Polarized positron beams are a powerful tool for exploration of the fine structure of matter, particularly for probing nuclear constituents1 and for testing the validity of the Standard Model2 of particle physics via weak and electromagnetic interactions. The natural decay of some radioisotopes generates polarized positrons with polarization up to 40%,3 but the flux is too low for further acceleration and applications. Positrons can also be polarized in a storage ring by spin-flips at photon emissions (the Sokolov–Ternov effect),4–8 but this is a slow process lasting at least several minutes and can only be realized in large-scale storage ring facilities. At particle accelerators, polarized e⁻e⁺ pairs are commonly produced by scattering of circularly polarized gamma photons in a high-Z material via the Bethe–Heitler process,9–11 but the luminosity of the positrons is limited because of constraints on the target thickness,12 and the required intense flux of sufficiently energetic photons with a high degree of circular polarization is challenging to produce.13

Recently, the rapid development of petawatt (PW) laser technology14–19 and laser wakefield acceleration20,21 have stimulated a considerable amount of interest in the development of a polarized positron source via the nonlinear Breit–Wheeler (NBW) process.22–29 Positrons created in a strong laser field can be polarized owing to the energetically preferred orientation of the positron spin along the local magnetic field. However, the challenge is that in a symmetric laser field (e.g., in a monochromatic laser field), the polarization of positrons created in different laser half-cycles oscillates following the laser magnetic field and averages out to zero for the total beam. Thus, to achieve net polarization of the created positrons, it is necessary to use an asymmetric laser field. For instance, a two-color laser field has recently been proposed to exploit the ultrafast generation of highly polarized electron30,31 and positron beams.24 Another efficient method for laser-driven generation of polarized electrons (or positrons) has been also demonstrated,23,32 in which the spin-dependent radiation reaction in an elliptically polarized laser pulse is employed to split the electron (or positron) beam into two oppositely transversely polarized parts. Laser-driven positron generation schemes provide a promising avenue for the development of high-current, highly polarized positron sources.

Usually the gamma photon that creates a pair in the NBW process is generated by Compton scattering of the incoming electron beam off a counterpropagating laser field. In most studies, the gamma photon has been assumed to be unpolarized, and the NBW probability has been averaged over the photon polarization.23,24,30,31 In reality, the intermediate gamma photon is partially polarized, which has consequences for further pair production processes.25,33,34 In particular, the decrease in pair density with the inclusion of photon polarization has been shown in analytical QED calculations33 averaged by the lepton spins, which is confirmed by more accurate spin-resolved Monte Carlo simulations.34 Moreover, highly polarized gamma photons can be obtained with polarized seed electrons,13 which in further NBW process may create highly polarized positrons, as has been shown in Monte Carlo simulations25 with the use of a simplified pair production probability summed up over final spin states of either electron or positron. A recent study has shown that arbitrarily polarized lepton beams can be obtained by controlling the polarization of gamma-ray beams and laser configurations in the NBW process.35 Therefore, including photon polarization in the description of the NBW process is mandatory for reliable prediction of the parameters of a laser-driven polarized positron source. The study of fully polarization resolved NBW is also of pure fundamental interest, providing insight into correlations of electron, positron, and photonspolarization in strong-field pair production processes.

An accurate analytical description of strong-field QED processes is possibly only in the case of a plane wave laser field.22,36–40 For QED processes in more realistic scenarios, including focused laser fields and laser–plasma interaction, a Monte Carlo method has been developed41–43 that is based on the local constant field approximation (LCFA)44–50 applicable to intense laser–plasma41–43,51/ultrarelativistic electron interactions.13,24,25,34 Recently, the QED Monte Carlo method has been generalized to include the spin of involved leptons24,32 and the polarization of emitted or absorbed photons.13,25,34,52 A numerical approach suitable for treating polarization effects beyond the LCFA and plane wave approximations at intermediate laser intensities has been developed53,54 for strong-field pair production processes within the semiclassical Baier–Katkov formalism.

In this paper, we investigate the interaction of an ultrarelativistic electron beam colliding head-on with an ultraintense laser pulse and focus on the effects of photon polarization in NBW pair production processes. To provide an accurate analysis of the produced pair polarization, we employ fully polarization-resolved NBW probabilities, i.e., resolved in incoming photon polarization as well as in the created electron and positron polarizations. The probabilities are derived using the Baier–Katkov QED operator method44,55 within the LCFA and have been included in a recently developed laser–electron beam simulation code.25 We consider a scheme where the initial electrons are transversely polarized and the laser field is elliptically polarized. With the fully polarization-resolved Monte Carlo method, we find that the polarization of the produced positrons is highly dependent on the polarization of the parent photons. In particular, we find that the polarization of positrons is reduced by 35%, since the emitted photons are partially polarized along the electric field direction, and that the angular distribution of positron polarization exhibits an abnormal twist near the small-angle region, which originates from pair production of highly polarized photons at the high-energy end of the spectrum.

In this section, we analyze the correlation of photon and positron/electron polarization based on fully polarization-resolved probabilities and briefly elaborate on the Monte Carlo method used for our simulation.

