Retrieving transient magnetic fields of ultrarelativistic laser plasma via ejected electron polarization

Interaction of an ultrastrong short laser pulse with non-prepolarized near-critical density plasma is investigated in an ultrarelativistic regime, with an emphasis on the radiative spin polarization of ejected electrons. Our particle-in-cell simulations show explicit correlations between the angle resolved electron polarization and the structure and properties of the transient quasistatic plasma magnetic field. While the magnitude of the spin signal is the indicator of the magnetic field strength created by the longitudinal electron current, the asymmetry of electron polarization is found to gauge the island-like magnetic distribution which emerges due to the transverse current induced by the laser wave front. Our studies demonstrate that the spin degree of freedom of ejected electrons could potentially serve as an efficient tool to retrieve the features of strong plasma fields.

Interaction of an ultrastrong short laser pulse with non-prepolarized near-critical density plasma is investigated in an ultrarelativistic regime, with an emphasis on the radiative spin polarization of ejected electrons. Our particle-in-cell simulations show explicit correlations between the angle resolved electron polarization and the structure and properties of the transient quasistatic plasma magnetic field. While the magnitude of the spin signal is the indicator of the magnetic field strength created by the longitudinal electron current, the asymmetry of electron polarization is found to gauge the island-like magnetic distribution which emerges due to the transverse current induced by the laser wave front. Our studies demonstrate that the spin degree of freedom of ejected electrons could potentially serve as an efficient tool to retrieve the features of strong plasma fields.
Generally, detection of plasma magnetic fields requires an external probe beam, where the field information is imprinted in the velocity space of charged particles [30][31][32][33][34][35] or the rotated polarization vector of the optical beam [36][37][38][39]. However, these conventional methods are inapplicable for scenarios with unprecedented field strength, ultrashort time scale (∼fs), and overcritical plasma density [40]. Furthermore, the spin, an intrinsic property of particles, offers a new degree of freedom of information, which is widely utilized in exploring magnetization of solids [41], nucleon structure [42], and phenomena beyond the standard model [43]. In extreme laser fields there is a strong coupling of the electron spin to the laser magnetic field [44][45][46][47][48], which may yield radiative spin polarization (SP) [49][50][51][52], i.e., polarization of electrons due to spin-flip during photon emissions. Even though in the oscillating field the electron net SP is suppressed, fast polarization of a lepton beam with laser pulses becomes possible when the symmetry of the monochromatic field is broken, such as in an elliptically polarized, or in two-color laser pulses [53][54][55][56][57]. Because of collective effects, more complex spin dynamics occurs in strong laser field interaction with plasma. Consequently, the question arises if it is possible to employ the spin signal of spontaneously ejected particles from plasma to retrieve information on transient plasma fields.
In this Letter, based on particle-in-cell (PIC) simulations, we investigate the ultrarelativistic dynamics of a short strong pulse interacting with a non-prepolarized near-critical density plasma, see Fig. 1. Special attention is devoted to describing the spin dynamics of plasma electrons, being strongly disturbed by the radiative spinflips modulated by the quasistatic plasma magnetic field (QPMF). The latter is commonly transient with a time scale as short as the driving pulse duration while being quasistatic with respect to the fast oscillating laser field. We show that the angle dependent SP of ejected electrons carries signatures of the inhomogeneous QPMF. The signal of SP of ejected electrons can be used to predict the strength of the leading order antisymmetic QPMF created by the longitudinal current. A more detailed analysis reveals that the asymmetry of SP of two outgoing divergent electron bunches characterizes the secondary QPMF, which is induced by a transverse transient current and generally neglected in previous studies [58][59][60][61][62]. The sum of these two part QPMFs gives rise to a nonlinear island-like magnetic structure [see Fig. 2 Our results demonstrate that the spin degree of freedom of ejected electrons from ultrarelativistic plasmas can be employed in principle as a tool to retrieve information on the QPMF structure and properties.
In 2D PIC simulations, a near-critical density target is irradiated by a linearly polarized pulse (with the transverse electric field along y). Our main example adopts a peak intensity of 1.7 × 10 23 W/cm 2 , equivalent to the normalized field amplitude a 0 = 350 given the laser wavelength λ 0 = 1µm. The pulse has a 2.6 µm focal spot size and 18 fs duration (FWHM intensity measure). The target has thickness l 0 = 10 µm and electron (carbon) density n e = 5n c (n i = n e /6), where n c = m e ω 2 0 /4πe 2 is the plasma critical density for a laser frequency ω 0 = ck 0 ; m e (e) the electron mass (charge); c the speed of light. The dynamics of spin precession is governed by the Thomas-Bargmann-Michel-Telegdi equation and spin-dependent photon emissions have been implemented in the EPOCH code [63], see [64].
