Zheng Gong,∗ Karen Z. Hatsagortsyan,† and Christoph H. Keitel Max-Planck-Institut fur Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany
(Dated: November 2, 2021)
Understanding and interpretation of the dynamics of ultrarelativistic plasma is a challenge, which calls for the development of methods for in situ probing the plasma dynamical characteristics. We put forward a new probing method, harnessing polarization properties of γ-photons spontaneously emitted from a non-prepolarized plasma irradiated by a circularly polarized strong laser pulse. We show that the temporal and angular pattern of γ-photon linear polarization is explicitly correlated with the instantaneous dynamics of the radiating electrons, which provides information on the laser- plasma interaction regime. Furthermore, with the γ-photon circular polarization originated from the electron radiative spin-flips, the plasma susceptibility to quantum electrodynamical processes is gauged. Our study demonstrates that the polarization signal of emitted γ-photons from ultrarel- ativistic plasma can be a versatile information source, which would be beneficial for the research fields of laser-driven plasma, accelerator science, and laboratory astrophysics.
The successful decoding of field properties nearby the event horizon [1] re-stimulates the interest in measure- ments based on photon polarization [2]. While polarized light is vulnerable to magneto-optic disturbance [3], the high-frequency γ-photon is robust during penetration of the plasma depth [4]. Previously, the celestial γ-ray emis- sion was observed to understand the star-forming galax- ies [5], accretion flows around black holes [6], and active galactic nuclei [7]. In contrast to the routinely detected quantities of arrival time, direction, and energy, the γ- photon polarization (GPP), provides new insights on the relativistic jet geometry [8] and magnetic field configu- ration [9], which allows to identify the cosmic neutrino scattering [10], dark matter annihilation [11], and acceler- ation mechanisms surrounding crab pulsars [12]. Whilst there has been progress in the development of polarime- ters [13], further studies of GPP originated from distant astronomical dynamics are hindered by the need to per- form observation on artificial satellites [14]. In this re- spect the modeling and simulation of the astrophysical GPP in a laboratory would be indispensable.
This goal can be accomplished by using ultrarelativis- tic plasma created in the cutting-edge laser facilities with the intensity of 1023 W/cm2 [15–17]. The later is not only favorable for examining nonlinear quantum electro- dynamics [18–20], but also capable of producing highly polarized γ-photons [21]. This energetic and overdense state, associated with sufficiently strong fields [22–24], radiative particle trapping [25, 26] and e−e+ pair cas- cades [27–30], cannot be measured by the conventional techniques, e.g. optical probes or charged particle radio- graphy [31–33]. Moreover, understanding of GPP phe- nomena in astrophysics would need to relate it to the electron in situ transient dynamics.
This Letter aims to find the distinct relationship be- tween the specific spatial-temporal features of the GPP and the time-resolved motion of plasma electrons, and in this way deduce the dynamical properties of plasma
and the regime of interaction. Using 3D particle-in-cell (PIC) simulations, we investigate polarization-resolved γ-photon emission in an ultrarelativistic plasma driven by a circularly polarized laser pulse [Fig. 1]. The collec- tive orientation of the γ-photon linear polarization (LP) resembles a spiral shape with the rotation tendency de- termined by the acceleration status of the radiating elec- trons. We define the spiral ratio, which quantifies the degree of rotation tendency, and can serve as a diag- nostic to gauge the transient acceleration gradient and self-generated magnetic fields exerted on the radiating electrons. The inspection of the angle dependence of the spiral ratio allows to distinguish between different scenarios of the laser-plasma interaction. Furthermore, the γ-photon circular polarization (CP), originated from the accumulated longitudinal polarization of plasma elec- trons, ascribed to the quantum electrodynamical (QED) radiative spin-flips, is shown to provide a measure of the quantum strong-field parameter of the electrons and of the susceptibility of the ultrarelativistic plasma to QED processes.