Here, we provide probabilities of a polarized photon emission with a polarized electron. Let us assume that the polarization of the emitted photon is $\vec{e}=a_1{\vec{e}}_1+a_2{\vec{e}}_2$, where$${\vec{e}}_1=\vec{s}-\vec{n}(\vec{n}\cdot \vec{s}),{\vec{e}}_2=\vec{n}\times {\vec{e}}_1.$$(1)$\vec{n}=\vec{k}/|\vec{k}|$ and $\vec{s}=\vec{w}/|\vec{w}|$ are the unit vectors along the photon emission and acceleration directions, respectively. The photon-polarization-resolved emission probability is13$$\begin{array}{l}\hfill d{W}_r=\frac{1}{2}(d{W}_{11}+d{W}_{22})+\frac{1}{2}{\xi }_1(d{W}_{11}-d{W}_{22})\\ \hfill & -i\frac{1}{2}{\xi }_2(d{W}_{21}-d{W}_{12})+\frac{1}{2}{\xi }_3(d{W}_{11}-d{W}_{22})\hfill \\ \hfill & =\frac{1}{2}(F_0+{\xi }_1{F}_1+{\xi }_2{F}_2+{\xi }_3{F}_3),\hfill \end{array}$$(2)where ξ_i (i = 1, 2, 3) are the Stokes parameters with respect to the axes $({\vec{e}}_1,{\vec{e}}_2,\vec{n})$, and$$\begin{array}{l}\hfill F_0=\frac{\alpha }{2\sqrt{3}\pi {\gamma }^2}d\omega \left\{\left[\frac{{\epsilon }^2+{\epsilon }^{\prime 2}}{{\epsilon \epsilon }^{\prime }}{\mathrm{K}}_{2/3}(z_q)-{\int }_{z_q}^{\infty }dx{\mathrm{K}}_{1/3}(x)\right]\right.\\ \hfill & +\left[2{\mathrm{K}}_{2/3}(z_q)-{\int }_{z_q}^{\infty }dx{\mathrm{K}}_{1/3}(x)\right]({\vec{\zeta }}_i\cdot {\vec{\zeta }}_f)\hfill \\ \hfill & -{\mathrm{K}}_{1/3}(z_q)\left[\frac{\omega }{\epsilon }({\vec{\zeta }}_i\cdot \vec{b})+\frac{\omega }{{\epsilon }^{\prime }}({\vec{\zeta }}_f\cdot \vec{b})\right]\hfill \\ \hfill & \left.+\frac{{\omega }^2}{{\epsilon }^{\prime }\epsilon }\left[{\mathrm{K}}_{2/3}(z_q)-{\int }_{z_q}^{\infty }dx{\mathrm{K}}_{1/3}(x)\right]({\vec{\zeta }}_i\cdot \vec{\widehat{v}})({\vec{\zeta }}_f\cdot \vec{\widehat{v}})\right\},\hfill \end{array}$$(3)$$\begin{array}{ll}\hfill F_1=& \frac{\alpha }{2\sqrt{3}\pi {\gamma }^2}d\omega \left\{\frac{{\epsilon }^2-{\epsilon }^{\prime 2}}{2{\epsilon }^{\prime }\epsilon }{\mathrm{K}}_{2/3}(z_q)\left[\vec{\widehat{v}}\cdot ({\vec{\zeta }}_f\times {\vec{\zeta }}_i)\right]\right.\hfill \\ \hfill & +\left[\frac{\omega }{{\epsilon }^{\prime }}({\vec{\zeta }}_i\cdot \vec{s})+\frac{\omega }{\epsilon }({\vec{\zeta }}_f\cdot \vec{s})\right]{\mathrm{K}}_{1/3}(z_q)\hfill \\ \hfill & \left.-\frac{{\omega }^2}{2{\epsilon }^{\prime }\epsilon }{\int }_{z_q}^{\infty }dx{\mathrm{K}}_{1/3}(x)\left[({\vec{\zeta }}_i\cdot \vec{s})({\vec{\zeta }}_f\cdot \vec{b})+({\vec{\zeta }}_i\cdot \vec{b})({\vec{\zeta }}_f\cdot \vec{s})\right]\right\},\hfill \end{array}$$(4)$$\begin{array}{ll}\hfill F_2=& -\frac{\alpha }{2\sqrt{3}\pi {\gamma }^2}d\omega \left\{\frac{{\epsilon }^2-{\epsilon }^{\prime 2}}{2{\epsilon }^{\prime }\epsilon }{\mathrm{K}}_{1/3}(z_q)\left[\vec{s}\cdot ({\vec{\zeta }}_f\times {\vec{\zeta }}_i)\right]\right.\hfill \\ \hfill & +\left[-\frac{{\epsilon }^2-{\epsilon }^{\prime 2}}{{\epsilon }^{\prime }\epsilon }{\mathrm{K}}_{2/3}(z_q)+\frac{\omega }{\epsilon }{\int }_{z_q}^{\infty }dx{\mathrm{K}}_{1/3}(x)\right]({\vec{\zeta }}_i\cdot \vec{\widehat{v}})\hfill \\ \hfill & +\left[-\frac{{\epsilon }^2-{\epsilon }^{\prime 2}}{{\epsilon }^{\prime }\epsilon }{\mathrm{K}}_{2/3}(z_q)+\frac{\omega }{{\epsilon }^{\prime }}{\int }_{z_q}^{\infty }dx{\mathrm{K}}_{1/3}(x)\right]({\vec{\zeta }}_f\cdot \vec{\widehat{v}})\hfill \\ \hfill & \left.+\frac{{\omega }^2}{2{\epsilon }^{\prime }\epsilon }{\mathrm{K}}_{1/3}(z_q)\left[({\vec{\zeta }}_i\cdot \vec{\widehat{v}})({\vec{\zeta }}_f\cdot \vec{b})+({\vec{\zeta }}_i\cdot \vec{b})({\vec{\zeta }}_f\cdot \vec{\widehat{v}})\right]\right\},\hfill \end{array}$$(5)$$\begin{array}{ll}\hfill F_3=& \frac{\alpha }{2\sqrt{3}\pi {\gamma }^2}d\omega \left\{{\mathrm{K}}_{2/3}(z_q)+\frac{{\epsilon }^2+{\epsilon }^{\prime 2}}{2{\epsilon }^{\prime }\epsilon }{\mathrm{K}}_{2/3}(z_q)({\vec{\zeta }}_i\cdot {\vec{\zeta }}_f)\right.\hfill \\ \hfill & -\left[\frac{\omega }{{\epsilon }^{\prime }}({\vec{\zeta }}_i\cdot \vec{b})+\frac{\omega }{\epsilon }({\vec{\zeta }}_f\cdot \vec{b})\right]{\mathrm{K}}_{1/3}(z_q)\hfill \\ \hfill & +\frac{{\omega }^2}{2{\epsilon }^{\prime }\epsilon }\left[-{\mathrm{K}}_{2/3}(z_q)({\vec{\zeta }}_i\cdot \vec{\widehat{v}})({\vec{\zeta }}_f\cdot \vec{\widehat{v}})+{\int }_{z_q}^{\infty }dx{\mathrm{K}}_{1/3}(x)\right.\hfill \\ \hfill & \left.\left.\times \left[({\vec{\zeta }}_i\cdot \vec{b})({\vec{\zeta }}_f\cdot \vec{b})-({\vec{\zeta }}_i\cdot \vec{s})({\vec{\zeta }}_f\cdot \vec{s})\right]\right]\right\}.\hfill \end{array}$$(6)

