Matter and Radiation at Extremes, Volume. 9, Issue 3, 037204(2024)

Fully polarized Compton scattering in plane waves and its polarization transfer

Suo Tang1... Yu Xin1, Meng Wen2, Mamat Ali Bake3,a) and Baisong Xie4 |Show fewer author(s)
Author Affiliations
  • 1College of Physics and Optoelectronic Engineering, Ocean University of China, Qingdao, Shandong 266100, China
  • 2Department of Physics, Hubei University, Wuhan 430062, China
  • 3Xinjiang Key Laboratory of Solid State Physics and Devices, School of Physics Science and Technology, Xinjiang University, Urumqi 830017, China
  • 4Key Laboratory of Beam Technology of the Ministry of Education, and College of Nuclear Science and Technology, Beijing Normal University, Beijing 100875, China
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    Fully polarized Compton scattering from a beam of spin-polarized electrons is investigated in plane-wave backgrounds in a broad intensity region from the perturbative to the nonperturbative regimes. In the perturbative regime, polarized linear Compton scattering is considered for investigating polarization transfer from a single laser photon to a scattered photon, and in the high-intensity region, the polarized locally monochromatic approximation and locally constant field approximation are established and are employed to study polarization transfer from an incoming electron to a scattered photon. The numerical results suggest an appreciable improvement of about 10% in the scattering probability in the intermediate-intensity region if the electron’s longitudinal spin is parallel to the laser rotation. The longitudinal spin of the incoming electron can be transferred to the scattered photon with an efficiency that increases with laser intensity and collisional energy. For collision between an optical laser with frequency ∼1 eV and a 10 GeV electron, this polarization transfer efficiency can increase from about 20% in the perturbative regime to about 50% in the nonperturbative regime for scattered photons with relatively high energy.

    I. INTRODUCTION

    Beams of high-energy photons have a broad range of applications in experiments in atomic physics,1 medical physics,2 nuclear physics,3 and particle physics,4 even including searches for new physics beyond the Standard Model.5 One of the simple ways to produce a high-energy photon source is to collide a beam of high-energy electrons with a strong laser pulse. The associated physical process is often referred to as nonlinear Compton scattering (NLCS),6–8 which is generally characterized by two factors: (i) the collisional energy parameter η = k · p/m2 and (ii) the normalized intensity ξ = |e|El/0 of the laser pulse, where p is the electron momentum, k, El, and ω0 are the laser wave vector, field intensity and frequency, respectively, and m and e are the electron rest mass and charge. Natural units = c = 1 are used throughout.

    The energy parameter η characterizes the importance of quantum recoil in the scattering process and the fraction of the energy taken by the scattered photon from the incoming particles. Evidence of quantum radiation recoil has recently been observed in laser–plasma experiments.9,10 The intensity parameter ξ indicates the nonlinearity of the process: if ξ ≪ 1, the process is linear and the total cross section can be written as the sum of the cross sections from each single laser photon; if ξO(1), multiple laser photons can be scattered simultaneously to a single γ photon, and the nonlinear effect will dominate the process. This multiphoton effect, up to the fourth harmonic, has been seen in the E144 experiment11 by colliding a beam of 46.6 GeV electrons with an intense laser pulse.12 If ξ ≫ 1, the electron will be strongly dressed, and the scattering will go into the fully nonperturbative regime. Current laser–particle experiments such as E320 at FACET-II13–15 and LUXE at DESY16–18 are designed to measure the transition of NLCS from the multiphoton regime to the fully nonperturbative regime.

    NLCS has been investigated theoretically for various types of fields: constant crossed backgrounds,19 monochromatic plane waves,19–22 finite pulses,23–30 and plasma fields, using numerical Monte Carlo simulations.31,32 The polarization properties of NLCS have recently been examined by specifying a particular polarization basis for the electron and photon.33–40 It has been proved that the scattering probability and the polarization of the scattered photon depend not only on the polarization of the background field,33 but also on that of the incoming electron.35 NLCS has also been suggested to cause strong polarization of high-energy electrons by introducing effective asymmetry into the laser field.41 However, a particular polarization basis, even one that does not precess during the interaction, cannot thoroughly reveal the polarization dependence of the process or reflect the full information about the polarization of the outgoing particles, which needs to be comprehensively described by density matrix theory,42,43 which is one of main theoretical tools used in the present paper. Density matrix theory has been successfully implemented to describe the polarization properties of the nonlinear Breit–Wheeler process44 and the spin evolution of the scattered electron in NLCS.35

    In this paper, we focus on the polarization of the scattered photons by a beam of high-energy polarized electrons and discuss how the polarization transfers from the incoming electron beam to that of the emitted photons in different intensity regions. Study of this polarized single-vertex process is relevant to the correct factorization of high-order processes by appropriate consideration of the polarization of the intermediate particle, such as the intermediate photon in the “two-step” trident process45 and the intermediate electron in double Compton scattering.46 The fully polarized local approximations for NLCS are also derived and benchmarked with the full QED calculations. This is partly motivated by modern-day high-energy laser–particle experiments, such as E32013–15 and LUXE,16–18 in which an electron beam with energy O(10 GeV) can be provided to collide with the laser pulse in the intermediate ξO(1) and even ultrarelativistic ξO(10) intensity regions. The production of a high-energy polarized photon beam is useful for fine measurements of real photon–photon scattering47 and Breit–Wheeler pair production.48

    The rest of the paper is organized as follows. In Sec. II, we outline a general theoretical model for fully polarized NLCS and apply this model explicitly in plane-wave backgrounds, deriving the energy spectrum and polarization of the scattered photon. We then analyze the scattering process in the perturbative regime in Sec. III and in the high-intensity region in Sec. IV using local approximations, namely, the locally monochromatic approximation and the locally constant field approximation. We also investigate the polarization transfer from the electron to the scattered photon in Sec. V in a broad intensity region from perturbative to nonperturbative. Finally, we present the main conclusions of the paper in Sec. VI.

