Manipulation of $\gamma$ ray polarization in Compton scattering

High-brilliance high-polarization $\gamma$ rays based on Compton scattering are of great significance in broad areas, such as nuclear, high-energy, astro-physics, etc. However, the transfer mechanism of spin angular momentum in the transition from linear, through weakly into strongly nonlinear processes is still unclear, which severely limits the simultaneous control of brilliance and polarization of high-energy $\gamma$ rays. In this work, we investigate the manipulation mechanism of high-quality polarized $\gamma$ rays in Compton scattering of the ultrarelativistic electron beam colliding with an intense laser pulse. We find that the contradiction lies in the simultaneous achievement of high-brilliance and high-polarization of $\gamma$ rays by increasing laser intensity, since the polarization is predominately contributed by the electron (laser photon) spin via multi-photon (single-photon) absorption channel. Moreover, we confirm that the signature of $\gamma$-ray polarization can be applied for observing the nonlinear effects (multi-photon absorption) of Compton scattering with moderate-intensity laser facilities.

Recently, the rapidly advanced high-power laser technique [28][29][30][31] has promoted the research of high-energy highbrilliance polarized γ rays [27,32,33], where the interaction mechanism transitions from linear into the nonlinear regime [34][35][36][37].At intermediate laser intensity, polarized γ rays can be generated by spin-nonpolarized (SNP) electron beams via weakly nonlinear CS (NLCS) [38][39][40][41].Furthermore, the stronger laser field scattered with initially spin-polarized (SP) electron beams enables the generation of more brilliant highpolarization γ rays in strongly NLCS [42].Importantly, in the near future, experiments such as LUXE at DESY [43] and E320 at FACET-II [44] will be performed using the conventionally accelerated ε e ∼ 10 GeV electron beam in collision with dozens up to hundreds of TW (corresponding to the laser invariant intensity a 0 ∼ 10) laser pulse to probe the transition from linear to strongly nonlinear QED regime.Moreover, all-optical PW up to 10 PW laser facilities have also been commissioned or will be online [45][46][47][48][49].However, there is no charted transfer mechanism of spin angular momentum (SAM) in the transition from linear, through weakly into strongly nonlinear CS.Therefore, controlling the brilliance and polarization of high-energy γ rays is still a great challenge.
In this Letter, the manipulation of γ-ray polarization in CS employing the S(N)P ultrarelativistic electron beam is investigated.We find that the contradiction lies in the simultaneous achievement of high-brilliance and high-polarization of γ rays by increasing laser intensity, since the polarization is predominately contributed by the electron (laser photon) spin via multiphoton (single-photon) absorption channel (see Figs. 1 and 2).And, SNP electrons in high-intensity laser pulse can also generate high-brilliance high-polarization γ photons (see Fig. 3).The transfer mechanism of SAM is also valid in the generation of vortex γ photons due to the same radiation dynamics as the plane-wave γ photons [50].Moreover, the polarization of γ photons radiated by the electrons with different initial spin states proceeds in different ways in LCS and strongly NLCS respectively, which can give a clear evidence of the nonlinear effects in CS with moderate-intensity laser facilities (see Fig. 4).
When electrons scatter with a laser pulse, they may absorb single or multiple low-energy laser photons and then emit a high-energy γ photon via CS.The polarization-dependent transition rate summing over the final electron spin in a circularly arXiv:2306.14936v3[physics.plasm-ph]20 Jul 2023 polarized (CP) monochromatic field is given by [35,51] where, W 0 = αm 2 e a 2 0 /(8ε eff ), a 0 = |e| E/(m e ω L ) is the laser invariant intensity, ε eff = ε e + a 2 0 ω L /Λ the effective energy of initial electron in the laser field, δ = (k L •k γ )/(k L • p) ≈ ε γ /ε e the energy ratio parameter, δ n = nΛ/(1+a 2 0 +nΛ) the cutoff-energy fraction of emitted photon absorbing n laser photons, i.e., n-th Compton edge [35], Λ = 2(k L • p)/m 2 e the invariant variable, p, k L and k γ the four-momenta of the initial electron, laser photon and emitted photon, respectively, α the fine structure constant, e and m e the charge and rest mass of the electron, and ω L and E the frequency and amplitude of the laser field, respectively.Relativistic units with c = = 1 are used throughout.F kn (k = 1, 2, 3, 4) in Eq. ( 1) are detailed in [52].h L , h e and h γ are helicities of the laser, initial electron, and emitted photon, respectively.The helicity of emitted γ photon is determined by [35,51] where, F k = ∞ n=1 F kn (k = 1, 2, 3, 4, respectively).The polarization of γ photon hardly changes during the subsequent propagation [51][52][53].