When an electron interacts with a circularly polarized electromagnetic wave A = a0e
iξey + εa0e i(ξ−π/2)ez with
the normalized field amplitude a0, ellipticity ε = 1, rel- ative phase ξ = ω0t − k0x, and laser frequency ω0 = ck0, the electron motion can be characterized by py ∼ a0mece
iξ, pz ∼ εa0mece i(ξ−π/2), y ∼ a0e
i(ξ−π/2)/Γk0, and z ∼ −εa0e
iξ/Γk0, with Γ ≡ γe − (px/mec) the dephasing value. The laser fields are the dominating terms in governing the electron dynamics, while the self- generated azimuthal magnetic fieldBφ = κb(−yez+zey) sustained by the longitudinal current [34] is introduced as a perturbation. The radially quasi-static electric field is neglected due to the ion motion compensating the charge separation [35]. The key parameter determining the polarized γ-photon emission and electron radiative spin-flips is the strong-field invariant quantum parame- ter χe,ph ≡ (e~/m3
ec 4)|Fµνpν | with the field tensor Fµν
and the momentum pν of the electron or photon, respec- tively. In the moderate QED regime χe . 1, the direction
ar X
iv :2
11 1.
00 56
3v 1
1
2
FIG. 1. (a) The schematic for γ-photon emission from the plasma (with y < 0 clipped) penetrated by a laser pulse, where the red lines present typical electron trajectories. (b) The LP orientation in the ideal condition. (c) The differ- ence δφ between a⊥ and aa. The analytically predicted LP orientation for the electron undergoing acceleration (d) and deceleration (e). (f) and (g) The simulated γ-photon LP ori- entation, with the LP degree PLP and the normalized number distribution dNph/ sin θdθdφ. (f) and (g) is for the time at t = 10 and 40 fs, respectively.
of the emitted γ-photon LP is primarily parallel with the acceleration direction perpendicular to the electron mo- mentum, i.e. a⊥ ≡ a − (a · v)v, where the hat symbol denotes the unit vector. The polarization orientation can be derived as
a⊥,y ≈− (1− cos θ) sin θ sinφ
ε − κb sin θ cos θ cosφ
εΓ
γe ,
a⊥,z ≈ ε(1− cos θ) sin θ cosφ+ εκb sin θ cos θ cosφ
Γ
γe ,
(1)
z) 1/2/px] the polar an-
gle and φ = arctan 2(pz, py) the azimuthal angle. In the ideal condition that the terms of electron acceleration and plasma self-generated fields are negligible, i.e. −β ·E = 0 and κb = 0, the orientation of the γ-photon LP would be along the azimuthal direction aa = (− sinφ, cosφ)
[Fig. 1(b)], which collectively resembles multiple concen- tric rings with each polarization segment along the az- imuthal direction. To characterize the deviation of the realistic orientation a⊥ from the azimuthal one aa, we define the spiral ratio R := (a⊥ × aa) · ex = sin δφ, where δφ ∈ [−90, 90] is the relative angle between a⊥ and aa [Fig. 1(c)]. The spiral ratio can be expressed as
R ≈ −β ·E√ [Γ + (γeκb cos θ/Γ)]
2 + (β ·E)2
. (2)
If the electron is undergoing acceleration with −β ·E > 0 (deceleration with −β · E < 0), the spiral ratio R > 0 (R < 0) corresponds to the counter-clockwise (clockwise)
spiral tendency in the angular distribution of γ-photon LP orientation as shown in Fig. 1(d) [Fig. 1(e)].
To examine the GPP features, we performed 3D PIC simulations, where an over-critical density slab is illu- minated by a right-handed (ε = 1) circularly polar- ized pulse. In the main example, the laser intensity I0 ≈ 1.68 × 1023W/cm
2 is equivalent to the normalized
field amplitude a0 ≈ 350 for the wavelength λ0 = 1µm. The pulse has a duration τ0 = 25 fs and focal spot size 2.6µm (FWHM intensity measure). The slab has a thick- ness l0 = 10µm and consists of electrons and carbon ions with the number density ne = 30nc and ni = 5nc, respec- tively. nc = meω
2 0/4πe
2 is the plasma critical density. The models of radiative spin-flips, spin-dependent pho- ton emission, and photon polarization effects have been incorporated in the EPOCH code [36], see the Supple- mental Materials [37].
Inside the laser-driven plasma channel, the electrons tend to form a helical density structure [38], undergo betatron acceleration [39], and radiate multi-MeV pho- tons [40]. The orientation of the emitted γ-photon LP exhibits the counter-clockwise spiral tendency [Fig. 1(f)] corresponding well with the analytical prediction for the accelerating electron [Fig. 1(d)]. The angular averaged spiral ratio is R ≈ 0.51 and the γ-photon LP degree PLP
close to 20%. If the low-energy photons are filtered out, the LP degree can be improved to PLP ∼ 50%, compara- ble with the scheme based on the Compton backscat- tering process [21]. The clockwise spiral tendency in Fig. 1(g) indicates the deceleration of plasma electrons happening at time t = 40 fs.