When the pulse impinges on the target, a fraction of bulk electrons is expelled outwards by the laser ponderomotive force to form a plasma channel [65]. Meanwhile, the peripheral electrons are prone to be injected [66] and subsequently polarized inside the channel due to spinflips during photon emissions, see Fig. 1(b). Since the ion reaction partially compensates the transverse charge separation [67], the quasistatic electric field E y is negligible in this scenario. Thus the deflection of the accelerated electrons in transverse direction is predominantly governed by the azimuthal QPMF B z , which is presumably sustained by the longitudinally forward moving electron current j x . The simulation results in Fig. 1(a) show that the electrons with a positive (negative) final angle θ f mainly originate from the plasma region of y < 0 (y > 0). As the magnetic field B z ∼ −µ 0 |j 0 |y is antisymmetric, created by the nearly uniform current j x ∼ −|j 0 |, the electrons exiting the plasma area with a final angle θ f > 0 mostly experience a positive B z [see Fig. 1(a)] and vice versa. This leads to oppositely SP ejected electron bunches: s z < 0 ( s z > 0) for the electron bunch of θ f > 0 (θ f < 0). The spatial evolution of SP in Fig. 1(b) manifests that two groups of electrons are firstly polarized and confined inside the channel, and then intersect with each other towards the opposite transverse direction. This procedure is also unveiled by the evolution of SP s z in the transverse phase space (y, θ) in Fig. 1(c), where θ = arctan(p y /p x ) denotes the direction of electron momentum. The clockwise rotation of s z indicates that the QPMF not only generates spatial dependent SP but also deflects the electrons to form an angle dependent polarization distribution of ejected electrons. In Fig. 1(d), asymmetry exists for both electron SP and number angular distributions. Specifically, the averaged SP (final angle) with a positive θ f is s The magnitude of the SP signal is characterized by the parameter δ s z ≡ s + − s − . According to Fig. 1(e), SP is insignificant for low-energy electrons because of damped radiative spin-flips. Therefore, the criterion of ε e > 4a 0 m e c 2 is adopted here to filter out the low-energy noise. To reveal more subtle features of QPMF B z , we introduce also the spin (angle) asymmetry characteristics via the absolute difference: The QPMF B z is determined by electric currents via ∂B z /∂y = µ 0 j x and ∂B z /∂x = −µ 0 j y (with the vacuum permeability µ 0 ). In general, inside a laser-driven plasma channel, the current is dominated by the longitudinal one j x and the transverse current j y is neglected [58][59][60][61][62]. However, the magnetic field in our simulation shows an irregular structure, with multiple islands associated with the current kinks and vortices, see Fig 2(b). The latter indicates that the transverse current j y is important in characterizing the exact form of B z . Let us divide QPMF into two parts B z = B z,1 + B z,2 , where the leading part B z,1 is induced by j x , while the secondary B z,2 by j y : ∂B z,1 /∂y = µ 0 j x and ∂B z,2 /∂x = −µ 0 j y . The leading part B z,1 ∼ −µ 0 |e|n e cy with antisymmetric feature B z,1 (−y) = −B z,1 (y) is ubiquitously utilized in previous studies [58][59][60][61][62]. Now, we focus on the secondary B z,2 . Considering the electron velocity v y = p y /(γm e c) and momentum p y ∼ A y = a 0 cos(ξ +φ 0 ) where ξ = ω 0 t−k 0 x and φ 0 the carrier envelop phase (CEP), we obtain j y ≈ −|e| The δ(t−x/v g ) function indicates that the transverse current is predominantly contributed by the electron density n 2 δ(t − x/v g ) piled up at the front edge of the plasma channel nearby the region x ∼ v g t, where the electron's transverse velocity is significant. Here, v g (v ph ) is the laser group (phase) velocity in plasma, and the Lorentzfactor γ ∼ a 0 is assumed. With ∂B z,2 /∂x = −µ 0 j y , the secondary magnetic field can be estimated: where k 2 = k 0 (v ph − v g )/v g . The analytically predicted B z = B z,1 + B z,2 is shown in Fig. 2(a), which agrees qualitatively with the simulated B z in Fig. 2(b). The asymmetric periodic island-like structure of QPMF B z stems from the nontrivial current vortex (∇ × j) x,y = 0 generated by the transverse current of electrons plough away by the laser beam front.
As we are interested in the relation of the electron SP to the magnetic field structure, and considering the polarization attributable to the spin-flip during a photon emission, we analyze the probability of this process P(χ ph ) in Fig. 2(c) for typical parameters of our PIC simulations. Here, the electron with an initial γ e = 2000 normally crosses the uniform magnetic field B 0 = 10 4 T, and the electron quantum invariant parameter is χ e ∼ 0.1, with χ e,ph ≡ (e /m 3 e c 4 )|F µν p ν | and the momentum p ν of the electron or photon, respectively. As Fig. 2(c) illustrates, the electron spin-flips exclusively take place when emitting an energetic photon with χ ph close to χ e , while the photon emission probability is peaked at ω c ∼ χ e γ e m e c 2 (at χ e < 1), i.e., the peak of the spin-flip process is shifted with respect to the photon emission to higher χ e 's, see Fig. 2  (κ > 1) or by the laser field (κ < 1). The evolution of SP in Fig. 2(e) demonstrates that the laser field dominated regime (κ < 1) mostly contributes to the final electron SP. A distinguishable feature between the κ ≶ 1 regimes is the angle θ of the electron's instantaneous momentum when the spin-flip occurs. As the angular dependent spin-flip shows in Fig. 2(f), the κ < 1 regime applies at backward emissions, while κ > 1 for forward ones.