The transfer mechanism of SAM in CS from linear to strongly nonlinear processes is illustrated in Fig. 1.For a 0 O(0.1), there is a distinct edge at the end of the first harmonic followed by smaller probabilities of higher harmonics [see Figs.dtdδ is smaller, where W rad is the radiation probability after summing over the emitted photon helicity and averaging over the initial electron spin with d 2 W rad dtdδ = 2W 0 ∞ n=1 F 1n , and δ 1 is the first Compton edge of the emitted photon.Therefore, for a 0 O(0.1), the electron absorbs almost only one laser photon and radiates a γ photon with δ ≤ δ 1 , i.e., the scattering process is LCS.For a 0 ∼ O(1), the process of absorbing dozens of laser photons appears with d 2 W rad dtdδ ≈ 10 −2.99 of the average number of absorbed laser photons n = 10 (corresponding to δ ≈ 0.4), termed as weakly NLCS.As a 0 increases to O (10), due to W rad ∝ a 2 0 , the electron will have a greater probability of absorbing thousands of laser photons to emit a higher-energy γ photon with d 2 W rad dtdδ ≈ 10 −1.22 of n = 1000 (corresponding to δ ≈ 0.55), described as strongly NLCS.Therefore, electrons will radiate more brilliant γ rays in higherintensity laser field.Due to δ n ∝ 1/a 2 0 for a given n, the harmonic cutoffs decrease as a 0 increases, e.g., δ 1 ≈ 0.190 for a 0 = 0.1 while δ 1 ≈ 0.0023 for a 0 = 10.Besides, for a certain a 0 the gaps of harmonic cutoffs ∆δ n = δ n − δ n−1 ∝ 1/n 2 decrease with absorbing more laser photons [see examples of four lines corresponding to the first four cutoffs δ n=1,2,3,4 in Fig. 1(a)].Therefore, the harmonic structure is clearly visible in linear and weakly nonlinear CS spectra, however, which becomes smoother in strongly NLCS.Importantly, the impact of electron spin on the transition rate weakens as the laser intensity increases.For instances, for a 0 = 0.1, the radiation probability of the longitudinally spin-polarized (LSP) electrons (satisfying h L h e = −1) at the first edge can be increased by ) is the radiation probability of LSP (SNP) electrons after summing over the emitted photon polarization in Eq.( 1).
The electron spin plays an increasingly significant role in the transfer of SAM with stronger nonlinear effects.The entanglement term F 2 F 1 of the laser helicity and electron spin in the photon helicity Eq.( 2) mainly operates at the end of the first few harmonics and decreases as a 0 increases.While F 2 F 1 can reach to −0.68 at δ = 0.7 for a 0 = 0.1, it is irrelevance due to the relatively low radiation probability with d 2 W rad dtdδ ≈ 10 −25 [see Figs.1(a) and 1(c)].Therefore, the helicity of emitted γ photon is mainly related to the independent laser helicity term F 3 and electron spin term F 4 .For emitted photons with δ δ 1 at any a 0 , the laser helicity contribution |F 3 | |F 3 |+F 4 ≈ 1 and correspondingly the electron spin contribution , regardless of whether the scattering process is linear or not, the average helicity of γ photons via the single-photon channel The transfer mechanism of SAM points a direction for manipulating the polarization of γ rays.For LSP (h e = −1) electrons scattered with the CP (h L = 1) laser, in LCS (a 0 0.1), the emitted photons are almost completely radiated by the single-photon absorption channel and helicities are mainly contributed by the laser [see Fig. 2(a)].In the low-energy parts of the first harmonic, γ photons are radiated in the forward scattering with average helicity h γ h L = 1, while near the edge h γ −h L = −1 via the backward scattering [54].As a 0 increases, the interaction process transitions into the NLCS regime, due to δ 1 ∝ 1/a 2 0 , the energy regions of h γ ∼ 1 of radiated photons via the single-photon channel tend toward lower-energy areas with smaller δ.And for a certain a 0 1, γ photons with high energy (δ δ 100 ) obtain increasingly high h γ transferred by the electron spin.For the case of h e = 1, the helicity behavior is similar to that of h e = −1 but slightly different especially at the end of the first few harmonics with a 0 1 due to the entanglement term of the laser helicity and electron spin [52].If electrons are SNP (h e = 0), h γ of γ photons is only provided by the laser helicity.Therefore, h γ of high-energy γ photons via multi-photon absorption channels (n 10) gradually decreases as a 0 increases, for instance, at δ = 0.3, h γ ≈ −0.40 and −0.03 for a 0 = 1 and 10, respectively [see Fig. 2(b)].