    II. THEORETICAL MODEL

    The incoming electron can be described generally as|p,spin=|p|spinwhere |p⟩ is the pure eigen-momentum state, and |spin⟩ is the spin state of the electron, defined with the spin basis |±s⟩ quantized in a specified axis s. In principle, the spin of the initial electron could stay in either a pure or mixed state, and can be described by a density matrix as ρi=|c+|2ρi,1,2ρi,2,1|c|2,where |c+|2 + |c|2 = 1 and for a pure state |spin=c++s+cs, ρi,1,2=ρi,2,1*=c+c*.

    The final state |f=Ŝ|p,spin of the scattering process is composed of a scattered photon and a radiation-recoiled electron and can be written in the form|ff|=Ŝ|p,spinp,spin|Ŝ,where Ŝ is the operator of the scattering process. By taking the trace over the momentum space of the outgoing particles, we can obtain the polarization density matrix of the final state:ρf=Vd3(2π)3Vd3q(2π)3e;q,+se;q,sγ;,ε+γ;,ε|ff|×|e;q,+s|e;q,s|γ;,ε+|γ;,ε,where V is the volume factor, γ;,ελ denotes the eigenstate of the scattered γ photon with momentum and polarization ɛλ, and e;q,ςs denotes the eigenstate of the recoiled electron with momentum q and spin ς/2 along the specified quantum axis s′. ρf includes not only the full information about the polarization of the outgoing particles, but also the physics of the polarization transfer from the incoming particles to the outgoing particles.

    By summing over the polarizations of the recoiled electron, we arrive at the polarization density matrix of the scattered photon:ργ=1Pς=±σ,σ=±ρi,σσVd3(2π)3Vd3q(2π)3,+;q,ς|,;q,ς|×Ŝp,σsp,σsŜ|,+;q,ς|,;q,ς.Similarly, by summing over the polarizations of the scattered photon, we arrive at the polarization density matrix of the recoiled electron:ρe=1Pλ=±σ,σ=±ρi,σσVd3(2π)3Vd3q(2π)3,λ;q,+|,λ;q,|×Ŝp,σsp,σsŜ|,λ;q,+|,λ;q,.

    Both of these are normalized by the total probability of the scattering process from an initially polarized electron:P=ς,λ=±σ,σ=±ρi,σσVd3q(2π)3Vd3(2π)3,λ;q,ς|×Ŝ|p,σsp,σs|Ŝ|,λ;q,ς,where the abbreviation ,λ;q,ςγ;,ελe;q,ςs has been used.

    In principle, the selection for the (spin) polarization basis of the incoming/outgoing particles is arbitrary, and cannot affect any physical results. Below, we will present the deviation of the scattering matrix element ,λ;q,ςŜp,σ in plane-wave backgrounds with lightfront polarization bases. After calculating the scattering matrix element, we can figure out not only the dependence of the scattering process on the incoming electron spin, but also the polarization of the outgoing particles. Here, we are interested in the polarization of the scattered photon (ργ) because of the broad application of polarized high-energy photon sources.49–51 Similar considerations can also be applied to the polarization of the recoiled electron (ρe)35 and to the polarization correlation between the two outgoing particles.40

    A. Polarized NLCS in plane waves

    We consider NLCS in a typical scenario for current laser–particle experiments, in which a beam of high-energy electrons collides nearly head-on with an intense laser pulse to scatter a beam of γ photons. For weakly focused pulses,52–55 the background field can be simplified as a plane wave with scaled vector potential a = |e|A(ϕ) = [0, ax(ϕ), ay(ϕ), 0] depending on the laser phase ϕ = k · x. Here, for simplicity, we specify the laser wave vector k = ω0(1, 0, 0, −1), and the vector potential asaμ(ϕ)=mξ0(0,cosϕ,csinϕ,0)f(ϕ)where f(ϕ) is the pulse envelope, c=±1 denote circular polarizations with opposite rotation, and c=0 denotes linear polarization.

    The S-matrix of the scatting process can be written asSλ,qς;pσ=,λ;q,ς|Ŝ|p,σs=ied4xΨ̄q,ς(x)A/,λ*Ψp,σ(x),with the electron Volkov wave functionΨp,σ(x)=mVp01k/a/2kpup,σ×expipx+iϕdϕ2pa+a22kpand the field of the scattered photonA,λμ=2π0Vελμeix.Here, we define the electron’s bispinor up,σ based on the light-front helicity quantum axisSpμ=pμmmkpkμ,which is antiparallel to the laser propagation direction in the electron rest frame:s=p++mp+(p0+m)p1,p2,p3+mp0+mp++m,where p+ = p0 + p3. This definition becomes clear in the geometry of a near head-on collision, i.e., with p1,2p3 and p3m, leading to s ≈ (0, 0, 1). The electron polarization Ξ = (Ξx, Ξy, Ξz) can be expressed using the density matrix asρi=1212×2+σ̂Ξwhere σ̂=(σ̂1,σ̂2,σ̂3) are the Pauli matrices. |Ξ| = 1 corresponds to a pure state polarized in the direction Ξ, and |Ξ| < 1 corresponds to a mixed state polarized in the direction Ξ/|Ξ|. In terms of the Chiral (Weyl) representation,56–58 the bispinor up,σ can be written explicitly as44up,+=m2p+10p+/m(p1+ip2)/m,up,=m2p+(ip2p1)/mp+/m01,clearly satisfying the relations (p/−m)up,σ = 0 and ūp,σup,σ=δσ,σ.

    The polarization basis of the scattered photon is given asε±μ=ϵ±μϵ±kkμ,to satisfy  · ɛ± = 0 and k · ɛ± = 0, where ϵ±=(ϵx±iϵy)/2 and ϵx = (0, 1, 0, 0), ϵy = (0, 0, 1, 0). The subscript “±” indicates the rotation of the state along the laser antipropagation direction, under the assumption of small-angle scattering. Using this polarization basis, the photon polarization (5a) can be written in terms of the Stokes parameters in the formργ=121+Γ3Γ1iΓ2Γ1+iΓ21Γ3,where the linear polarization degree Γ1 denotes the preponderance of the polarization along the x axis over that along the y axis, Γ2 denotes the preponderance of the polarization at 45° to the x-axis over that at 135° to the x axis, and Γ3 is the degree of circular polarization.

