as the radial momentum ${p}_{\mathrm{r}}$ remains nearly constant. The scaling relationship between ${p}_{\varphi }$ and $\delta$ provides a potential method for diagnosing electron azimuthal momentum through $\gamma$ -ray polarization state in the laboratory. Experimentally, detection of the polarization of CV $\gamma$ rays would involve using a multi-aperture collimator to sample the $\gamma$ rays into several beamlets at different azimuthal positions, and then measuring the local polarization of each beamlet with established techniques such as Compton polarimetry^[ Reference Wan, Wang, Guo, Chen, Shaisultanov, Xu, Hatsagortsyan, Keitel and Li ^{93 –} Reference Ilie ^{95 ]}. The CV polarization structure can thus be reconstructed from such measurements (see Supplementary Material for details). The average polarization degree ${P}_{\gamma }$ rises with $\mid {p}_{\varphi}\mid$ and reaches 70%, since a larger $\mid {p}_{\varphi}\mid$ suppresses the polarization cancellation^[ Reference Cao, Xue, Liu, Li, Hu, Liu, Dou, Wan, Zhao, Yu and Li ^{63 ]}. A higher electron charge ${Q}_{\mathrm{b}}$ strengthens the CTR field intensity, leading to an enhancement of ${\mathbf{E}}^{\prime}\propto {Q}_{\mathrm{b}}$ . This subsequently increases ${\chi}_{\mathrm{e}}\approx {\gamma}_{\mathrm{e}}\mid {\mathbf{E}}^{\prime}\mid /{E}_{\mathrm{c}}$ and the radiation probability^[ Reference Li, Shaisultanov, Chen, Wan, Hatsagortsyan, Keitel and Li ^{58 ,} Reference Xue, Dou, Wan, Yu, Wang, Ren, Zhao, Zhao, Xu and Li ^{60 ]}. As shown in Figure 4(b), increasing ${Q}_{\mathrm{b}}$ leads to an increase in both the cutoff energy ${\varepsilon}_{\mathrm{m}}\propto {\chi}_{\mathrm{e}}{\gamma}_{\mathrm{e}}$ and the number of radiated photons ${N}_{\gamma }$ , while ${P}_{\gamma }$ decreases (as discussed in Section 4). Similarly, increasing electron beam energy ${\varepsilon}_{\mathrm{e}}$ also increases ${\chi}_{\mathrm{e}}$ , leading to consequently increasing ${N}_{\gamma }$ and ${\varepsilon}_{\mathrm{m}}$ while decreasing ${P}_{\gamma }$ , as illustrated in Figure 4(c). Conversely, an increase in the angle spread $\Delta \theta$ broadens the electron transverse momentum distribution, enhancing the polarization cancellation^[ Reference Cao, Xue, Liu, Li, Hu, Liu, Dou, Wan, Zhao, Yu and Li ^{63 ]}. This leads to a reduction in the polarization degree, whereas the cutoff energy ${\varepsilon}_{\mathrm{m}}$ and the total photon yield ${N}_{\gamma }$ remain stable, as shown in Figure 4(d).

Figure 4 (a) Average polarization degree ${P}_{\gamma }$ (green line) and polarization angle $\delta$ (blue line) as a function of azimuthal momentum ${p}_{\varphi }$ . Effects of (b) charge of the electron beam ${Q}_{\mathrm{b}}$ , (c) energy of the electron beam ${\varepsilon}_{\mathrm{e}}$ , (d) beam angle spread $\Delta \theta$ , (e) thickness of the foil and (f) density of the foil on the average polarization degree ${P}_{\gamma }$ (green line), number of the emitted photons ${N}_{\gamma }$ (blue line) and cutoff energy ${\varepsilon}_{\mathrm{m}}$ (red line).