Here, $\vec{\widehat{v}}=\vec{v}/v$, $\vec{b}=\vec{\widehat{v}}\times \vec{s}$, z_q = 2ω/(3χ_eε′), ε and ε′ are the energies of the emitting particle before and after emission, respectively, ${\vec{\zeta }}_i$ and ${\vec{\zeta }}_f$ are the spin vectors before and after emission, respectively, and ω is the emitted photon energy.

The emission probability and the polarization of the emitted photon both depend on the initial electron spin ${\vec{\zeta }}_i$. For instance, the emission probability is larger for the spin-down electrons, with respect to the magnetic field direction in the rest frame of the electron, than for the spin-up electrons, as shown in Fig. 1(a). The dependence of the polarization of the photon on ${\vec{\zeta }}_i$ is more remarkable, as shown in Fig. 1(b). For the low-energy region, the Stokes parameter ξ₃ ∼ 0.5 regardless of the initial electron spin. However, with an increase in the energy of the emitted photons, ξ₃ increases to ξ₃ = 1 for the spin-up electrons, while it decreases to ξ₃ = −1 for the opposite case.

Here, we provide the probability of polarized electron–positron pair production with a polarized photon. The polarization of the photon is defined as follows:$$\begin{array}{l}\hfill \vec{e}=a_1{\vec{e}}_1+a_2{\vec{e}}_2,\end{array}$$(7)$$\begin{array}{ll}\hfill {\vec{e}}_1& =\frac{\vec{E}-\vec{n}(\vec{n}\cdot \vec{E})+\vec{n}\times \vec{B}}{|\vec{E}-\vec{n}(\vec{n}\cdot \vec{E})+\vec{n}\times \vec{B}|},\hfill \end{array}$$(8)$$\begin{array}{ll}\hfill {\vec{e}}_2& =\vec{n}\times {\vec{e}}_1,\vec{n}=\frac{\vec{k}}{|\vec{k}|}.\hfill \end{array}$$(9)The pair production rate of the polarized photon takes the form$$\begin{array}{ll}\hfill d{W}_p& =\frac{1}{2}(d{W}^{(11)}+d{W}^{(22)})+\frac{1}{2}{\xi }_1(d{W}^{(11)}-d{W}^{(22)})\hfill \\ \hfill & -i\frac{1}{2}{\xi }_2(d{W}^{(21)}-d{W}^{(12)})+\frac{1}{2}{\xi }_3(d{W}^{(11)}-d{W}^{(22)})\hfill \\ \hfill & =\frac{1}{2}(G_0+{\xi }_1{G}_1+{\xi }_2{G}_2+{\xi }_3{G}_3),\hfill \end{array}$$(10)where ξ_i = G_i/G₀ (i = 1, 2, 3) are the Stokes parameters, and$$\begin{array}{l}\hfill G_0=\frac{\alpha m^{2}d\epsilon }{2\sqrt{3}\pi {\omega }^2}\left\{\left[{\int }_{z_p}^{\infty }dx{\mathrm{K}}_{1/3}(x)+\frac{{\epsilon }_{+}^2+{\epsilon }^2}{{\epsilon }_{+}\epsilon }{\mathrm{K}}_{2/3}(z_p)\right]\right.\\ \hfill & +\left[{\int }_{z_p}^{\infty }dx{\mathrm{K}}_{1/3}(x)-2{\mathrm{K}}_{2/3}(z_p)\right]({\vec{\zeta }}_{-}\cdot {\vec{\zeta }}_{+})\hfill \\ \hfill & \backslash !\backslash !\backslash !+\left[\frac{\omega }{{\epsilon }_{+}}({\vec{\zeta }}_{+}\cdot \vec{b})-\frac{\omega }{\epsilon }({\vec{\zeta }}_{-}\cdot \vec{b})\right]{\mathrm{K}}_{1/3}(z_p)\hfill \\ \hfill & +\left[\frac{{\epsilon }_{+}^2+{\epsilon }^2}{{\epsilon \epsilon }_{+}}{\int }_{z_p}^{\infty }dx{\mathrm{K}}_{1/3}(x)\right.\hfill \\ \hfill & \left.\left.