Deep understanding of the impact of photon polarization on pair production is essential for the efficient generation of laser-driven polarized positron beams and demands a complete description of polarization effects in strong-field QED processes. Employing fully polarization-resolved Monte Carlo simulations, we investigate correlated photon and electron (positron) polarization effects in the multiphoton Breit–Wheeler pair production process during the interaction of an ultrarelativistic electron beam with a counterpropagating elliptically polarized laser pulse. We show that the polarization of ee+ pairs is degraded by 35% when the polarization of the intermediate photon is resolved, accompanied by an ∼13% decrease in the pair yield. Moreover, in this case, the polarization direction of energetic positrons at small deflection angles can even be reversed when high-energy photons with polarization parallel to the laser electric field are involved.

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 ee+ 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 e=a1e1+a2e2, where

e1=sn(ns),e2=n×e1.
(1)

n=k/|k| and s=w/|w| are the unit vectors along the photon emission and acceleration directions, respectively. The photon-polarization-resolved emission probability is13 

dWr=12(dW11+dW22)+12ξ1(dW11dW22)i12ξ2(dW21dW12)+12ξ3(dW11dW22)=12(F0+ξ1F1+ξ2F2+ξ3F3),
(2)

where ξi (i = 1, 2, 3) are the Stokes parameters with respect to the axes (e1,e2,n), and

F0=α23πγ2dωε2+ε2εεK2/3(zq)zqdxK1/3(x)+2K2/3(zq)zqdxK1/3(x)(ζiζf)K1/3(zq)ωε(ζib)+ωε(ζfb)+ω2εεK2/3(zq)zqdxK1/3(x)(ζiv̂)(ζfv̂),
(3)
F1=α23πγ2dωε2ε22εεK2/3(zq)v̂(ζf×ζi)+ωε(ζis)+ωε(ζfs)K1/3(zq)ω22εεzqdxK1/3(x)(ζis)(ζfb)+(ζib)(ζfs),
(4)
F2=α23πγ2dωε2ε22εεK1/3(zq)s(ζf×ζi)+ε2ε2εεK2/3(zq)+ωεzqdxK1/3(x)(ζiv̂)+ε2ε2εεK2/3(zq)+ωεzqdxK1/3(x)(ζfv̂)+ω22εεK1/3(zq)(ζiv̂)(ζfb)+(ζib)(ζfv̂),
(5)
F3=α23πγ2dωK2/3(zq)+ε2+ε22εεK2/3(zq)(ζiζf)ωε(ζib)+ωε(ζfb)K1/3(zq)+ω22εεK2/3(zq)(ζiv̂)(ζfv̂)+zqdxK1/3(x)×(ζib)(ζfb)(ζis)(ζfs).
(6)

Here, v̂=v/v, b=v̂×s, zq = 2ω/(3χeε′), ε and ε′ are the energies of the emitting particle before and after emission, respectively, ζi and ζf 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 ζi. 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 ζi 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.

FIG. 1.

(a) Photon emission probability log10Wi (arbitrary units) and (b) Stokes parameter ξ3i (i ∈ ↑, ↓) vs emitted photon energy δe = ωγ/εi for χe = 10. i denotes the electron spin before the emission with respect to the magnetic field direction.

FIG. 1.

(a) Photon emission probability log10Wi (arbitrary units) and (b) Stokes parameter ξ3i (i ∈ ↑, ↓) vs emitted photon energy δe = ωγ/εi for χe = 10. i denotes the electron spin before the emission with respect to the magnetic field direction.

Close modal

Here, we provide the probability of polarized electron–positron pair production with a polarized photon. The polarization of the photon is defined as follows:

e=a1e1+a2e2,
(7)
e1=En(nE)+n×B|En(nE)+n×B|,
(8)
e2=n×e1,n=k|k|.
(9)