    With the polarization bases (13) and (14), it is easy to derive the scattering matrix (7) and obtain for a spin-polarized electron the scattering probability P and the Stokes parameters Γ1,2,3 of the scattered photon:P=F̂[2hsΔ21+iΞzgsw1×w2+s(ΞyΔxΞxΔy)],Γ1=F̂Pw1xw2xw1yw2y+ΞyΔxs1s+ΞxΔys1s,Γ2=F̂Pw2xw1y+w1xw2yΞxΔxs1s+ΞyΔys1s,Γ3=F̂Pihsw1×w2+Ξz(2gsΔ2s)s2Ξx(w1x+w2x)s2Ξy(w1y+w2y),with the integral operator given asF̂=α(2πη)201sds1sd2rdϕ1dϕ2×expiϕ2ϕ1dϕπp(ϕ)m2η(1s),wherehs=1+(1s)22(1s),gs=1(1s)22(1s),Δ=i[a(ϕ1)a(ϕ2)]2m,wj=ra(ϕj)m,w1×w2=w1,xw2,yw1,yw2,x,andπpμ(ϕ)=pμ+aμkμpakp+a22kpis the instantaneous momentum of the electron in the laser pulse. The scattering probability and the photon’s Stokes parameters are parameterized by the three components of the photon momentum: s = k · /k · p is the fraction of the lightfront momentum taken by the scattered photon from the incoming electron, and r = (rx, ry), where rx,y = x,y/smpx,y/m are the photon momenta in the plane perpendicular to the laser propagation direction. The momentum of the scattered photon can be written as = 0(1, sin θ cos ψ, sin θ sin ψ, cos θ), where θ and ψ are the scattering angle and azimuthal angle, respectively. In nearly head-on collisions, s0/p0 is actually the energy fraction taken by the scattered photon and (rx, ry) = (/ω0) tan(θ/2) (cos ψ, sin ψ) represents the angular spread of the scattered photon around the direction of the incoming electron.

    After integrating over the transverse momentum r, we arrive atP=T̂[2Δ2hs1+iΞzgsa1×a2s(ΞxΔyΞyΔx)],Γ1=T̂Pa1xa2xa1ya2y+ΞyΔxs1s+ΞxΔys1s,Γ2=T̂Pa1xa2y+a1ya2xΞxΔxs1s+ΞyΔys1s,Γ3=T̂P[ihsa1×a2+Ξz(2gsΔ2s)sΞxbxsΞyby],with the integral operatorT̂=iα2πη01dsdφdϑϑexpisϑΛ2η(1s),wherea1,2=aa(ϕ1,2)m,b=a1+a22,the Kibble mass20 Λ is given byΛ=1+a2m2a2m2,and φ = (ϕ1 + ϕ2)/2 and ϑ = ϕ1ϕ2 are the average phase and interference phase, respectively.59–61 The window average ⟨f⟩ is calculated as f=φϑ/2φ+ϑ/2dϕf(ϕ)/ϑ.

    As we can see, the scattering probability P and the polarization Γ1,2,3 of the scattered photon depend not only on the form of the laser background, but also on the spin Ξ of the incoming electron. The transverse spin Ξx,y affects the probability and linear polarization Γ1,2 by coupling with the first order of the magnetic field, as ΔyBx and Δx ∼ − By, while the longitudinal spin Ξz contributes to the probability by coupling with the rotation of the fields, a(ϕ1)×a(ϕ2), and transfers to the circular polarization Γ3 of the scattered photon with a factor determined by the field and photon energy s. For a laser pulse with low asymmetry (between the positive and negative parts), the transverse spin can only lead to asymmetry in the transverse distributions of the scattered photons, such as d2P/drx dry and Γ1,2,3(rx, ry), but cannot affect the energy spectra, such as dP/ds and Γ1,2,3(s), since the first order of the field will go to zero after the phase integral over φ (as we will see later in the LMA results). Here, we point out that Γ1,2,3(rx, ry) and Γ1,2,3(s) can be obtained from (16) and (17), respectively, by removing the corresponding integrals in the operators F̂ and T̂ and normalizing with d2P/drxdry and dP/ds, respectively.

    III. LINEAR COMPTON SCATTERING

    Inverse Compton scattering (ICS) has been widely used in experiments for beams of high-energy γ photons, because of its narrow bandwidth62 and high polarization degree.28 It is essentially linear Compton scattering and can be simply obtained by colliding a beam of high-energy electrons with a weak laser pulse, i.e., ξ0 ≪ 1.

    The fully polarized expression for ICS photons can be obtained from (16) via a perturbative approximation in which only O(ξ2) terms are kept.27 After integrating over the azimuthal angle ψ, we can obtain the photon spectrum and polarization Stokes parameters:dPds=2παη2s1srdr|ã(v)|2m2×hs2r2(r2+1)2+Ξzς3(v)gs1r2r2+1,Γ1(s)=2πα/η2dP/dss1srdr(r2+1)2|ã(v)|2m2ς1(v),Γ2(s)=2πα/η2dP/dss1srdr(r2+1)2|ã(v)|2m2ς2(v),Γ3(s)=2πα/η2dP/dss1srdr|ã(v)|2m2×ς3(v)hs1r2r2+1+Ξzgs2sr2(r2+1)2.

    Here, ã(v)=dϕa(ϕ)eivϕ/(2π) is the Fourier frequency spectrum of the laser pulse and weights the contribution from the component with frequency v = s(r2 + 1)/2η(1 − s), which is actually the relation between the energy of the scattered photon and the incident frequency including the Compton effect, and indicates that the higher-frequency component could be scattered into a broader region of scattering angle rθ and photon energy s. The classical Stokes parameters of the laser photon are used:ς1(v)=|ãx(v)|2|ãy(v)|2|ã(v)|2,ς2(v)=ãx*(v)ãy(v)+ãx(v)ãy*(v)|ã(v)|2,ς3(v)=iãx(v)ãy*(v)ãx*(v)ãy(v)|ã(v)|2.For the carrier-frequency photon v = 1, we can have the linear Compton line sl = 2η/(2η + 1 + r2) and its energy edge sl,1 = 2η/(2η + 1). In the classical limit, i.e., → 0 leading to η = ℏk · p/m2 → 0, we can then get the well-known limited energy 0=4(p0/m)2ω0 of linear Compton scattering from the laser carrier photons.63