Above analytical discussions are based on the single-photon emission of electrons with ε e = 10 GeV in a monochromatic plane-wave field.However, the transfer mechanism also holds true for other electron energies.The slight difference lies in the positions of harmonic edges and average number of absorbed laser photons [52].For linearly polarized (LP) γ photons generated in the LP laser, the competition between the laser and electron for controlling the γ-photon polarization also exists [52].If electrons are SP, one can increase the laser intensity to generate high-brilliance high-polarization γ rays.However, for SNP electrons, due to the energy regions δ δ 1 ∝ ε e for a certain a 0 , apart from the high-intensity laser, higher-energy electrons are also required to produce the similar high-quality γ rays, where the polarization is mainly contributed by the laser via the single-photon absorption channel.Importantly, when scattering process satisfies the rotational symmetry around the collision axis, e.g., in the case of CP laser, the angular momentum conservation holds and one could expect the generation of vortex γ photons with intrinsic orbital angular momentum (OAM) via multi-photon absorption channels.In the monochromatic CP laser, γ photons corresponding to the n-th harmonic carry OAM l γ = nh L + h e − h e − h γ with the final electron helicity h e .Since the radiation dynamics of the vortex γ photons are the same as those of the plane-wave γ photons [50], the transfer mechanism of SAM discussed above is still valid.However, the electrons are required to possess coherence in the transverse plane with axial symmetry in order to attain vortex γ photons.
The manipulation mechanism paves a clear path for highbrilliance high-polarization γ-ray sources.And the generation of CP γ rays via different CS mechanisms in realistic laser pulse is detailed in Fig. 3.The focused Gaussian left-hand CP laser pulse (h L = 1) is employed with peak intensity a 0,peak = 1, wavelength λ 0 = 0.8 µm, pulse duration τ = 10 T 0 with the period T 0 , and focal radius w 0 = 5 µm.For the head-on colliding cylindrical electron beam, the initial kinetic energy ε e = 10 GeV, energy spread ∆ε e /ε e = 6%, radius r e = 2 µm with transversely Gaussian profile, longitudinal length l e = 3 µm, polar angle θ e = 180 • with angular divergence ∆θ e = 0.2 mrad, and the electron number N e = 5 × 10 6 .Such electron bunches can be obtained via laser wakefield acceleration [55][56][57][58].In the laser pulse, the intensity sensed by electrons changes in realtime and is strongest at about T = 35 T 0 where copious γ rays are radiated [see Fig. 3(a)].For SNP electrons, the helicities of γ photons come from the laser.In the front of the laser pulse with T 23 T 0 and a 0 0.1, electrons only radiate a small amount of high polarized γ photons due to the low radiation probability of LCS regime [see Figs.For 23 T 0 T 35 T 0 , a 0 gradually increases to 1 and since δ 1 ∝ 1/a 2 0 the energy regions of h γ −h L = −1 decrease from ε γ 2 GeV to 1 GeV.And there are high-energy γ photons radiated by electrons absorbing dozens of laser photons and h γ of GeV γ photons decreases from ∼ −1 to ∼ −0.43 due to the decline of the transfer efficiency of laser helicity.As T continues to increase, a 0 begins to decreases and the trends of radiated spectra and helicities of γ photons are basically symmetrical with those of T 35 T 0 .Therefore, the final h γ of γ photons is the comprehensive result due to the laser pulse effect, and the clearly first few harmonics are smoothed out and h γ of γ photons with ε γ 2 GeV is about −0.53 which is higher than h γ ≈ −0.43 of the plane wave with a 0 = 1 [see Fig. 2(b) and the green-solid line in Fig. 3(h)].For LSP elec- trons [59][60][61], the helicities of γ photons contributed by the CP laser are almost identical to the case of SNP electrons for all energy regions radiated at the both ends of the laser pulse with T 23 T 0 and T 49 T 0 and ε γ 1 GeV radiated at the middle of the laser pulse with 23 T 0 T 49 T 0 .However, at the middle of the laser pulse the γ photons with ε γ 1 GeV receive helicities from the scattering electrons and h γ linearly falls (rises) as ε γ increases for h e = −1 (h e = 1) [see Fig. 3(c), the blue-dash-dotted line in Fig. 3(h), and [52]].The relative deviation of energy spectra ∆ γ can reach to 30% (-30%) for h e = −1 (h e = 1) [see the red dash-dotted line in Fig. 3(g)].