The pair production rate of the polarized photon takes the form

dWp=12(dW(11)+dW(22))+12ξ1(dW(11)dW(22))i12ξ2(dW(21)dW(12))+12ξ3(dW(11)dW(22))=12(G0+ξ1G1+ξ2G2+ξ3G3),
(10)

where ξi = Gi/G0 (i = 1, 2, 3) are the Stokes parameters, and

G0=αm2dε23πω2zpdxK1/3(x)+ε+2+ε2ε+εK2/3(zp)+zpdxK1/3(x)2K2/3(zp)(ζζ+)\!\!\!+ωε+(ζ+b)ωε(ζb)K1/3(zp)+ε+2+ε2εε+zpdxK1/3(x)(ε+ε)2εε+K2/3(zp)(ζv̂)(ζ+v̂),
(11)
G1=αm2dε23πω2ε+2ε22ε+εK2/3(zp)v̂(ζ+×ζ)+ωε(ζ+s)ωε+(ζs)K1/3(zp)ω22ε+εzpdxK1/3(x)(ζb)(ζ+s)+(ζs)(ζ+b),
(12)
G2=\!\!\!\!αm2dε23πω2ω22ε+εK1/3(zp)s(ζ×ζ+)+ωε+zpdxK1/3(x)+ε+2ε2ε+εK2/3(zp)(ζ+v̂)+ωεzpdxK1/3(x)ε+2ε2ε+εK2/3(zp)(ζv̂)ε+2ε22ε+εK1/3(zp)(ζv̂)(ζ+b)+(ζb)(ζ+v̂),
(13)
G3=\!\!\!αm2dε23πω2K2/3(zp)+ε+2+ε22ε+εK2/3(zp)(ζζ+)+ωεζ+b+ωε+ζbK1/3(zp)(ε+ε)22ε+εK2/3(zp)(ζv̂)(ζ+v̂)+ω22ε+εzpdxK1/3(x)(ζb)(ζ+b)(ζs)(ζ+s).
(14)

Here, v̂=v/v is the velocity direction of the produced pairs, which satisfies v̂v̂+n in the relativistic case, s=w/|w| is the acceleration direction of the produced particles, and b=v̂×s is the magnetic field direction in the rest frame of the electron/positron. The parameter zp = 2ω/(3χγεε+), 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 χγ=|e|(Fμνkν)2/m3, with Fμν and kν being the electromagnetic field strength tensor and the photon momentum four-vector, respectively. Averaging over the photon polarization yields the spin-resolved pair production probability44,56,57dWp(0)=12G0.

Summing over the final spin states of the electron, one can obtain the polarization of the positron depending on the photon polarization:25,44

ζ+f,ξ=ξ1f3ωεs+ξ2v̂ωε+f1+ε+2ε2εε+f2+ωε+ξ3ωεbf3f1+ε2+ε+2εε+f2ξ3f2.
(15)

In the case of an unpolarized photon,23 

ζ+f,0=ωε+bf3f1+ε2+ε+2εε+f2,
(16)

where f1=zpdxK1/3(x), and f2 = K2/3(zp), f3 = K1/3(zp). Equations (15) and (16) show that the photon polarization has significant effects on the positron polarization ζ+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).

FIG. 2.

(a) and (b) Pair production probability dWζζ+ for a polarized photon with (a) ξ1 = ξ2 = 0, ξ3 = 1 and (b) ξ3 = −1. (c) Positron polarization ζ+=ζdWζdWζdWζ+dWζ vs ξ3 and δ+. In (a)–(c), χγ = 3. (d) Ratio of the photon-polarization-resolved and averaged pair production probabilities dWp(ξ)dWp(0)dWp(ξ)+dWp(0) vs ξ3 and δ+. (e) ζ¯+=iζ+(δi)dWp(δi)idWp(δi) vs ξ3 for χγ = 0.1 (blue solid curve), 1 (red dashed curve), and 10 (magenta dotted curve). (f) Relative differences ΔN = [N(0) − N(ξ)]/N(ξ) (solid curve) and Δζy=[ζy+(0)ζy+(ξ)]/ζy+(ξ) (dashed curve) vs χγ for ξ3 = 0.5.

FIG. 2.

(a) and (b) Pair production probability dWζζ+ for a polarized photon with (a) ξ1 = ξ2 = 0, ξ3 = 1 and (b) ξ3 = −1. (c) Positron polarization ζ+=ζdWζdWζdWζ+dWζ vs ξ3 and δ+. In (a)–(c), χγ = 3. (d) Ratio of the photon-polarization-resolved and averaged pair production probabilities dWp(ξ)dWp(0)dWp(ξ)+dWp(0) vs ξ3 and δ+. (e) ζ¯+=iζ+(δi)dWp(δi)idWp(δi) vs ξ3 for χγ = 0.1 (blue solid curve), 1 (red dashed curve), and 10 (magenta dotted curve). (f) Relative differences ΔN = [N(0) − N(ξ)]/N(ξ) (solid curve) and Δζy=[ζy+(0)ζy+(ξ)]/ζy+(ξ) (dashed curve) vs χγ for ξ3 = 0.5.