    Equation (18) describe the linear scattering of laser photons by a polarized electron. In Fig. 1, we present a comparison between the results from the ICS expression (18a) and those from the full QED expression (16a) for the angular-resolved spectra emitted by a high-energy electron (η = 0.2, corresponding to an electron energy of 16.8 GeV and a laser frequency ω0 = 1.55 eV16) colliding with a circularly polarized laser pulse with intensity ξ0 = 0.1, rotation c=+1, and envelope f(ϕ) = cos2[ϕ/(2N)] if − < ϕ < and f(ϕ) = 0 otherwise, where N = 16, corresponding to an FWHM duration of about 21.3 fs. As shown in Fig. 1(a) for an unpolarized electron, the scattered photons come from the laser carrier photons (v = 1) predominantly via linear Compton scattering, with the peak around the linear Compton line sl (black dashed curve) being much higher than the peaks from multiphoton scattering. Thus, the scattering process in this low-intensity region can be approximated well as a linear process with the ICS spectrum shown in Fig. 1(b).

    Comparison between the results of nonlinear (NLC) and inverse (ICS) Compton scattering in the collision between an electron with η = 0.2 and a 16-cycle laser pulse with intensity ξ0 = 0.1 and rotation c=+1. (a) and (b) 2D photon spectra d2P/ds dθ from an unpolarized electron calculated using (16a) and (18a), respectively. The black dashed curves correspond to the linear Compton line sl = 2η/(2η + 1 + r2) for laser carrier frequency v = 1. (c) Energy spectra dP/ds emitted by an electron with longitudinal spin Ξz = 0, ±1. The black dashed line indicates the location of the linear Compton edge sl,1 = 2η/(2η + 1).

    Figure 1.Comparison between the results of nonlinear (NLC) and inverse (ICS) Compton scattering in the collision between an electron with η = 0.2 and a 16-cycle laser pulse with intensity ξ0 = 0.1 and rotation c=+1. (a) and (b) 2D photon spectra d2P/ds dθ from an unpolarized electron calculated using (16a) and (18a), respectively. The black dashed curves correspond to the linear Compton line sl = 2η/(2η + 1 + r2) for laser carrier frequency v = 1. (c) Energy spectra dP/ds emitted by an electron with longitudinal spin Ξz = 0, ±1. The black dashed line indicates the location of the linear Compton edge sl,1 = 2η/(2η + 1).

    As can be seen in (18a), the ICS probability depends on the longitudinal spin Ξz of the electron by coupling with the circular polarization ς3(v) of the laser photon: an electron with longitudinal spin parallel to the circular polarization of the laser photon (Ξzς3 = +1) has a higher probability than one with Ξzς3 = −1 around the linear Compton edge sl,1 (in “helicity” language, an electron and photon with opposite helicity have higher scattering probability). This is demonstrated in Fig. 1(c), where the ICS result can again be seen to describe the linear Compton part of the full QED spectrum (17) well, but is lacking the part corresponding to the higher-order processes, where ς3(v)=c for the long-duration pulse used here. The higher-energy part (s > sl,1) of the ICS spectrum comes from the linear scattering of the higher-frequency (v > 1) components in the laser pulse. This dependence on Ξz has been widely utilized for precise electron beam polarimetry by measuring the asymmetry between the scattering probabilities in the laser pulse with opposite rotation.64 The dependence of the scattering process on the transverse spin Ξx,y is averaged out after integration over the azimuthal angle.

    We can also see from (18) that the polarizations ς1,2,3(v) of the laser photon can be transferred straightforwardly to the respective polarizations Γ1,2,3 of the scattered photon as follows:Γ1(s,r)/ς1(v)=1hs(r2+1)22r2,Γ2(s,r)/ς2(v)=1hs(r2+1)22r2,Γ3(s,r)/ς3(v)=hs(1r4)hs(r2+1)22r2,with the relation s = 2/(2 + 1 + r2) being satisfied, where Ξz = 0 has been used. Figure 2 plots the polarization transfer from a laser photon to a scattered photon via the linear Compton scattering by an unpolarized electron in linear [c=0, Fig. 2(a)] and circular [c=+1, Fig. 2(b)] backgrounds. As can be seen, the efficiencies Γ1,3(s, r)/ς1,3(v) of polarization transfer depend significantly on the scattering angle and the energy of the emitted photon. For narrow-angle scattering θ → 0, the polarization of the incoming photon can be transferred more efficiently to the polarization of the scattered photon: for the linear polarization in Fig. 2(a), the efficiency, Γ1(s, r → 0)/ς1(v) ≈ 1/hs ≈ 1 at s → 0, decreases rapidly with increasing photon energy s, which is scattered from the higher-frequency component, whereas the efficiency decreases much more slowly in the circular polarization case in Fig. 2(b), with Γ3(s, r)/ς3(v) ≈ 1 at θ < 5 μrad. With increasing scattering angle, the transfer efficiency declines monotonically for linear polarization, and changes its sign in the circular case, which may be attributed to the polarization angular momentum transferred to the orbital angular momentum of the outgoing particles. For a fixed incoming photon, the transfer efficiency improves with decreasing scattering angle for larger scattered photon energy. These properties result in the characteristics of the polarization spectra Γ1,3(s) shown in Fig. 2(c). In the low-energy region s → 0, the photons are scattered with a large angle θ [black dashed lines in Fig. 2(a) and 2(b)], with a low polarization Γ1(s) → 0 in the linear background and a high polarization Γ3(s) → −1 opposite to the laser polarization in the circular case. With the energy increased to the Compton edge sl,1, the photons are scattered with a high polarization degree Γ1,3 → 1 in both linear and circular laser fields. For photons with higher energy s → 1, scattered from a field component with frequency v ≫ 1, the polarization will decrease to zero. (The discussion for Γ2 is exactly the same as for Γ1.)

    Polarization transfer from a laser photon to a scattered photon via the linear Compton process (18) of scattering by an unpolarized electron: (a) Γ1(s, r)/ς1(v) in a linearly polarized field; (b) Γ3(s, r)/ς3(v) in a circularly polarized field. (c) Polarization spectra Γ1,3(s) of the scattered photon in the linear (a) and circular (b) backgrounds. The same parameters as in Fig. 1 are used, except for the linearly polarized field (c=0).