The detailed particle tracking further confirms these conclusions. In the κ > 1 regime Figs. 3(a),(b), the position of spin-flip with κ > 1 is closely correlated with the spatial distribution of QPMF B z . The time evolution of p x illustrates that the spin-flip happens after the electron starts an efficient acceleration and its velocity aligns longitudinally θ 1, resulting in (1−cos θ)B l < B z . For the laser dominant regime κ < 1, the electron trajectory and momentum evolution [in Fig. 3(c)(d)] demonstrate that the typical spin-flip occurs at the electron's temporarily backward motion, when (1 − cos θ) ∼ 1 and B l > B z .
It should be noted that even in the laser dominant regime, the QPMF B z is still the key factor for the SP. The reason is that the laser field has oscillating character. Although it can cause spin-flips, its net contribution to the final SP is negligible. The laser magnetic field B l acts as a catalyst to enhance the electron spin-flips by increasing χ e and net spin-flips contributing to the final SP are still determined by B z [64]. We may estimate , and A * (χ e ) ≈ 0.18χ e (at 0.01 < χ e < 0.4) [64]. The electrons with final angle θ f > 0 (θ f < 0) are mainly exposed to the QPMF B z > 0 (B z < 0) at the region y < 0 (y > 0), and the overall SP with θ f > 0 (θ f < 0) would be s z < 0 (s z > 0) which are illustrated as the solid black lines in Fig. 3(e)(f).
Thus, we calculate the electron's SP magnitude δ s z being correlated with the leading order QPMF B z,1 : where γ e ∼ a 0 is used, and η ≈ 4 × 10 −8 accounts for the deviations from the radiative spin evolution. With B z,1 ≈ (a 0 /4π)(n e /n c )(m e ω 0 /|e|), we find the SP scaling δ s z ∝ a 5/2 0 , as well as the relation B z,1 ≈ [−(δ s z /η)(n e /4πn c ) 2 ] 1/5 . In Fig. 4(a)(b),(c), the analytically predicted scalings of δ s z and B z,1 are in good accordance with the 2D simulation results.
Finally, we show how with the help of the SP asymmetry signal ∆ s z defined above, the secondary QPMF can be retrieved. In the ∆ s z signal the contribution of the B z,1 is cancelled, and ∆ s z ≈ 2 (B z,2 /|B l |)A(χ e )dt. Since B z,2 ∼ b 2 sin(k 2 x + φ 0 ) is oscillating along the longitudinal position (along the laser CEP), the overall effect of B z,2 imprinted on the signal of ∆ s z is oscillating as well. Taking into account the results for δ s z and B z,2 , we find for the asymmetry signal where b 2 is the amplitude of B z,2 . The oscillating dependence of ∆ s z on the laser CEP φ 0 is reproduced by the simulation results in Fig. 4(d). We see that the amplitude of the SP asymmetry signal ∆ s z a is directly related to the secondary QPMF b 2 : Therefore, the SP signals of δ s z and ∆ s z allow to retrieve the strength of the leading and secondary QPMFs. In addition, the combination of ∆ s z and ∆ θ , allows to predict the concrete spatial structure of B z , see Fig. 5(a). Based on the sign of ∆ s z and ∆ θ , the analytically estimated magnetic island structures agree well with the simulation results, see [64]. We define the limitations of the presented field retrieval model. Firstly, it is applicable when no more than two QPMF islands exist at y ≶ 0, with a criterion l 0 < l island ∼ 1.7λ 0 (a 0 n c /n e ) 1/2 [64]. Secondly, to exclude the influence of depolarization, the ejected electrons should experience a half-period of betatron oscillation inside the channel, with a criterion 0.5l os < l 0 < l os ∼ 5λ 0 (a 0 n c /n e ) 1/4 [64]. Consequently, the valid range of the model is 0.5l os < l 0 < min{l is , l os } shown as the white area in Fig. 5(b)(c). Our method based on the electron radiative polarization will be efficient in the quantum radiation dominated regime at α f a 0 χ e 1 (approximately at a 0 300) and χ e 0.1 [26], with a SP signal within the precision of the electron polarimetry of ∼ 0.4% [68]. The requirement might be relieved at alternative setups [29], e.g., in multiple colliding laser pulses [69], where new schemes for the magnetic field retrieval may be needed.
To confirm the robustness of our scheme, we also inves-tigated the role of experimental imperfections and uncertainties, in particular, the asymmetry in the driving laser pulse, and the ramp-up and -down of the plasma density profile [64]. The simulation results indicate that the presented scheme is robust to moderate imperfections of such practical issues. It should be noted that distinguishing more complex field structures, e.g., the three-island structure like that in Fig. 5(d), could be achievable with modifications of the retrieval method, see an example in [64], which however needs further exploration.
In conclusion, the ejected electron spin provides a new degree of freedom to extract information on the structure and magnitude of different components of the transient plasma fields. Our results open a new avenue for the electron spin-based plasma diagnostics in extreme conditions, which are prevalent in astrophysical environments and are expected in near future laser facilities.