Close modal

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 ζdWζζdWζ. 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 χγ = 1, then the polarization of the intermediate photons is around ξ3 = 0.5. The dependences of the relative differences ΔN and Δζy on χγ are shown in Fig. 2(f). For ξ3 = 0.5, the relative difference in positron number ΔN decreases with increasing χγ, while the relative difference in positron polarization Δζy increases with χγ 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/ = 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 τp = 8 T, and the ellipticity ε = 0.03. The electron beam consists of Ne = 6 × 106 electrons, with the beam length Le = 5λ0, beam radius re = 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.

FIG. 3.

Scheme for producing polarized positrons via nonlinear Compton scattering of an initially transversely polarized electron off a strong elliptically polarized laser pulse. The energetic gamma photons in the region θyγ<0 have high polarizations up to ξ3 = 1, resulting in a reduction in or even reversal of polarization of positrons as θy+ decreases.

FIG. 3.

Scheme for producing polarized positrons via nonlinear Compton scattering of an initially transversely polarized electron off a strong elliptically polarized laser pulse. The energetic gamma photons in the region θyγ<0 have high polarizations up to ξ3 = 1, resulting in a reduction in or even reversal of polarization of positrons as θy+ decreases.

Close modal

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 ζ¯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 ζ¯y near θy = 0 is distorted when the intermediate photon polarization is considered, as shown in Fig. 4(f). The angular distribution of ζ¯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 [ζ¯y(0)ζ¯y(ξ)]/ζ¯y(ξ)35%.

FIG. 4.

(a) Polarized positron density distribution d2N+/dθx+dθy+ (rad−2) and (b) averaged polarization degree of y component ζ̄y vs θx+=px/pz (rad) and θy+=py/pz (rad) for photon-polarization-unresolved pair production. (c) and (d) Corresponding plots for photon-polarization-resolved pair production. (e) Polarized positron density distribution dN/y (rad−1) vs θy+ (rad) for unresolved (solid curve) and resolved (dashed curve) photon polarization. (f) Averaged polarization degree ζ̄y vs θy+ (rad) for unresolved (solid curve) and resolved (dashed curve) photon polarization.

FIG. 4.

(a) Polarized positron density distribution d2N+/dθx+dθy+ (rad−2) and (b) averaged polarization degree of y component ζ̄y vs θx+=px/pz (rad) and θy+=py/pz (rad) for photon-polarization-unresolved pair production. (c) and (d) Corresponding plots for photon-polarization-resolved pair production. (e) Polarized positron density distribution dN/y (rad−1) vs θy+ (rad) for unresolved (solid curve) and resolved (dashed curve) photon polarization. (f) Averaged polarization degree ζ̄y vs θy+ (rad) for unresolved (solid curve) and resolved (dashed curve) photon polarization.

Close modal

To understand the effects of photon polarization on pair production, we investigate the polarization of emitted photons in θyγ>0 and θyγ<0 separately, as shown in Fig. 5. The photon density emitted within the angular region θyγ<0 is larger than within the region θyγ>0, especially in the high-energy region, as shown in Fig. 5(a). When an electron with ζi=b counterpropagates to the laser field, the direction of the instantaneous quantization axis is n=ζf/|ζf|,25 with

ζf=(2f2f1)ζiωεbf3+ωεε(f2f1)(ζiv̂)v̂ε2+ε2εεf2f1ωεζibf3,
(17)

which is mostly along By and changes sign every half-cycle. For an electron with initial spin ζy = 1, its spin is parallel (spin-up) and antiparallel (spin-down) to the quantization axis in the half-cycles with By > 0 and By < 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 By > 0 and By < 0 propagate with py < 0 and py > 0, respectively, because the oscillation phase of py has a π delay with respect to By. Therefore, the electrons emit photons with θyγ>0 at By > 0 and photons with θyγ<0 at By < 0. The latter case has a higher emitted photon number than the former, owing to the larger emission probability Wr↓ > Wr↑, 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 θy > 0 comes mainly from photon emission at By > 0, and has polarization −1 < ξ3 < 0.5, while the radiation with θy < 0 comes from photon emission at By < 0, which has polarization 0.5 < ξ3 < 1, as shown in Fig. 5(b).