-\frac{{({\epsilon }_{+}-\epsilon )}^2}{{\epsilon \epsilon }_{+}}{\mathrm{K}}_{2/3}(z_p)\right]({\vec{\zeta }}_{-}\cdot \vec{\widehat{v}})({\vec{\zeta }}_{+}\cdot \vec{\widehat{v}})\right\},\hfill \end{array}$$(11)$$\begin{array}{ll}\hfill G_1=& \frac{\alpha m^{2}d\epsilon }{2\sqrt{3}\pi {\omega }^2}\left\{-\frac{{\epsilon }_{+}^2-{\epsilon }^2}{2{\epsilon }_{+}\epsilon }{\mathrm{K}}_{2/3}(z_p)\left[\vec{\widehat{v}}\cdot ({\vec{\zeta }}_{+}\times {\vec{\zeta }}_{-})\right]\right.\hfill \\ \hfill & +\left[\frac{\omega }{\epsilon }({\vec{\zeta }}_{+}\cdot \vec{s})-\frac{\omega }{{\epsilon }_{+}}({\vec{\zeta }}_{-}\cdot \vec{s})\right]{\mathrm{K}}_{1/3}(z_p)\hfill \\ \hfill & \left.-\frac{{\omega }^2}{2{\epsilon }_{+}\epsilon }{\int }_{z_p}^{\infty }dx{\mathrm{K}}_{1/3}(x)\left[({\vec{\zeta }}_{-}\cdot \vec{b})({\vec{\zeta }}_{+}\cdot \vec{s})+({\vec{\zeta }}_{-}\cdot \vec{s})({\vec{\zeta }}_{+}\cdot \vec{b})\right]\right\},\hfill \end{array}$$(12)$$\begin{array}{ll}\hfill G_2=\backslash !\backslash !\backslash !\backslash !& \frac{\alpha m^{2}d\epsilon }{2\sqrt{3}\pi {\omega }^2}\left\{-\frac{{\omega }^2}{2{\epsilon }_{+}\epsilon }{\mathrm{K}}_{1/3}(z_p)\left[\vec{s}\cdot ({\vec{\zeta }}_{-}\times {\vec{\zeta }}_{+})\right]\right.\hfill \\ \hfill & +\left[\frac{\omega }{{\epsilon }_{+}}{\int }_{z_p}^{\infty }dx{\mathrm{K}}_{1/3}(x)+\frac{{\epsilon }_{+}^2-{\epsilon }^2}{{\epsilon }_{+}\epsilon }{\mathrm{K}}_{2/3}(z_p)\right]({\vec{\zeta }}_{+}\cdot \vec{\widehat{v}})\hfill \\ \hfill & +\left[\frac{\omega }{\epsilon }{\int }_{z_p}^{\infty }dx{\mathrm{K}}_{1/3}(x)-\frac{{\epsilon }_{+}^2-{\epsilon }^2}{{\epsilon }_{+}\epsilon }{\mathrm{K}}_{2/3}(z_p)\right]({\vec{\zeta }}_{-}\cdot \vec{\widehat{v}})\hfill \\ \hfill & \left.-\frac{{\epsilon }_{+}^2-{\epsilon }^2}{2{\epsilon }_{+}\epsilon }{\mathrm{K}}_{1/3}(z_p)\left[({\vec{\zeta }}_{-}\cdot \vec{\widehat{v}})({\vec{\zeta }}_{+}\cdot \vec{b})+({\vec{\zeta }}_{-}\cdot \vec{b})({\vec{\zeta }}_{+}\cdot \vec{\widehat{v}})\right]\right\},\hfill \end{array}$$(13)$$\begin{array}{ll}\hfill G_3=\backslash !\backslash !\backslash !& \frac{\alpha m^{2}d\epsilon }{2\sqrt{3}\pi {\omega }^2}\left\{-{\mathrm{K}}_{2/3}(z_p)+\frac{{\epsilon }_{+}^2+{\epsilon }^2}{2{\epsilon }_{+}\epsilon }{\mathrm{K}}_{2/3}(z_p)({\vec{\zeta }}_{-}\cdot {\vec{\zeta }}_{+})\right.\hfill \\ \hfill & +\left[-\frac{\omega }{\epsilon }\left({\vec{\zeta }}_{+}\vec{b}\right)+\frac{\omega }{{\epsilon }_{+}}\left({\vec{\zeta }}_{-}\vec{b}\right)\right]{\mathrm{K}}_{1/3}(z_p)\hfill \\ \hfill & -\frac{{({\epsilon }_{+}-\epsilon )}^2}{2{\epsilon }_{+}\epsilon }{\mathrm{K}}_{2/3}(z_p)({\vec{\zeta }}_{-}\cdot \vec{\widehat{v}})({\vec{\zeta }}_{+}\cdot \vec{\widehat{v}})\hfill \\ \hfill & \left.+\frac{{\omega }^2}{2{\epsilon }_{+}\epsilon }{\int }_{z_p}^{\infty }dx{\mathrm{K}}_{1/3}(x)\left[({\vec{\zeta }}_{-}\cdot \vec{b})({\vec{\zeta }}_{+}\cdot \vec{b})-({\vec{\zeta }}_{-}\cdot \vec{s})({\vec{\zeta }}_{+}\cdot \vec{s})\right]\right\}.\hfill \end{array}$$(14)

