Polarized light from the transportation of a matter-antimatter beam in a plasma

TL;DR

This study shows that a relativistic electron-positron beam in a magnetized plasma produces polarized synchrotron radiation, with polarization degrees depending on plasma conditions, providing insights into gamma-ray burst observations.

Contribution

It reveals how polarized radiation arises from pair beams in plasma and links circular polarization to energy dissipation during filamentation instability.

Findings

Maximum circular polarization of ~18% observed.

Circular polarization linked to asymmetric energy dissipation.

Polarization depends on beam-to-background plasma density ratio.

Abstract

A relativistic electron-positron beam propagating through a magnetized electron-ion plasma is shown to generate both circularly and linearly polarized synchrotron radiation. The degrees of circular and linear polarizations depend both on the density ratio of pair beam to background plasma and initial magnetization, and a maximum degree of circular polarization $\langle P_\textrm{circ}\rangle \approx 18\%$ is found to occur for a tenuous pair beam. We demonstrate that the generation of circularly polarized radiation is intrinsically linked to asymmetric energy dissipation of the pair beam during the filamentation instability dynamics in the electron-ion plasma. These results can help in understanding the recent observations of circularly polarized radiation from gamma-ray-bursts.

Equations

$$⟨P_{\mathrm{circ}}^{\text{th}}⟩=0.8415\frac{⟨{\gamma }^4{⟩}_{e_b^{-}}-⟨{\gamma }^4{⟩}_{e_b^{+}}}{⟨{\gamma }^4{⟩}_{e_b^{-}}+⟨{\gamma }^4{⟩}_{e_b^{+}}}.$$

Full text

Radiation from extreme astrophysical scenarios as e.g gamma-ray bursts (GRBs) occurring in supernova explosions, pulsar wind nebulae and active galactic nuclei is of immense interest as it holds vital clues about stellar dynamics and acceleration of cosmic rays Mundell et al. (2013); Wiersema et al. (2014); Band et al. (1993); Hededal (2005). In particular, the radiation observed in the afterglow of GRBs, which are the most luminous events in the Universe, is central to understanding the role played by various nonlinear processes occurring in the astrophysical plasmas providing extremely efficient energy conversion of the matter into radiation and vice-versa.

GRBs are one of the unsolved problems in astrophysics, and they are capable of releasing enormous amount of energy ($E_{k}\sim 10^{51}$ erg) in a small volume of radius $\sim 10^{3}$ km  Piran (1997). Thus, one can expect a copious amount of pair-plasma ($e^{-},e^{+}$) being generated, which is termed as a fireball that propagates in the plasma made of baryonic matter Piran (2005). This can give rise to the onset of plasma instabilities, notably the Weibel instability (WI) Weibel (1959) that can generate a strong magnetic field, converting the kinetic energy of the streaming particle into electromagnetic fields Silva et al. (2003); Muggli et al. (2013). Both the pair-plasma and the relativistic ejecta of supernova explosion are capable of launching internal and external (with the interstellar medium) collisionless shocks, respectively. These shocks can accelerate the baryonic matter to high-energy and can be responsible for the ultra-high energy cosmic rays. The presence of a strong magnetic field can cause ultra-relativistic particles to emit synchrotron radiation. Indeed, the radiation signatures from GRBs indicate the presence of strong equipartition magnetic field and a power law distribution of the cosmic rays Band et al. (1993); Begelman et al. (1984); Blandford and Znajek (1977). Moreover, the radiation is deemed to be linearly polarized Rybicki and Lightman (1985); Jackson (1998); Medvedev and Loeb (1999). However, recent observation of the circular polarized radiation Wiersema et al. (2014) in the afterglow of the GRBs calls for studying the polarization properties of the radiation from GRBs and the role of plasma composition and dynamics on the synchrotron radiation emission Mundell et al. (2013); [U.Sinha; J.Martins; J.Vieira; K.M.Schoeffler; R.A.Fonseca; andL.Silva; (tobesubmitted).Seealso;][.TheapproachoftrackingparticletrajectoriesfromPICsimulationtocomputethepolarizedradiationwasfirstemployedhere; whilethephysicalconfigurationandunderlyingphysicalprocessesaredifferentinbothmanuscripts.]Sinha:2015aa; Sagiv et al. (2004).