FIG. 5.

(a) and (b) log10dNγ/γ and averaged Stokes parameter ξ̄3, respectively, of gamma photons emitted in θyγ>0 (solid line) and θyγ<0 (dashed line) vs photon energy ωγ in |θx|, |θy| < 10 mrad. (c) and (d) Angular distributions of d2Nγ/dθxγdθyγ (rad−2) and ξ̄3, respectively, vs θxγ=kx/kz and θyγ=ky/kz. (e) and (f) Angular distributions of d2Nγ/dθxγdθyγ (rad−2) and ξ̄3, respectively, for photons with energy ε > 7.5 GeV [shaded red in (b)].

FIG. 5.

(a) and (b) log10dNγ/γ and averaged Stokes parameter ξ̄3, respectively, of gamma photons emitted in θyγ>0 (solid line) and θyγ<0 (dashed line) vs photon energy ωγ in |θx|, |θy| < 10 mrad. (c) and (d) Angular distributions of d2Nγ/dθxγdθyγ (rad−2) and ξ̄3, respectively, vs θxγ=kx/kz and θyγ=ky/kz. (e) and (f) Angular distributions of d2Nγ/dθxγdθyγ (rad−2) and ξ̄3, respectively, for photons with energy ε > 7.5 GeV [shaded red in (b)].

Close modal

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), ξ¯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 θyγ<0 have ξ¯3<0, while in the region θyγ<0, the polarization is ξ¯3>0. Since the pairs are produced mostly by energetic photons, the distinct polarization properties of high-energy photons in θyγ<0 and θyγ>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 Ay(tp) at the instant of creation tp: pf = pi + eAy(tp), where pi is the momentum inherited from the parent photon, with vanishing average value 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, n=ζ+f,ξ/|ζ+f,ξ|=b. Therefore, the positrons produced at Ay(tp) < 0 acquire a final momentum pfeAy(tp) > 0 and are polarized along ζy > 0, since the instantaneous SQA is along y > 0 at tp. Similarly, one has ζy < 0 when pf < 0.

To reveal the origin of the abnormal polarization features in the small-angle region, we artificially turn on pair production of photons with θyγ>0 and θyγ<0 separately. As shown in Fig. 5(b), the photons with θyγ<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 θyγ<0 has p¯i<0, the positron distribution is slightly shifted toward θy+<0. Meanwhile, the polarization of hard photons at the θyγ>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 ζ¯y of positrons produced by photons with θyγ>0 increases monotonically with θy+, and the angular distribution shifts slightly toward θy+>0. The pair production probability is inversely proportional to ξ3, and the photons with θyγ<0 have larger ξ¯3 than those with θyγ>0, and therefore more positrons are produced at θyγ>0, as shown in Fig. 6(a). Thus, the domination of pair production of photons in θyγ<0 and p¯i<0 results in a decrease in the polarization of the positron beam and an asymmetric angular distribution of polarization.

FIG. 6.

(a) Polarized positron density distribution dN/dθy+ (rad−1) and (b) averaged polarization degree ζ̄y vs positron polar angle θy+ (rad) for positrons produced by photons with θyγ>0 (solid curves) and θyγ<0 (dashed curves).

FIG. 6.

(a) Polarized positron density distribution dN/dθy+ (rad−1) and (b) averaged polarization degree ζ̄y vs positron polar angle θy+ (rad) for positrons produced by photons with θyγ>0 (solid curves) and θyγ<0 (dashed curves).

Close modal

When the initial electrons are unpolarized as in Ref. 23, the photons are equally distributed in θyγ>0 and θyγ<0, with polarization ξ¯30.51%, 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 Δζ¯y=[ζ¯y(ξ)ζ¯y(0)]/ζ¯y(ξ)34%, both of , which are close to the values obtained with our scheme, namely, ΔN ≈ 13% and Δζ¯y35%. This is because the photons emitted are polarized with ξ¯30.51, regardless of the initial spin of the electrons. Specifically, since the photon polarization reads13 

ξ3=f2ωε(ζib)f3f1+ε2+ε2εεf2ωε(ζib)f3,
(18)

and

f1+ε2+ε2εεf2ωεf3
(19)

for most emitted photons, the average polarization

ξ¯3(ζi)ξ¯3(0)=f2f1+ε2+ε2εεf20.51,
(20)

owing to the change in sign of the term ζib 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.