Here, $\vec{\widehat{v}}=\vec{v}/v$ is the velocity direction of the produced pairs, which satisfies ${\vec{\widehat{v}}}_{-}\approx {\vec{\widehat{v}}}_{+}\approx \vec{n}$ in the relativistic case, $\vec{s}=\vec{w}/|\vec{w}|$ is the acceleration direction of the produced particles, and $\vec{b}=\vec{\widehat{v}}\times \vec{s}$ is the magnetic field direction in the rest frame of the electron/positron. The parameter z_p = 2ω/(3χ_γεε₊), where ω, ε, and ε₊ are the energies of the parent photon and the produced electron and positron, respectively, and the quantum strong-field parameter ${\chi }_{\gamma }=|e|\sqrt{{(F_{\mu \nu }{k}^{\nu })}^2}/m^3$, with F_μν and k^ν being the electromagnetic field strength tensor and the photon momentum four-vector, respectively. Averaging over the photon polarization yields the spin-resolved pair production probability44,56,57$d{W}_p(0)=\frac{1}{2}{G}_0$.

Summing over the final spin states of the electron, one can obtain the polarization of the positron depending on the photon polarization:25,44$${\vec{\zeta }}_{+}^{f,\xi }=\frac{{\xi }_1{f}_3\frac{\omega }{\epsilon }\vec{s}+{\xi }_2\vec{\widehat{v}}\left(\frac{\omega }{{\epsilon }_{+}}{f}_1+\frac{{\epsilon }_{+}^2-{\epsilon }^2}{{\epsilon \epsilon }_{+}}{f}_2\right)+\left(\frac{\omega }{{\epsilon }_{+}}-{\xi }_3\frac{\omega }{\epsilon }\right)\vec{b}{f}_3}{f_1+\frac{{\epsilon }^2+{\epsilon }_{+}^2}{{\epsilon \epsilon }_{+}}{f}_2-{\xi }_3{f}_2}.$$(15)In the case of an unpolarized photon,23$${\vec{\zeta }}_{+}^{f,0}=\frac{\frac{\omega }{{\epsilon }_{+}}\vec{b}{f}_3}{f_1+\frac{{\epsilon }^2+{\epsilon }_{+}^2}{{\epsilon \epsilon }_{+}}{f}_2},$$(16)where $f_1={\int }_{z_p}^{\infty }dx{\mathrm{K}}_{1/3}(x)$, and f₂ = K_2/3(z_p), f₃ = K_1/3(z_p). Equations (15) and (16) show that the photon polarization has significant effects on the positron polarization ${\vec{\zeta }}_{+}^f$. The longitudinal polarization of positrons is completely missing if the photon polarization is averaged over, while the transverse polarization either increases or decreases, as determined by ξ₁ and ξ₃.

The correlation of the electron and positron polarizations in the pair production is analyzed in Fig. 2. In the case ξ₁ = ξ₂ = 0 and ξ₃ > 0, the probabilities of e⁺e⁻ co-polarization are higher than those of counter-polarization with respect to the magnetic field direction, i.e., dW_↑↑, dW_↓↓ > dW_↓↑, dW_↑↓, as shown in Fig. 2(a). On the other hand, when ξ₃ < 0, the probability of producing an electron with spin down and a positron with spin up dominates, i.e., dW_↓↑ > dW_↓↓, dW_↑↑, dW_↑↓, as shown in Fig. 2(b).

After integration over the electron spin, one obtains the dependence of the positron spin on the positron energy and the polarization of the parent photon, as shown in Fig. 2(c). For ξ₃ < 0, the positron polarization degree decreases gradually with increasing positron energy. The domination of dW_↓↑ results in a high polarization degree of positrons with spin up through the whole spectrum. For a photon with ξ₃ > 0, the polarization of the produced positron decreases dramatically to a negative value with increasing energy, resulting in a smaller averaged polarization compared with the case of ξ₃ = 0. In particular, when ξ₃ ∼ 1, as shown in Fig. 2(a), the probability ${\sum }_{{\zeta }_{-}}d{W}_{{\zeta }_{-}\uparrow }\approx {\sum }_{{\zeta }_{-}}d{W}_{{\zeta }_{-}\downarrow }$. The polarizations of positrons in high-energy (δ₊ > 0) and low-energy (δ₊ < 0) regions cancel each other, producing unpolarized positrons after energy integration. Therefore, if the parent photon is polarized along the laser polarization direction, i.e., ξ₃ = 1, then the produced pairs are unpolarized. However, if the polarization of the parent photon is orthogonal to the laser polarization, i.e., ξ₃ = −1, the produced pairs have a high degree of polarization, with positrons spin-up and electrons spin-down. After integration over the positron energy, one obtains the relation between the positron polarization and that of its parent photon, as shown in Fig. 2(e). The polarization of the positron decreases monotonically with increasing ξ₃, which provides a way of estimating the polarization of the intermediate photons during nonlinear Compton scattering. For instance, if the polarization of positrons is measured to be 37% at χ_γ = 1, then the polarization of the intermediate photons is around ξ₃ = 0.5. The dependences of the relative differences ΔN and Δζ_y on χ_γ are shown in Fig. 2(f). For ξ₃ = 0.5, the relative difference in positron number ΔN decreases with increasing χ_γ, while the relative difference in positron polarization Δζ_y increases with χ_γ increases.

The photon polarization affects not only the polarization of produced pairs, but also the pair density, as shown in Fig. 2(d). When ξ₃ > 0, the pair production probability is smaller than in the case where photon polarization is unresolved, i.e., ξ₃ = 0. In contrast, photons with ξ₃ < 0 yield more pairs than unpolarized photons.