The influence of foil parameters is also investigated. As the foil thickness $d$ increases, the number of emitted photons ${N}_{\gamma }$ decreases slightly, while both the polarization degree ${P}_{\gamma }$ and cutoff energy ${\varepsilon}_{\mathrm{m}}$ remain almost unchanged, as shown in Figure 4(e). They spend more time traversing a thicker foil, thereby reducing the time for interacting with the CTR field propagating along the $+x$ direction and resulting in fewer $\gamma$ -ray emissions at the same time. In contrast, the density of the target has a negligible impact on ${N}_{\gamma }$ , ${P}_{\gamma }$ and ${\varepsilon}_{\mathrm{m}}$ , according to Figure 4(f), since it has little influence on the CTR field^[ Reference Sampath, Davoine, Corde, Gremillet, Gilljohann, Sangal, Keitel, Ariniello, Cary, Ekerfelt, Emma, Fiuza, Fujii, Hogan, Joshi, Knetsch, Kononenko, Lee, Litos, Marsh, Nie, O'Shea, Peterson, San Miguel Claveria, Storey, Wu, Xu, Zhang and Tamburini ^{81 ]}.

We further investigate the interaction of the rotating electron beam with multiple foils. The results reveal a more complex polarization structure that can be controlled efficiently by varying the number of foils. For example, we use the electron beam with ${p}_{\varphi }=20{m}_{\mathrm{e}}c$ and the foil separation of 4 μm. Here, all other parameters are identical to those in Section 3. As shown in Figures 5(a) and 5(b), the radial polarization component of the emitted $\gamma$ rays increases with the polar angle $\theta$ for a given azimuthal angle $\varphi$ . This $\theta$ -dependent polarization structure becomes clearer upon increasing the foil number, owing to the broader angular distribution of the emitted $\gamma$ rays. For both ${N}_{\mathrm{foil}}=3$ (Figure 5(a)) and ${N}_{\mathrm{foil}}=7$ (Figure 5(b)), the polarization degree ${P}_{\gamma }$ remains above 50%. (Figure 5(c)) displays the angular distribution of polarization angle of $\gamma$ rays for ${N}_{\mathrm{foil}}=7$ . The results reveal a clear decrease of $\delta$ from nearly $90{}^{\circ}$ to approximately $10{}^{\circ}$ with increasing polar angle $\theta$ . Specifically, the polarization is predominantly azimuthal at small $\theta$ , whereas it changes to predominantly radial at large $\theta$ .

Figure 5 (a), (b) Angle-resolved distribution ${\log}_{10}\left(\mathrm{d}{N}_{\gamma }/\mathrm{d}\Omega \right)$ (background heatmap) and average polarization ${P}_{\gamma }$ of the emitted $\gamma$ rays, where (a) corresponds to electrons traversing three foils and (b) corresponds to seven foils. (c) Angle-resolved distribution of the $\gamma$ -ray polarization angle $\delta$ , after traversing seven foils. (d) Evolution of the average radial momentum of electrons (blue line) and the number of radiated photons (red line).

Figure 5(d) reveals the origin of this polarization structure. Upon interacting with each foil, electrons gain radial momentum and emit $\gamma$ rays through the CTR fields generated at each foil. As ${p}_{\mathrm{r}}$ accumulates with increasing the foil number, ${p}_{\varphi }/{p}_{\mathrm{r}}$ gradually decreases. At ${N}_{\mathrm{foil}}=7$ , ${p}_{\mathrm{r}}$ eventually exceeds ${p}_{\varphi }$ . This leads to a corresponding reduction in the polarization angle $\delta$ and an increase in the polar angle of the emitted $\gamma$ rays. Consequently, $\gamma$ rays generated at lower ${N}_{\mathrm{foil}}$ exhibit larger $\delta$ and concentrate around $\theta =0^{\circ}$ in the angular distribution, while those generated at higher ${N}_{\mathrm{foil}}$ have smaller $\delta$ and appear at larger $\theta$ . This gives rise to the polarization structure observed in Figures 5(a) and 5(b). The results demonstrate that the polarization is tunable by simply changing the foil number, thereby making it feasible in experiments.