The time-resolved electron energy spectrum dNe/dεe [Fig. 2(a)] demonstrates that the acceleration primarily takes place earlier at t . 35 fs, whereas the deceleration occurs later at t & 35 fs. Near the acceleration saturation time tεes ∼ 35 fs, the maximum energy reaches 1 GeV. The time-resolved spiral ratio R(t) explicitly correlates with the electron acceleration status, where R > 0 (R < 0) corresponds to the electrons being predominantly accel- erated (decelerated) [Fig. 2(b)]. As a consequence, the moment of the spiral ratio changing sign, defined as the reversal time tRs , should be equal to tεes .
To confirm the robustness of the notion of the rever- sal time tRs for any regime of interaction, we study the same scenario with varied laser intensity 150 6 a0 6 550 and plasma density 10 6 ne/nc 6 50. In the consid- ered first regime ne = 10nc, the pulse readily penetrates through the plasma, and the electron acceleration termi- nates when the pulse exits the slab’s rear surface. The saturation time is estimated as ts,10nc ∼ l0/vg. In the second regime with ne = 30nc, the acceleration satura- tion time is approximated when the electron slides out of one laser period, i.e. ts,30nc
∼ λ0/(vph − vx). Here, vph ∼ c[1 − (ne/a0nc)]
1/2 (vg ∼ c[1 − (ne/a0nc)] −1/2) is
the group (phase) velocity of the pulse propagating in- side the relativistically transparent plasma [41]. In the
3
FIG. 2. (a) The time evolution of electron energy spectra. (b) The time evolution of R and the kinetic energy Ee of all elec- trons (with γe > a0). (c) The dependence of tRs (circle) and tεes (cross) on a0, where the lines denote the analytical esti- mation. (d) The dependence of the simulated (grey bar) and analytically derived (dashed line) R on εph, and the red line shows the photon energy spectrum. (e) R(θ) obtained from the main PIC simulation (circles) and the test particle simu- lation with κb = 0 (triangles), while the analytical prediction is illustrated by the lines color-coded with κb. (f) θ∗(κb) ob- tained from simulations (markers) and the numerical solution of ∂R(θ∗)/∂θ = 0 (lines). The error bars in (c) and (f) are from the statistic uncertainties in post processing.
third regime with ne = 50nc, the pulse bores a hole on the front side of the overdense plasma and its termina- tion is determined by the laser reflection time ts,50nc
∼ τ0/(1−βHB) ∼ τ0(1+Π1/2), where βHB ∼ Π1/2/(1+Π1/2) the hole boring velocity and Π ∼ I0/(minic
3) the dimen- sionless piston parameter [42]. Even if the mechanisms are distinct among the above three regimes with differ- ent plasma density, the acceleration saturation time tεes is well reproduced by the reversal time tRs [Fig. 2(c)]. Therefore, the measurement of the reversal time tRs can allow to distinguish the actual interaction regime. Given the parameter normalization, an effect similar to the con- sidered one could be observed in the astronomical sce- nario of a strong radio wave interacting with the rare plasma background, where the time scale of the evolving R would be prolonged to a µs level and could be identi- fied by the observatories [43].
Otherwise, as the timing accuracy of ∼ 10 fs is still un- available for the current γ-photon polarimetry, it is mean- ingful to investigate the time-averaged spiral ratio R and its dependence on the emission angle. A nontrivial peak of the spiral ratio at an intermediate angle θ∗ is found [Fig. 2(e)], which can be calculated from the analyti-
cal expression R(θ) ∼ (a0 sin θ)/{[Γ + (γeκb cos θ/Γ)] 2
+ (a0 sin θ)2}1/2, approximating −β ·E ∼ a0 sin θ, and as- suming a nonvanishing plasma field κb 6= 0. The simu- lated R(θ) behaves as a right-skewed distribution peaked at θ∗ ≈ 24, whereas the R(θ) of a test particle simula- tion with κb = 0 is monotonically decreasing with the rise
FIG. 3. (a) R(θ) obtained from simulations for three cases, whose typical electron dynamics is illustrated in (c). (b) The angle dependence of −β ·E calculated from the tracked elec- trons (histograms) and R(θ) (lines), respectively. (c) The time evolution of angle θ with blue-red color denoting −β ·E (determining the sign of R), and γ-photon emissions marked by yellow circles. The green lines show the evolution of γe.