To simulate the nonlinear Compton scattering of a strong laser pulse at an ultrarelativistic electron beam, we modified the three-dimensional Monte Carlo method25 by employing the fully polarization-resolved pair production probabilities given in Sec. II B. The developed Monte-Carlo method includes the correlation of electron and positron spins, providing a complete description of polarization effects in strong-field QED processes. In each simulation step, one calculates the total emission rate to determine the occurrence of photon emission and the pair production rate to determine the pair production event, using a common QED Monte Carlo stochastic algorithm.41–43 The spins of electron/positron after emission and creation are determined by the polarization-resolved emission probability of Sec. II A and the pair production probability of Sec. II B, respectively, according to the stochastic algorithm. The electron/positron spin instantaneously collapses into one of its basis states defined with respect to the instantaneous spin quantization axis (SQA), which is chosen according to the properties of the scattering process. Moreover, since the probability of no emission is also polarization-resolved and asymmetric along an arbitrary SQA, it is necessary to include the spin variation between emissions induced by radiative polarization, as well as the spin precession governed by the Thomas–Bargmann–Michel–Telegdi equation;58–60 for more details, see Ref. 25. The polarization of the emitted photon is determined using a similar stochastic procedure, which has also been used in laser–plasma simulation codes.61

Recently, various schemes have been proposed to produce transversely polarized positrons via strong lasers, but these have neglected photon polarization. Here, we proceed to investigate photon polarization effects on the transverse polarization of positrons with the fully polarization-resolved Monte Carlo method, and compare the result with that obtained using the unpolarized photon model.

A PW laser with intensity ξ₀ = |e|E₀/mω = 100 (I = 10²² W/cm²) counterpropagates with a relativistic electron beam with an energy of ε₀ = 10 GeV, with the setup shown in Fig. 3. The wavelength of the laser is λ₀ = 1 µm, the beam waist size w = 5λ₀, the pulse duration τ_p = 8 T, and the ellipticity ε = 0.03. The electron beam consists of N_e = 6 × 10⁶ electrons, with the beam length L_e = 5λ₀, beam radius r_e = λ₀, energy divergence Δε = 0.06, and angular divergences Δθ = 0.3 mrad and Δϕ = 1 mrad. The initial electron beam is fully polarized along the y direction.

The impact of the photon polarization on pair production is elucidated in Fig. 4. The positron density decreases when the photon polarization is resolved in both photon emission and pair production processes, as shown in Figs. 3(a), 3(c), and 3(e). The difference in positron density between using a polarization-resolved or unresolved treatment is approximately (N(0) − N(ξ))/N(ξ) ≈ 12.6%. More importantly, the y component of polarization ${\overline{\zeta }}_y$ decreases dramatically at small angles, even showing a reversal of polarization direction [see Fig. 4(d)]. As a consequence, the symmetric angular distribution of ${\overline{\zeta }}_y$ near θ_y = 0 is distorted when the intermediate photon polarization is considered, as shown in Fig. 4(f). The angular distribution of ${\overline{\zeta }}_y$ oscillates around the small-angle region, instead of exhibiting a monotonic increase as in the photon-polarization-averaged case. The average polarization of the positron beam decreases by $[{\overline{\zeta }}_y(0)-{\overline{\zeta }}_y(\xi )]/{\overline{\zeta }}_y(\xi )\approx 35\%$.

To understand the effects of photon polarization on pair production, we investigate the polarization of emitted photons in ${\theta }_y^{\gamma }>0$ and ${\theta }_y^{\gamma }<0$ separately, as shown in Fig. 5. The photon density emitted within the angular region ${\theta }_y^{\gamma }<0$ is larger than within the region ${\theta }_y^{\gamma }>0$, especially in the high-energy region, as shown in Fig. 5(a). When an electron with ${\vec{\zeta }}_i=\vec{b}$ counterpropagates to the laser field, the direction of the instantaneous quantization axis is $\vec{n}={\vec{\zeta }}^f/|{\vec{\zeta }}^f|$,25 with$${\vec{\zeta }}^f=\frac{(2f_2-f_1){\vec{\zeta }}_i-\frac{\omega }{{\epsilon }^{\prime }}\vec{b}{f}_3+\frac{\omega }{{\epsilon }^{\prime }\epsilon }(f_2-f_1)({\vec{\zeta }}_i\cdot \vec{\widehat{v}})\vec{\widehat{v}}}{\frac{{\epsilon }^2+{\epsilon }^{\prime 2}}{{\epsilon }^{\prime }\epsilon }{f}_2-f_1-\frac{\omega }{\epsilon }{\vec{\zeta }}_i\cdot \vec{b}{f}_3},$$(17)which is mostly along B_y and changes sign every half-cycle. For an electron with initial spin ζ_y = 1, its spin is parallel (spin-up) and antiparallel (spin-down) to the quantization axis in the half-cycles with B_y > 0 and B_y < 0, respectively. The emission probability is larger when the electron is spin-down before the emission, as shown in Fig. 5(a), which is in accordance with the analysis of the spin-resolved probability given in Fig. 1(a). Further, the photon emission direction is parallel to the momentum of the emitting particle, and the photons emitted when B_y > 0 and B_y < 0 propagate with p_y < 0 and p_y > 0, respectively, because the oscillation phase of p_y has a π delay with respect to B_y. Therefore, the electrons emit photons with ${\theta }_y^{\gamma }>0$ at B_y > 0 and photons with ${\theta }_y^{\gamma }<0$ at B_y < 0. The latter case has a higher emitted photon number than the former, owing to the larger emission probability W_r↓ > W_r↑, as explained above. More importantly, the electrons with spin up (spin down) have a higher probability to emit photons with −1 < ξ₃ < 0.5 (0.5 < ξ₃ < 1). Therefore, the radiation with θ_y > 0 comes mainly from photon emission at B_y > 0, and has polarization −1 < ξ₃ < 0.5, while the radiation with θ_y < 0 comes from photon emission at B_y < 0, which has polarization 0.5 < ξ₃ < 1, as shown in Fig. 5(b).