FIG. 7.

(a) and (b) log10dNγ/γ and averaged Stokes parameter ξ̄3, respectively, of gamma photons emitted in θyγ>0 (solid curves) and θyγ<0 (dashed curves) vs photon energy ωγ in |θx|, |θy| < 10 mrad. (c) and (d) Polarized positron density distribution dN/y (mrad−1) and averaged polarization degree ζ̄y, respectively, vs θy (mrad) for unresolved (dashed curves) and resolved (solid curves) photon polarization.

FIG. 7.

(a) and (b) log10dNγ/γ and averaged Stokes parameter ξ̄3, respectively, of gamma photons emitted in θyγ>0 (solid curves) and θyγ<0 (dashed curves) vs photon energy ωγ in |θx|, |θy| < 10 mrad. (c) and (d) Polarized positron density distribution dN/y (mrad−1) and averaged polarization degree ζ̄y, respectively, vs θy (mrad) for unresolved (dashed curves) and resolved (solid curves) photon polarization.

Close modal

With experimental feasibility in mind, it is interesting to find how ΔN and Δζ¯y depend on the laser parameters. The produced positron number N+Nγχγτp, where Nγ is the gamma-photon number and χγa0ε0 is the quantum strong-field parameter. Thus, N+ increases with increasing a0 and τp, as shown in Figs. 8(a) and 8(c). Moreover, since the relative difference ΔN is inversely proportional to χγ, as shown in Fig. 2(f), ΔN decreases as a0 increases, owing to the increase in χγ. On the other hand, since the χγ barely changes with τp, the variation of ΔN is negligible in Fig. 8(c). Meanwhile, Δζ¯y of the produced positrons increases with χγ, as shown in Fig. 2(f). However, the positron polarization decreases after the instant of creation ti, owing to further radiative polarization in the laser field, which affects the scaling law of Δζ¯y. The time evolution of the positron polarization due to radiative polarization is given by62 

ζ¯y(t)ζ¯y(ti)eΨ1(χe)t,
(21)

with

Ψ1(χe)=0u2du(1+u)3K2/323uχe,
(22)

where u = ω/(ε0ω). Therefore, the decrease in ζ¯y is proportional to ζ¯y(ti), χe, and the interaction time tfti. As the laser intensity increases, the decrease in polarization is more dramatic, owing to the increase in χe, as shown in Fig. 8(b). Since the ζ¯y(ti) 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, Δζy 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 ζ¯y(ti), and therefore Δζy is smaller for longer laser pulses.

FIG. 8.

(a) and (c) Positron number N+ in the case of unresolved photon polarization (dashed lines) and in the resolved case (solid lines) and their relative difference ΔN (dotted lines) vs laser intensity a0 (at a0 = 50, 100, and 150) and laser pulse duration tp (at tp = 3, 5, and 8 T), respectively. (b) and (d) Positron polarization ζ¯y in the case of unresolved photon polarization (dashed lines) and in the resolved case (solid lines) and their relative difference Δζ¯y (dotted lines) vs laser intensity a0 (at a0 = 50, 100, and 150) and laser pulse duration tp (at tp = 3, 5, and 8 T), respectively. Other parameters are the same as in Fig. 4.

FIG. 8.

(a) and (c) Positron number N+ in the case of unresolved photon polarization (dashed lines) and in the resolved case (solid lines) and their relative difference ΔN (dotted lines) vs laser intensity a0 (at a0 = 50, 100, and 150) and laser pulse duration tp (at tp = 3, 5, and 8 T), respectively. (b) and (d) Positron polarization ζ¯y in the case of unresolved photon polarization (dashed lines) and in the resolved case (solid lines) and their relative difference Δζ¯y (dotted lines) vs laser intensity a0 (at a0 = 50, 100, and 150) and laser pulse duration tp (at tp = 3, 5, and 8 T), respectively. Other parameters are the same as in Fig. 4.

Close modal

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 ξ¯30.51. The hard photons in the angular region θyγ<0 have even higher polarization ξ¯31, causing the energetic positrons produced in the small-angle region to reverse the polarization direction. If one separates the intermediate hard gamma photons within θyγ>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.

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.

We have no conflicts of interest to disclose.

The data that support the findings of this study are available from the corresponding author [Y.-Y. Chen], upon reasonable request.

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