of θ [Fig. 2(e)]. The peak position θ∗ is determined by the gradient of self-generated magnetic field κb = ∂Bφ/∂r. Figure 2(f) shows that the θ∗(κb) obtained from PIC sim- ulations corresponds well with the numerical solution of θ∗(κb) (with the approximation γe ∝ a0). Thus, the de- tected peak position θ∗ provides an estimate for the ex- perimental gradient of the plasma magnetic field. Addi- tionally, Fig. 2(f) illustrates that the functional relation- ship θ∗(κb) clearly distinguishes the three regimes with different plasma density discussed above.
Note that the spiral ratio is insensitive to the pho- ton energy εph, as the estimation from Eq.(2) shows: R(εph) ∼ 0.24(εph/mec
2)1/2/[η + (εph/mec 2)1/2], with
η ∼ 102, and the signal R is applicable for the majority of photons since the photon distribution function expo- nentially decays at high energies [Fig. 2(d)].
Even more detailed information on the electron dy- namics can be deduced by closer inspection of the angle- resolved spiral ratio R(θ). For instance, in the regime a0 = 350; ne = 30nc the spiral ratio R(θ) > 0 holds over the whole range of angle θ [Fig. 3(a)], when the laser field is phase matched to the betatron oscillations and the electrons are efficiently accelerated [38, 39]. The rep- resentative time evolution of an electron’s angle θ con- firms the acceleration dominance over the deceleration [Fig. 3(c)]. A different scenario is seen in the regime a0 = 150; ne = 30nc, when the spiral ratio R(θ) is neg- ative at θ . 40. This highlights a typical electron dy- namics, where acceleration takes place at a relative large angles 40 < θ, while the deceleration at small angles θ . 40. Conversely, in the case a0 = 150; ne = 1nc, the electron tends to be accelerated at a collimated direction while decelerated at a large divergent angle. Based on Eq. (2), the acceleration gradient can be expressed as −β ·E ∼ sign(R)[Γ+κb cos θ(1−cos θ)−1]/(R−2−1)1/2, which is approximately ∼ 30mecω0/|e| ≈ 1014 V/m. The angle dependent spiral ratio R(θ) correlated with the electron acceleration gradient [Fig. 3(b)] could be applied to explore the new interaction mechanisms with higher
4
FIG. 4. (a) The angular dependent γ-photon CP degree PCP (histograms) and photon number distribution dNph/dθ (lines). (b) The electron energy density V = εene, and the distributions of electron spin polarization sx, sy, and sz. The dependence of (c) χe and (d) PCP on a0. (e) The cor- relation between PCP and χe. The lines in (c)(d)(e) present the analytical estimation. The inset in (d) and (e) zooms in on the shadow region. (f) The number distribution of energetic electrons Ne versus χe, Eeff , and γe.
acceleration efficiency and collimated photon emission.
Apart from the LP, the emitted γ-photons can be par- tially circularly polarized [Fig. 4(a)]. In strong fields a0 1, the photon formation length lf ∼ c/a0ω0 is much less than the laser wavelength [18] and during a photon emission the electron does not experience the entire ro- tating structure of the laser field. Consequently, the CP of γ-photons cannot be inherited from the spin angu- lar momentum of the driving pulse. It rather originates from the spin of polarized plasma electrons. The latter is caused by the radiative spin flips, which are governed by the QED quantum strong-field parameter χe. Thus, the CP of emitted γ-photons can be a characteristic of the QED properties of the laser-driven plasma.