The different helicity behaviors of γ rays in linear and nonlinear CS regimes also represent a clear signal for testing the nonlinear effects of CS (see Fig. 4).In LCS, the first cutoff δ 1 = Λ/(1 + Λ) ≈ 4ε e ε L /(m 2 e + 4ε e ε L ) and higher harmonics (n > 1) disappear.Therefore, there is a distinct edge at ε γ,edge = δ 1 ε e = 1.9 GeV, which is smaller than 2.5 GeV of the numerical result due to the initial electron energy spread [see the red-solid line in Fig. 4(c)].Furthermore, the helicities of emitted photons are almost completely derived from the laser, and due to the opposite scattering direction, h γ at both ends of the spectra are anti-parallel.And because of the radiation reaction, ε γ,edge and the turning points from h γ ≈ 1 to −1 decrease as T increases [see Figs.4(a) and 4(b)].As a result, for 10 MeV ε γ 1 GeV, h γ is averaged to a smaller polarization degree [see the green-solid line in Fig. 4(d)].Therefore, the strongly nonlinear effects not only broaden the spectra but also change the helicity distributions of radiated γ photons compared with the LCS process [see Figs.4(c) and 4(d)].More significantly, if electrons are LSP, the nonlinear signals will be more sensitive.For instance, the electron spin hardly affects the γ-photon helicity in the LCS process, while, in strongly NLCS, LSP electrons will absorb abundant laser photons and transfer SAM to high-energy γ photons [see Fig. 4(d)].
In conclusion, we have investigated the transfer mechanism of SAM in CS from linear, through weakly into strongly nonlinear regimes to achieve the simultaneous manipulation of high-brilliance and high-polarization of γ rays, paving a way for high-quality polarized γ-beam sources.Moreover, detecting SAM of particles can help us to observe the nonlinear effects of strong field QED processes with currently feasible laser facilities.

FIG 2 .
FIG 2. Average helicity h γ of the emitted photons versus a 0 and δ for two different models of (a) the laser helicity h L = 1 and initial electron helicity h e = −1, and (b) h L = 1 and h e = 0, respectively.The laser and electron-beam parameters are the same as those in Fig.1.

1 FIG 3 .
FIG 3. (a)-(c) Generation of γ photons in the realistic laser pulse with peak intensity a 0,peak = 1 versus the interaction time T (T 0 ) and γ-photons energy ε γ with distributions of (a) and (b) the yields log 10 dNγ dTdεγ and average helicity h γ for h e = 0, and (c) h γ for h e = −1, respectively, where T 0 is the laser period.The magenta solid line in (a) indicates the laser invariant intensity a 0 in real-time versus T (T 0 ).(d)-(f) The physical representations and other laser and electronbeam parameters are the same as those in (a)-(c) respectively, except a 0,peak = 10.Comparisons of (g) energy spectra dN γ /dε γ with relative deviation ∆ γ and (h) h γ between the cases of a 0,peak = 1 and a 0,peak = 10 versus ε γ , respectively, where ∆ γ = (N he=−1 − N he=0 )/N he=0 and N = dN γ /dε γ .Other laser and electron-beam parameters are given in the text.

1 FIG 4 .
FIG 4. Distributions of γ photons predicted by LCS for SNP electrons (h e = 0) with (a) the yields log 10 dNγ dTdεγ and (b) average helicity h γ , respectively.(c) and (d) Comparisons of energy spectra dN γ /dε γ and h γ predicted by LCS and NLCS, respectively.The laser and electron-beam parameters are the same as those in Figs.3(d)-3(f).