    Figure 2.Polarization transfer from a laser photon to a scattered photon via the linear Compton process (18) of scattering by an unpolarized electron: (a) Γ1(s, r)/ς1(v) in a linearly polarized field; (b) Γ3(s, r)/ς3(v) in a circularly polarized field. (c) Polarization spectra Γ1,3(s) of the scattered photon in the linear (a) and circular (b) backgrounds. The same parameters as in Fig. 1 are used, except for the linearly polarized field (c=0).

    After integrating over the energy fraction s, we can obtain the probability and polarization of the scattered photons from the frequency component κ = vk for an electron with momentum p:P(β)=σ0D(v)vηβ214β2+2vη+lnβ2β+1(vη)2lnβ2+Ξzς3(v)β2+(vη)2β21+ββ1lnβ2,Γ1(β)=σ0D(v)P(β)1+β2βlnββ1ς1(v)(vη)2,Γ2(β)=σ0D(v)P(β)1+β2βlnββ1ς2(v)(vη)2,Γ3(β)=σ0D(v)P(β)ς3(v)β132+2β12β2β+1β1lnβ+Ξz2+10ββ(β1)2β+12β2+β26β7(β1)3lnβ,which depend only on the collisional energy parameter β=1+2vη=(pμ+κμ)2/m2, where σ0=2πα2λe2, λe = 1/m is the electron’s reduced Compton wavelength, and D(v)=v|ã(v)|2/(αλe2m2) is the areal number density of the photon beam with frequency 0. The unpolarized part of the scattering probability, (21a), is the well-known Klein–Nishina formula.65

    IV. POLARIZED LOCAL APPROXIMATIONS

    The full QED calculation is limited to the scenario where the background field can be approximated well as a plane wave.52–55 To investigate QED effects in more realistic situations, many numerical simulations have been performed based on local approximations, such as the locally constant field approximation (LCFA)31 and the locally monochromatic approximation (LMA).66,67

    The LCFA is generally used in cases with ultrahigh intensities ξ0 ≫ 1, in which the formation length of the scattering process is much shorter than the typical length of field variation,68 i.e., the laser wavelength, and the quantum interference effects induced by this typical field variation can be ignored, whereas the LMA is designed to catch this interference effect by treating the fast variation of the laser carrier frequency exactly, but neglecting the slow variation of the pulse envelope.69 However, owing to the massive computational requirements of the related Bessel functions (as we will show later), the application of the LMA is limited to the intermediate-intensity region, i.e., ξ0 ≲ 10.

    A. Polarized LCFA

    The LCFA results can be simply obtained by expanding the field and the Kibble mass in (17) to first and second orders, respectively, of the interference phase ϑ:aμ(ϕ1)=aμ(φ)+aμ(φ)ϑ/2,aμ(ϕ2)=aμ(φ)aμ(φ)ϑ/2,Λ=1+ϑ2ξ2(φ)/12,where a(φ) = m(0, −ξx, −ξy, 0), ξx,yξx,y(φ), and ξ = (ξx, ξy) is the local value of the electric field, followed by an analytical integration over ϑ. The detailed derivation can be found in Ref. 39.

    In the LCFA, the photon spectrum and the classical Stokes parameters are given as follows:dPds=αηdφ2τAi(τ)hs+Ai1(τ)+sAi(τ)ΞxξyΞyξx|ξ|τ,Γ1(s)=α/ηdP/dsdφAi(τ)ξx2ξy2τξ2Ai(τ)Ξyξx+Ξxξyτ|ξ|s1s,Γ2(s)=α/ηdP/dsdφAi(τ)2ξxξyτξ2+Ai(τ)ΞxξxΞyξyτ|ξ|s1s,Γ3(s)=α/ηdP/dsdφsAi1(τ)+gs2τAi(τ)Ξz,where τ = s2/3/[η|ξ|(1 − s)]2/3. These results can be easily compared with those in Ref. 40, which were obtained with the linear polarization basis for the scattered photon and the spin basis along the magnetic field. As can be seen, the LCFA splits the whole scattering process virtually into infinitesimally short intervals and integrates the results in each successive interval linearly by ignoring the interference between them.

    B. Polarized LMA

    The LMA result is obtained by considering the scattering process in a plane wave with a slowly varying pulse envelope (i.e., f′(ϕ) ≪ 1) and making use of the slowly varying envelope approximation for integrals:ϕ2ϕ1dϕf2(ϕ)(ϕ1ϕ2)f2(φ),ϕ2ϕ1dϕf(ϕ)cosϕ(sinϕ1sinϕ2)f(φ),ϕ2ϕ1dϕf(ϕ)sinϕ(cosϕ2cosϕ1)f(φ).The exponent in the integral operator F̂ in (16) can then be simplified asϕ2ϕ1dϕπp/m2η(1s)cir.κϕ1ζsin(ϕ1cψ)κϕ2+ζsin(ϕ2cψ),lin.κϕ1μsinϕ1+νsin2ϕ1κϕ2+μsinϕ2νsin2ϕ2,in the circular (“cir.”) and linear (“lin.”) cases, respectively, whereκ=s(r2+m*2)2η(1s),ζ=rsξaη(1s),μ=rxsξaη(1s),ν=sξa2[8η(1s)].m* is the effective mass of the electron, given by m*=(1+ξa2)1/2 in the circular background and m*=(1+ξa2/2)1/2 in the linear case, and ξaξ0f(φ) is the local amplitude of the field. As can be seen, the azimuthal angle ψ of the scattered photon plays the role of the carrier-envelope phase of the laser pulse in the circular case.70

    Inserting the expressions (23a) and (23b) into (16), integrating over the laser interference phase by ignoring the dependence of the local amplitude ξa(φ) on the averaged phase, and then integrating over the transverse momentum r, we arrive at the spectrum and Stokes parameters of the scattered photon in the circular case,dPds=αηdφn=n*+ξa2n2ζn2Jn2+Jn2Jn2hsJn2+cΞzgsξaJnJnm*2rn2rn,Γ1(s)=0,Γ2(s)=0,Γ3(s)=α/ηdP/dsdφn=n*+chsm*2rn2rnξaJnJn+Ξzgsξa2n2ζn2Jn2+Jn2Jn2sJn2,and in the linear case,dPds=αηdφn=n*+ππdψ2π[ξa2(Λ1,n2Λ0,nΛ2,n)hsΛ0,n2],Γ1(s)=α/ηdP/dsdφn=n*+ππdψ2π×[(rncosψΛ0,nξaΛ1,n)2rn2sin2ψΛ0,n2],Γ2(s)=0,Γ3(s)=α/ηdP/dsdφn=n*+ππdψ2π×[gs(Λ1,n2Λ0,nΛ2,n)ξa2sΛ0,n2]Ξz.