The fringes in the angular distribution seen in Figs. 5(c)–5(e) are due to the radiation in different laser cycles. As shown in Fig. 5(b), ${\overline{\xi }}_3$ in the high-energy region is positive for spin-down electrons and negative for spin-up electrons. However, since the radiation is dominated by low-energy photons with ξ₃ ∼ 0.51, the angular distribution of the photon polarization in Fig. 5(d) is also dominated by ξ₃ ∼ 0.51. Nevertheless, the low-energy emissions have been filtered, a correlation of photon polarization and emission angle can be seen in Figs. 4(e) and 4(f). As expected, the photons distributed in ${\theta }_y^{\gamma }<0$ have ${\overline{\xi }}_3<0$, while in the region ${\theta }_y^{\gamma }<0$, the polarization is ${\overline{\xi }}_3>0$. Since the pairs are produced mostly by energetic photons, the distinct polarization properties of high-energy photons in ${\theta }_y^{\gamma }<0$ and ${\theta }_y^{\gamma }>0$ break the symmetry of the angular distribution of polarization.

The separation of positron polarization along the propagation direction can be explained,32 taking into account that the final momentum of the created electron (positron) is determined by the laser vector potential A_y(t_p) at the instant of creation t_p: p_f = p_i + eA_y(t_p), where p_i is the momentum inherited from the parent photon, with vanishing average value ${\overline{p}}_i=0$. On the other hand, when the photon is linearly polarized with (0, 0, ξ₃), the SQA for pair production of such a photon is along the laser magnetic field direction, $\vec{n}={\vec{\zeta }}_{+}^{f,\xi }/|{\vec{\zeta }}_{+}^{f,\xi }|=\vec{b}$. Therefore, the positrons produced at A_y(t_p) < 0 acquire a final momentum p_f ≈ eA_y(t_p) > 0 and are polarized along ζ_y > 0, since the instantaneous SQA is along y > 0 at t_p. Similarly, one has ζ_y < 0 when p_f < 0.

To reveal the origin of the abnormal polarization features in the small-angle region, we artificially turn on pair production of photons with ${\theta }_y^{\gamma }>0$ and ${\theta }_y^{\gamma }<0$ separately. As shown in Fig. 5(b), the photons with ${\theta }_y^{\gamma }<0$ have ξ₃ ∼ 1 at the high-energy end of the spectrum. These hard photons have a higher probability of producing energetic positrons antiparallel with the magnetic field, as discussed in Sec. II B, resulting in a reversed polarization direction in the small-angle region and an overall decrease in the averaged polarization of positrons, as shown in Fig. 6(b). Moreover, since the parent photon with ${\theta }_y^{\gamma }<0$ has ${\overline{p}}_i<0$, the positron distribution is slightly shifted toward ${\theta }_y^{+}<0$. Meanwhile, the polarization of hard photons at the ${\theta }_y^{\gamma }>0$ side reaches ξ₃ ∼ −1 [Fig. 5(b)]. If these photons are collected to produce pairs, highly polarized positrons could be obtained owing to the overwhelming dominance of the spin-up positrons, dW_↓↑(ξ₃ = −1) ≫ dW_↓↑(ξ₃ = 0). In the present case, the ${\overline{\zeta }}_y$ of positrons produced by photons with ${\theta }_y^{\gamma }>0$ increases monotonically with ${\theta }_y^{+}$, and the angular distribution shifts slightly toward ${\theta }_y^{+}>0$. The pair production probability is inversely proportional to ξ₃, and the photons with ${\theta }_y^{\gamma }<0$ have larger ${\overline{\xi }}_3$ than those with ${\theta }_y^{\gamma }>0$, and therefore more positrons are produced at ${\theta }_y^{\gamma }>0$, as shown in Fig. 6(a). Thus, the domination of pair production of photons in ${\theta }_y^{\gamma }<0$ and ${\overline{p}}_i<0$ results in a decrease in the polarization of the positron beam and an asymmetric angular distribution of polarization.