The electron radiative spin flip is described by the equation dsR/dt ≈ (
√ 3αfmec
2/2π~γe)χeA∗(χe)b⊥ [37], where A∗(χe) ≈ 0.2χe (at 0.01 < χe < 0.4) [44], αf the fine structure constant, χe ∼ γeEeff/Es, γe ∼ a0 the Lorentz factor, Eeff ≈ a0(1−cos θ)mecω0/|e| the effective
field, Es = m2 ec
3/(|e|~) the Schwinger field, and b⊥ the unit vector transverse to the magnetic field in the labora- tory frame. The latter is along the transverse momentum direction p⊥, because py,z/mec ∼ |e|By,z/meω0. Consid-
ering the component of b⊥ along the x-axis, the net lon- gitudinal electron spin polarization is negative sx < 0 [Fig. 4(b)], leading to a negative CP degree PCP of the emitted γ-photons. As the divergence angle can be esti- mated as θ ∼ (Bφ/a0)1/2, through the balance between the transverse electric and magnetic forces [45], the in- variant parameter could be reformulated as χe ∼ ρa0, with ρ ≈ Bφ~ω0/(2mec
2) ∼ 10−4. At χe . 0.1, the
FIG. 5. (a) The interaction scheme of a nanowire array irradi- ated by a strong laser field. (b) The longitudinal current den- sity jx, where the arrow lines denote the jy,z and Ja = 17kA the Alfven current limit. The solid line represents a typical electron evolution in (b) the transverse coordinate, (c) px vs x, and (d) χe (dsR/dt) vs t, where the yellow circles refer to emitted backward γ-photons with εph > 10MeV
γ-photon CP degree can be calculated as [37]
PCP ∼ χph(2χe − χph)
(s · β), (3)
and the longitudinal electron spin polarization is derived as s · β ∼ −(αfmec
2/h)(χ2 e/γe). Taking into account
that the photon emission probability is peaked at χph ∼ χ2 e (for χe 1), we obtain the CP degree as PCP ∼ −(αfmec
2/h)ρ3a2 0. Then, the plasma QED degree χe
can be retrieved through the γ-photon CP degree PCP: χe ∼ 102(~ω0/2παfmec
2)1/2|PCP|1/2. As illustrated in Figs. 4(c)(d)(e), the analytically derived relation agrees reasonably with the PIC simulation results.
In the above setup, the QED parameter is moderate for the averaged value of energetic electrons: χe ≈ 0.05 [Fig. 4(f)]. Higher χe-parameters can be reached in ultrahigh-energy-density states sustained by a strong pulse interaction with nanowire arrays [46]. We analyzed the electron spin polarization and γ-photon CP in such a scenario [Fig. 5(a)] [37]. Here, the electrons, replenished by the return current within the interior of the nanorods [Fig. 5(b)], are prone to backscattering with the colliding pulse at a large angle θ [Fig. 5(c)(d)]. The QED param- eter is improved to χe ∼ 0.13 due to the enhancement of the effective field Eeff , rather than electron energy γe [Fig. 4(f)]. With the rise of χe, the γ-photon CP de- gree is improved from PCP ≈ −0.4% to −3.2%. The higher PCP correlated with the large angle scattering pro- cess helps distinguish the interaction scenarios associated with the violent plasma return current [47].
Concluding, we demonstrate that the polarization properties of γ-photons emitted from the energetic plasma driven by a circularly polarized laser pulse pro- vide extra degrees of freedom of information on the tran- sient electron acceleration dynamics, self-generated mag- netic fields, plasma QED status, and on the regimes of laser-plasma interaction. This information will help in better understanding of the underlying mechanisms of observed phenomena in broad high-intensity interac- tion scenarios including ion acceleration [48–51], elec-
5
tron direct acceleration [39], high-harmonic genera- tion [52, 53], brilliant photon emission [54], ultradense nanopinches [55], and e−e+ pair plasma cascades [56]. For astrophysics aspects, our results indicate that the γ-photon LP orientation modulated by the particle ac- celeration gradient is potentially connected to the cor- relation between the observed GPP and universe mag- netic fields [9]. Besides, the extra γ-photon CP degree originated from spin-polarized plasma electrons appears attractive to be explored for influence on dark matter annihilation [11] and cosmic neutron scattering proce- dure [10].
The PIC code EPOCH is funded by the UK EPSRC grants EP/G054950/1, EP/G056803/1, EP/G055165/1 and EP/ M022463/1. Z. G. would like to thank Pei-Lun He for fruitful discussions. The Supplemental Material includes Refs. [57–65].
∗ gong@mpi-hd.mpg.de † k.hatsagortsyan@mpi-hd.mpg.de
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Deciphering in situ electron dynamics of ultrarelativistic plasma via polarization pattern of emitted -photons
Abstract
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