    Here, Jn = Jn(ζn) is the Bessel function of the first kind with argument ζn = rna/[η(1 − s)], Jn=Jn(ζn) is its derivative, and Λj,n(μn, ν) are generalized Bessel functions defined asΛj,n(μn,ν)=ππdϕ2πei(nϕμnsinϕ+νsin2ϕ)cosjϕ,with j = 0, 1, 2, and un = rna cos ψ/[η(1 − s)]. rn=[2nη(1s)/sm*2]1/2 is the transverse momentum of the nth harmonic scattered with lightfront momentum s and indicates the energy edge sn=2nη/(2nη+m*2) of the nth harmonic. ⌈n*⌉ denotes the lowest integer greater than or equal to n*=sm*2/[2η(1s)]. The effective mass m*2>1 results in a redshift of the harmonic edge (compared with the linear Compton edge sl,1) that increases with the intensity.

    The LMA can only work well for pulses with long duration and would deviate far from the correct QED calculation for a few-cycle pulse, since the high-frequency contribution induced by the pulse envelope would then become important. In the circular background, the longitudinal spin of the incoming electron contributes to the spectrum by coupling with the field rotation cΞz(24a), while the contribution from the electron’s transverse spin Ξx,y is zero because the LMA field is symmetric in each direction. This property also leads to unpolarized linear polarization Γ1,2 = 0, even in the case of scattering by a transversely polarized electron. The photon’s circular polarization Γ3 stems from the rotation c of the background field, proportional to the field intensity ξa, and can also be transferred from the electron’s longitudinal spin Ξz. In the linear case [see (25)], the scattered photon is linearly polarized in the direction of the laser polarization, with the spectrum being free from the electron spin and with circular polarization being transferred from the longitudinal spin of the electron.

    C. Benchmarking and discussion

    We can clearly see the differences between the LMA [(24) and (25)] and LCFA (22):The LMA resolves the harmonic contribution, with the order depending on the local amplitude of the field, whereas the LCFA linearly integrates the local contributions determined by the local value of the field.The LMA can capture the rotation property of the laser field in (24), whereas the LCFA treats the laser as a series of infinitesimally short intervals of field linearly polarized in a direction varying with the local phase. Because of this, the LCFA cannot describe the contribution of the longitudinal spin Ξz to the probability, although it can still describe the longitudinal polarization transfer (Ξz → Γ3) as in the linear case of LMA.The LMA spectrum (24a) includes the contribution from the electron’s longitudinal spin Ξz, but hides the effect of transverse spin Ξx,y, which is zero after azimuthal integration in the LMA field, whereas in the LCFA spectrum (22a), the contribution of transverse spin is manifested by coupling with the magnetic field (ξx ∼ −By and ξyBx), which could be significant in an ultrashort pulse.

    Below, we benchmark the LMA and LCFA results with the full QED calculations for plane waves with circular polarization (c=+1; Figs. 3 and 4) and with linear polarization (c=0; Figs. 5 and 6). The collision energy parameter η = 0.2 is used, and the pulse envelope is again f(ϕ) = cos2[ϕ/(2N)] if − < ϕ < and f(ϕ) = 0 otherwise, where N = 16.

    (a) Energy spectra and (b) polarization of photons scattered by a high-energy electron with η = 0.2 and different longitudinal spins Ξz = 0, ±1 in a circularly polarized laser field with intensity ξ0 = 1, rotation c=+1, and envelope f(ϕ) = cos2[ϕ/(2N)] if |ϕ| Nπ and f(ϕ) = 0 otherwise, where N = 16. The black dashed lines indicate the edges of the first three harmonics, sn=2nη/(2nη+1+ξ02), with n = 1, 2, 3. In (a), the inset zooms in the energy spectra around the first harmonic to show the difference between the QED and LMA results.

    Figure 3.(a) Energy spectra and (b) polarization of photons scattered by a high-energy electron with η = 0.2 and different longitudinal spins Ξz = 0, ±1 in a circularly polarized laser field with intensity ξ0 = 1, rotation c=+1, and envelope f(ϕ) = cos2[ϕ/(2N)] if |ϕ| < and f(ϕ) = 0 otherwise, where N = 16. The black dashed lines indicate the edges of the first three harmonics, sn=2nη/(2nη+1+ξ02), with n = 1, 2, 3. In (a), the inset zooms in the energy spectra around the first harmonic to show the difference between the QED and LMA results.

    (a) Energy spectra and (b) polarization of photons scattered by a high-energy electron in a circularly polarized laser field with intensity ξ0 = 5. The black dashed lines indicate the edges of the first three harmonics, sn=2nη/(2nη+1+ξ02), with n = 1, 2, 3. The other parameters are the same as in Fig. 3.

    Figure 4.(a) Energy spectra and (b) polarization of photons scattered by a high-energy electron in a circularly polarized laser field with intensity ξ0 = 5. The black dashed lines indicate the edges of the first three harmonics, sn=2nη/(2nη+1+ξ02), with n = 1, 2, 3. The other parameters are the same as in Fig. 3.

    (a) Energy spectra and (b) polarization of photons scattered by an electron with η = 0.2 in a linearly polarized laser field with intensity ξ0 = 1. The black dashed lines indicate the edges of the first three harmonics, sn=2nη/(2nη+1+ξ02/2), with n = 1, 2, 3. In (a) and (b), the contributions from the electron spin Ξ are too small to show. (c) Longitudinal polarization transfer from incoming electron to scattered photon as a function of photon energy s. The other parameters are the same as in Fig. 3.