When the initial electrons are unpolarized as in Ref. 23, the photons are equally distributed in ${\theta }_y^{\gamma }>0$ and ${\theta }_y^{\gamma }<0$, with polarization ${\overline{\xi }}_3\approx 0.51\%$, as shown in Figs. 7(a) and 7(b). Consequently, the abnormal polarization feature in the small-angle region vanishes because of a lack of angle-dependent photon polarization with ξ₃ ∼ 1. As shown in Figs. 7(c) and 7(d), the produced positrons have a symmetric angular distribution of polarization. On the other hand, the positron density and average polarization decrease when the photon polarization is resolved, regardless of the spin of the initial electrons, as also shown in Figs. 7(c) and 7(d). The difference in positron density between using a polarization-resolved or unresolved treatment is approximately ΔN = [N(ξ) − N(0)]/N(ξ) ≈ 12%, and the difference in polarization is $\mathrm{\Delta }{\overline{\zeta }}_y=[{\overline{\zeta }}_y(\xi )-{\overline{\zeta }}_y(0)]/{\overline{\zeta }}_y(\xi )\approx 34\%$, both of , which are close to the values obtained with our scheme, namely, ΔN ≈ 13% and $\mathrm{\Delta }{\overline{\zeta }}_y\approx 35\%$. This is because the photons emitted are polarized with ${\overline{\xi }}_3\approx 0.51$, regardless of the initial spin of the electrons. Specifically, since the photon polarization reads13$${\xi }_3=\frac{f_2-\frac{\omega }{{\epsilon }^{\prime }}({\vec{\zeta }}_i\cdot \vec{b})f_3}{-f_1+\frac{{\epsilon }^2+{\epsilon }^{\prime 2}}{{\epsilon \epsilon }^{\prime }}{f}_2-\frac{\omega }{\epsilon }({\vec{\zeta }}_i\cdot \vec{b})f_3},$$(18)and$$-f_1+\frac{{\epsilon }^2+{\epsilon }^{\prime 2}}{{\epsilon \epsilon }^{\prime }}{f}_2\ll \frac{\omega }{\epsilon }{f}_3$$(19)for most emitted photons, the average polarization$${\overline{\xi }}_3({\vec{\zeta }}_i)\approx {\overline{\xi }}_3(0)=f_2/\left(-f_1+\frac{{\epsilon }^2+{\epsilon }^{\prime 2}}{{\epsilon \epsilon }^{\prime }}{f}_2\right)\approx 0.51,$$(20)owing to the change in sign of the term ${\vec{\zeta }}_i\cdot \vec{b}$ as the field oscillates. Therefore, the initial electron spin is not important for density/polarization decrease, but essential for polarization features in the small-angle region, which could be used as additional information for detecting photon polarization.

With experimental feasibility in mind, it is interesting to find how ΔN and $\Delta {\overline{\zeta }}_y$ depend on the laser parameters. The produced positron number N⁺ ∝ N_γχ_γτ_p, where N_γ is the gamma-photon number and χ_γ ∝ a₀ε₀ is the quantum strong-field parameter. Thus, N⁺ increases with increasing a₀ and τ_p, as shown in Figs. 8(a) and 8(c). Moreover, since the relative difference ΔN is inversely proportional to χ_γ, as shown in Fig. 2(f), ΔN decreases as a₀ increases, owing to the increase in χ_γ. On the other hand, since the χ_γ barely changes with τ_p, the variation of ΔN is negligible in Fig. 8(c). Meanwhile, $\Delta {\overline{\zeta }}_y$ of the produced positrons increases with χ_γ, as shown in Fig. 2(f). However, the positron polarization decreases after the instant of creation t_i, owing to further radiative polarization in the laser field, which affects the scaling law of $\mathrm{\Delta }{\overline{\zeta }}_y$. The time evolution of the positron polarization due to radiative polarization is given by62$${\overline{\zeta }}_y(t)\propto {\overline{\zeta }}_y(t_i)e^{-{\mathrm{\Psi }}_1({\chi }_e)t},$$(21)with$${\Psi }_1({\chi }_e)={\int }_0^{\infty }\frac{u^{2}du}{{(1+u)}^3}{\mathrm{K}}_{2/3}\left(\frac{2}{3}\frac{u}{{\chi }_e}\right),$$(22)where u = ω/(ε₀ − ω). Therefore, the decrease in ${\overline{\zeta }}_y$ is proportional to ${\overline{\zeta }}_y(t_i)$, χ_e, and the interaction time t_f − t_i. As the laser intensity increases, the decrease in polarization is more dramatic, owing to the increase in χ_e, as shown in Fig. 8(b). Since the ${\overline{\zeta }}_y(t_i)$ is larger in the case of unresolved photon polarization than in the resolved case, the rate of reduction rate in the former is larger than in the later. Therefore, Δζ_y decreases as a₀ increases. Similarly, as the laser pulse duration increases, the final polarization decreases, owing to enhancement of the radiative polarization. This effect is more noticeable in the case of unresolved photon polarization, owing to the higher ${\overline{\zeta }}_y(t_i)$, and therefore Δζ_y is smaller for longer laser pulses.

We have investigated the effects of photon polarization on pair production in the nonlinear Breit–Wheeler process via a newly developed Monte Carlo method employing fully polarization resolved quantum probabilities. We have shown that the longitudinal polarization of the produced positrons is induced solely by the photon polarization, while their transverse polarization can increase, decrease, or even be unchanged, depending on the polarization of the intermediate gamma photons. For the interaction of initially transversely polarized electrons and an elliptically polarized laser, both the polarization degree and density of the positrons are reduced when the polarization of the intermediate photons is taken into account. This is because the photons emitted during the nonlinear Compton process are partially polarized along the electric field direction, with ${\overline{\xi }}_3\approx 0.51$. The hard photons in the angular region ${\theta }_y^{\gamma }<0$ have even higher polarization ${\overline{\xi }}_3\sim 1$, causing the energetic positrons produced in the small-angle region to reverse the polarization direction. If one separates the intermediate hard gamma photons within ${\theta }_y^{\gamma }>0$, the polarization of positrons will be greatly enhanced owing to the dominance of dW_↓↑ probabilities throughout the spectrum. Our results confirm the important role of the intermediate photon polarization during strong-field QED processes and should be taken into account in the design and optimization of practical laser-driven polarized positron sources. Moreover, measurement of positron polarization in pair production processes can shed light on the intermediate interaction dynamics, particularly on the polarization properties of intermediate photons.

Acknowledgment. This work was supported by the National Natural Science Foundation of China (Grants No. 12074262) and the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning and the Shanghai Rising-Star Program.