Parallel to the observations of astrophysical processes from ground based and airborne telescopes, there has also been a growing interest in utilising powerful and highly energetic laser systems Boehly et al. (1995); Lobet et al. (2017, 2015); eli ; cil to study collisionless shocks, magnetic reconnection and plasma instability dynamics in a controlled laboratory experiment Huntington et al. (2015); Sarri et al. (2015, 2013); Lobet et al. (2015); Fox et al. (2013); Warwick et al. (2017). It is a rapidly evolving area of research known as laboratory astrophysics, which invokes the scale invariance principle to felicitate comparison between the laboratory experiments and the astrophysical scenarios. Recently, a breakthrough experiment was conducted in which a dense pair plasma beam of density $n_{b}\sim 10^{16}$cm*-3* and an average Lorentz factor, $\gamma_{\textrm{av}}\approx 15$, was produced in a laboratory Sarri et al. (2015). The generation of neutral high-density pair plasma is very encouraging as it can attempt to give insight into the transport of matter-antimatter beam in a plasma, mimicking the conditions of the fireball beam propagation in GRBs.

Motivated by these developments, in this Letter, we study the polarized radiation from propagation of a pair-plasma in a magnetized electron-ion plasma by 3D particle-in-cell (PIC) simulations. Here, the case of magnetized scenario is important as the polarization of GRBs indicate the presence of an ordered magnetic field  Mundell et al. (2013); Wiersema et al. (2014). The motives of our study are twofolds: First, it is analogous to the finite fireball beam propagation in a magnetized background plasma of baryonic matter and it can shed light on the observation of circular polarization in GRBs. Second, it compliments the ongoing experimental efforts in laboratory astrophysics, where the proton radiography technique is heavily used to map the field strength and particle densities Fox et al. (2013); Huntington et al. (2015). Polarized radiation from plasma particle can also serve an important role in deciphering the dynamics of plasma instabilities, complimenting the proton radiography technique.

Full 3D PIC simulations on the propagation of a cold relativistic pair beam ($e_{b}^{-},e_{b}^{+}$) in a preformed uniform electron-proton plasma are carried out using the electromagnetic PIC code SMILEI Derouillat et al. (2018). A simulation window of size $[16\times 32\times 32]({c/\omega_{\textrm{p}}})^{3}$ and a resolution of $[256\times 256\times 256]$ cells moving with the speed of light along the $x$-direction with absorbing boundary conditions for particles and fields in the transverse direction was used. Here $\omega_{\textrm{p}}=(4\pi n_{\textrm{0}}e^{2}/m_{\textrm{e}})^{1/2}$ is the electron plasma frequency of the ambient electron-proton plasma, $n_{\textrm{0}}$ the electron density, $e$ and $m_{\textrm{e}}$ are the electronic charge and mass respectively, and $c$ is the velocity of the light in vacuum. The pair beam with a Gaussian density profile in each direction was initialized at the center of the simulation window with FWHM $[2\times 8.5\times 8.5]({c/\omega_{\textrm{p}}})^{3}$. We used the simulation timestep of $\Delta t=0.0295\omega_{\textrm{p}}^{-1}$ and $[2\times 2\times 2]$ particles per cell per species 111Higher number, e.g. 100 particles per cell produced similar results.. The ambient plasma had realistic proton to electron mass ratio, $m_{\textrm{p}}/m_{\textrm{e}}=1836$ with temperature $\sim$5 keV. The ambient plasma was magnetized with a uniform magnetic field $\textbf{B}_{\textrm{0}}=B_{\textrm{0}}\hat{\textbf{e}}_{x}$, of strength $B_{\textrm{0}}=[0.1,0.2,0.3,0.4,0.5,0.75,1.0]$ (normalized to $m_{\textrm{e}}\omega_{\textrm{p}}c/e$). For plasma density of $n_{\textrm{0}}=10^{16}\,\textrm{cm}^{-3}$, it corresponds to $B_{0}\sim(3-30)$ T. These parameters are readily available for current laboratory astrophysics experiments and can be scaled up for comparison with astrophysical scenarios Ryutov (2018). The choice of parallel magnetic field can correspond to either the magnetic field of central engine of the GRB, interstellar magnetic field compressed at the shock front or field generated due to other dynamical processes Piran (2005); Sagiv et al. (2004); Medvedev and Loeb (1999).