    Figure 5.(a) Energy spectra and (b) polarization of photons scattered by an electron with η = 0.2 in a linearly polarized laser field with intensity ξ0 = 1. The black dashed lines indicate the edges of the first three harmonics, sn=2nη/(2nη+1+ξ02/2), with n = 1, 2, 3. In (a) and (b), the contributions from the electron spin Ξ are too small to show. (c) Longitudinal polarization transfer from incoming electron to scattered photon as a function of photon energy s. The other parameters are the same as in Fig. 3.

    (a) Energy spectra and (b) polarization of photons scattered by a high-energy electron in a linearly polarized laser field with intensity ξ0 = 5. The black dashed lines indicate the edges of the first three harmonics, sn=2nη/(2nη+1+ξ02/2), with n = 1, 2, 3. In (a) and (b), the contributions from the electron spin Ξ are too small to show. (c) Longitudinal polarization transfer from incoming electron to scattered photon as a function of photon energy s. The other parameters are the same as in Fig. 5.

    Figure 6.(a) Energy spectra and (b) polarization of photons scattered by a high-energy electron in a linearly polarized laser field with intensity ξ0 = 5. The black dashed lines indicate the edges of the first three harmonics, sn=2nη/(2nη+1+ξ02/2), with n = 1, 2, 3. In (a) and (b), the contributions from the electron spin Ξ are too small to show. (c) Longitudinal polarization transfer from incoming electron to scattered photon as a function of photon energy s. The other parameters are the same as in Fig. 5.

    In Fig. 3 for ξ0 = 1, clear harmonic structures and an appreciable Ξz contribution are exhibited by the energy spectra [dP/ds in Fig. 3(a)] and polarization [Γ3(s) in Fig. 3(b)] from the full QED calculations (17) for different longitudinal spins Ξz = 0, ±1. The contribution of the transverse spin Ξx,y in (17) (not shown in Fig. 3) is negligible because of the low field asymmetry. In this intermediate-intensity region, the photons can be effectively scattered into the energy region s > s1 with polarization parallel to the laser polarization.39 In the low-energy region s → 0, the photons are strongly polarized opposite to the laser polarization, Γ3(s)=c [see also the result in Fig. 2(c)]. As we can also see in Fig. 3(b), the photon polarization Γ3(s) can be affected considerably, especially in the high-energy region s → 1, by the longitudinal spin Ξz of the incoming electron. The linear polarization degree Γ1,2 (not shown) of the scattered photons is close to zero in this long-duration pulse.

    As can be seen, the LMA (24) is able not only to reproduce the location sn and amplitude of the harmonic structure, but also to describe exactly their dependence on Ξz. The only slight difference induced by the LMA is the presence of sharper harmonic peaks as shown in the inset of Fig. 3(a), because of the absence of finite-pulse effects.71 However, in the LCFA results, no harmonic structure is apparent, and as the LCFA lacks the rotation property of the background field, the Ξz dependence cannot be revealed in the LCFA spectrum in Fig. 3(a), which, as can be seen, is divergent in the low-energy limit s → 072 and is smaller than the full QED result in the high-energy region s → 1, because the high-frequency contribution from the field is ignored. The lack of the field rotation property also leads to an incorrect prediction of the polarization of the scattered photon, as shown in Fig. 3(b): for an unpolarized electron, the LCFA predicts a scattered photon with no polarization, Γ3 = 0 [see (22) (d)]. For a polarized electron with Ξz = ±1, the LCFA results differ considerably from those of the full QED calculations in the low-energy region (s → 0), where the influence of the background field is crucial. However, with increasing photon energy (s → 1), this difference becomes smaller as the polarization transferred from the incoming electron becomes dominant.

    As can be seen in Fig. 4 for ξ0 = 5, the harmonic peaks move to the low-energy limit s ≪ 0.1, and simultaneously, with increasing laser intensity, the importance of the electron’s longitudinal spin for the scattering probability, A=(PΞz=cPΞz=c)/PΞz=0, decreases to about 1.7% [Fig. 4(a)] from about 9.1% at ξ0 = 1 [Fig. 3(a)]. [A=14.6% at ξ0 = 0.1 in Fig. 1(c) for the perturbative result.] Therefore, the LCFA result, in the high-intensity region ξ0 ≳ 5, could be used to calculate the photon spectrum from arbitrarily polarized electrons. As is also evident from Fig. 4(b), the scattered photons can only be slightly polarized (|Γ3(s)|≪ 1) if Ξz = 0 for s > s1. (Effective polarization is only possible in a region with extremely low energy s < s1 ≪ 0.1, which is not the region we are interested in.) This is because at high intensities, the formation length of the scattering photons (s > s1) is much shorter than the laser wavelength, and the field rotation becomes less important during photon formation.68 The polarization Γ3(s) of the scattered photons is now determined by the longitudinal spin Ξz of the electron and thus can be described well by the LCFA results. The performance of the LMA is as good as in the case of ξ0 = 1, except that the calculations impose a much heavier computational load.

    A similar discussion to that above also applies in the case of the linear background shown in Fig. 5 for ξ0 = 1 and Fig. 6 for ξ0 = 5. The LMA works well in reproducing precisely the full QED results over a wide intensity region, whereas the LCFA can only reproduce the QED results at high intensities, with a loss of the details in the low-energy tail (s < s1). The only difference from the circular case is that in a linear background, the polarized contributions to the energy spectrum [see (17)] are always much smaller than the unpolarized contibutions because the long-duration pulse is used, and the scattered photons are linearly polarized in the same direction as the laser field, with the degree depending on the laser intensity ξ0 and photon energy s, and the photons’ circular polarization can only be transferred from the longitudinal spin of the incoming electron.

    V. HELICITY TRANSFER

    On the basis of the above discussions, we can clearly see the polarization transfer from incoming electron to scattered photon in the scattering process. This transfer can occur straightforwardly for the longitudinal spin Ξz of the electron, which is often called “helicity transfer.” The transfer from the electron’s transverse spin is much more complicated. By coupling with the first order of the field, the transverse spin changes the transverse distribution of the scattered photon and may affect both the linear and circular polarization of the scattered photon [see (17)]. For a long-duration pulse, there is no net transfer from the electron’s transverse spin to the photon polarization, as can be seen from the LMA results (24) and (25).

    To see how the helicity transfer Ξz → Γ3 occurs via the scattering process, we first look at the transfer in the case of linear Compton scattering, shown in Fig. 7(a), in which we consider linear Compton scattering of a circularly unpolarized photon (ς3 = 0) by a high-energy electron with longitudinal spin Ξz. As shown, the helicity transfer depends sensitively on the collisional energy β=(pμ+κμ)2/m2 [see (21d)]: the transfer is less efficient with Γ3z → 0 in the lower-energy region β → 1, but increases quickly to exceed 90% at β = 104, presumably corresponding to the collision between a p0O(1 GeV) electron and a κ0O(1 MeV) photon. For the parameter relevant to the LUXE experiment,16 namely, p0 = 16.5 GeV and κ0 = 1.55 eV with β ≈ 1.4, the transfer efficiency is about Γ3z = 16%; see the red circle in Fig. 7(a).

    Helicity transfer from incoming electron to scattered photon via (a) linear Compton scattering with change in collision energy β and (b) nonlinear Compton scattering with change in laser intensity ξ0. In (a), an electron with longitudinal spin Ξz scatters a circularly unpolarized photon ς3 = 0. In (b), a linearly polarized laser pulse collides with an electron with η = 0.2 and longitudinal spin Ξz. The same laser parameters as in Fig. 5 are used.

    Figure 7.Helicity transfer from incoming electron to scattered photon via (a) linear Compton scattering with change in collision energy β and (b) nonlinear Compton scattering with change in laser intensity ξ0. In (a), an electron with longitudinal spin Ξz scatters a circularly unpolarized photon ς3 = 0. In (b), a linearly polarized laser pulse collides with an electron with η = 0.2 and longitudinal spin Ξz. The same laser parameters as in Fig. 5 are used.

    With increasing laser intensity, the total efficiency of helicity transfer can also be increased, as can be inferred from Figs. 5(c) and 6(c): Transfer is almost forbidden, with Γ3(s)/Ξz = 0, if the photon is scattered into the low-energy limit s → 0, but becomes more efficient with increasing energy of the scattered photon, for example, Γ3(s)/Ξz = 1 as s → 1, and for higher intensities, the photons are more likely to be scattered into the high-energy region with the more efficient transfer. [The same effect can also be found in the circular cases in Figs. 3 and 4, except that in the low-energy region, Γ3(s) can also be affected by the polarization of the laser field.] Fig. 7(b) shows the increase in transfer efficiency with increasing laser intensity in a linearly polarized laser pulse scattered by an electron with η = 0.2. The total transfer efficiency Γ3z (the red dashed curve calculated by LMA) increases from about 16% in the perturbative regime ξ0 ≪ 1, corresponding to the red circle in Fig. 7(a), to about 19% at ξ0 = 10 in the nonperturbative regime. Actually, with a high-intensity (ξ0 ≫ 1) laser, the lower-energy photons are scattered into a much broader angular spread,39 which would limit its application in experiments. In Fig. 7(b), the helicity transfer for scattered photons with energy s > 0.1 is plotted. The transfer efficiency increases from about 23% in the perturbative regime ξ0 ≪ 1 to about 36% at ξ0 = 10 (solid magenta curve calculated by LMA) and further to about 47% at ξ0 = 102 in the full nonperturbative regime (dot-dashed magenta curve calculated by LCFA). [The LCFA significantly underestimates the total transfer efficiency with s > 0 (not shown) because of the divergent low-energy tail.]

    VI. CONCLUSION

    Fully polarized nonlinear Compton scattering from a beam of spin-polarized electrons has been investigated in pulsed plane-wave backgrounds. Compact expressions for the energy spectrum and polarization of the scattered photon depending on the electron spin have been derived and discussed for a broad intensity region from the perturbative to the nonperturbative regimes. In the perturbative regime ξ0 ≪ 1, to investigate polarization transfer from a single laser photon to the scattered photon, the polarized linear Compton scattering has been considered. To analyze helicity transfer from incoming electron to scattered photon in the high-intensity region, the polarized locally monochromatic approximation and the locally constant field approximation have been derived and benchmarked against the full QED calculations.

    The locally monochromatic approximation is able to calculate precisely the photon spectrum and polarization from differently polarized electrons over the whole intensity region. However, the locally constant field approximation is valid only in the high-intensity regime (ξ0 ≫ 1), with a divergent low-energy tail in the spectrum. In the intermediate-intensity regime (ξ0 ∼ 1), relevant to the upcoming laser–particle experiments, the LCFA completely lacks the spin contribution in circularly polarized laser pulses, which has been shown to affect the scattering probability by about 10%.

    Helicity transfer from electron to scattered photon has been investigated in a linearly polarized background. In the perturbative regime, the transfer is determined by the collisional energy between the incoming electron and photon, with the efficiency increasing from about 10% for collision between a 10 GeV electron and an optical laser photon with ω0 ∼ 1 eV, to about 90% for collision between a 1 GeV electron and a 1 MeV photon. With increasing laser intensity, the total transfer efficiency could also be increased, since the photons are then more likely to be scattered in the higher-energy region, in which helicity transfer is more efficient. For scattered photons with energy s > 0.1, the total transfer efficiency of about 23% in the perturbative regime ξ0 ≪ 1 could be increased to about 47% at ξ0 = 102 in the fully nonperturbative regime.

    ACKNOWLEDGMENTS

    Acknowledgment. The authors are supported by the National Natural Science Foundation of China (Grant Nos. 12104428, 12075081, 12375240, and 12265024). Numerical computations were performed on Hefei Advanced Computing Center.

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    Suo Tang, Yu Xin, Meng Wen, Mamat Ali Bake, Baisong Xie. Fully polarized Compton scattering in plane waves and its polarization transfer[J]. Matter and Radiation at Extremes, 2024, 9(3): 037204

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    Paper Information

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    Received: Jan. 5, 2024

    Accepted: Mar. 6, 2024

    Published Online: Jul. 2, 2024

    The Author Email: Bake Mamat Ali (mabake@xju.edu.cn)

    DOI:10.1063/5.0196125

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