When a pair beam ($e_{b}^{-},e_{b}^{+}$) propagates in a plasma, the beam electrons ($e_{b}^{-}$) repel and the beam positrons ($e_{b}^{-}$) attract the background plasma electrons respectively, violating the quasi-neutrality of the plasma. This causes the generation of an inductive electric field that, in turn, generates a return plasma current to neutralize the effect of external beam. Since the pair-beam has transverse size larger than the plasma skin-depth $d_{s}=c/\omega_{p}$, the return plasma current can penetrate, and flow inside the pair-beam. This configuration of the opposite beam currents is unstable in a plasma and it can split the pair-beam into smaller filaments of size $\sim d_{s}$ due to the WI or current filamentation instability (WI/CFI), leading to the generation of a strong transverse magnetic field at the expense of the pair-beam energy. Since in the case of an ambient electron-proton plasma, only the background electrons participate in the current neutralization as $m_{\textrm{p}}\gg m_{\textrm{e}}$, the $e_{b}^{+}$ and $e_{b}^{-}$ filaments experience deceleration and acceleration respectively, leading to the difference in their energy spectrum Huynh and Ryu (2016). Fig.1 (a-b) shows density of the $(e_{b}^{-},e_{b}^{+})$ beam species with initial Lorentz factor $\gamma_{\textrm{0}}=50$ and a peak beam density per species $n_{\textrm{b0}}=\textrm{0.1}n_{\textrm{0}}$ after propagating a distance of 1500 $c/\omega_{\textrm{p}}$ (8 cm for an ambient plasma with $n_{\textrm{0}}=10^{16}\textrm{cm}^{-3}$) in an initially magnetized ($B_{0}=\textrm{7.5}\,$T) electron-proton plasma. One can see the filamentation of the pair-beam due to the WI/CFI in panels (a) and (b) and associated magnetic field energy in panel(c). Fig. 1(d) shows the difference in average kinetic energies between the beam $e_{b}^{+}$ and $e_{b}^{-}$ which is $\langle\gamma\rangle_{e_{b}^{+}}-\langle\gamma\rangle_{e_{b}^{-}}\approx 10\,(\sim 5\textrm{MeV})$. This energy difference is predominantly caused by the beam energy loss in generating the magnetic field, but also due to return current neutralization dynamics. Fig.2 shows that for density ratio $n_{b0}/n_{0}=0.1$, the $e_{b}^{-}$ looses energy while $e_{b}^{+}$ gains energy as explained before. However, at density ratio $n_{b0}/n_{0}=1$, both species loose energy to generate the magnetic field due to WI/CFI, though the $e_{b}^{+}$ population still has higher average energy than the $e_{b}^{-}$ population. Indeed, the beam to magnetic field energy conversion ratios in both cases read $2\%$ for $n_{b0}/n_{0}=0.1$ and $4.7\%$ for $n_{b0}/n_{0}=1$. This suggests that $e_{b}^{-}$ population dominantly contributes to magnetic field generation. This can be simply understood by observing the fact that both the $e_{b}^{+}$ and return plasma currents are in the same direction and hence they are not as unstable as the oppositely moving $e_{b}^{-}$ and return plasma currents.

To analyze the spectrum and polarization of the radiation emitted from the beam particles, we extracted trajectories of 1000 beam $e_{b}^{-}$ and $e_{b}^{+}$ directly from the PIC simulations. The particles were randomly selected from those lying initially in the $xy$ plane at $z=\textrm{16}\,c/\omega_{\textrm{p}}$ 222The average kinetic energy of the selected particles were comparable to that of the entire respective beam species, (see Supplementary material). We develop a new post-processing code CASPER cas , to compute the radiation spectrum and its polarization employing the method of Fourier transform of the radiated electric field and the Stokes parameters Sironi and Spitkovsky (2009); Hededal (2005); Chen et al. (2013); Martins et al. (2009). A two dimensional virtual detector of size $[30000\times 30000](c/\omega_{p})^{2}$ is kept in the $yz$ plane at a distance $x=10^{5}(c/\omega_{p})$, having $50\times 50$ grid points. The frequency range $\omega=(10^{0}-10^{3})\omega_{p}$ (large enough for the spectrum) is analyzed with a resolution of $\Delta\omega=1$. To capture the influence of the filamentation, filaments merging and saturation of the WI/CFI magnetic fields, the radiation was calculated from the time $t_{i}=750\,\omega_{p}^{-1}$ to $t_{f}=1500\,\omega_{p}^{-1}$. Defining $\boldsymbol{\epsilon}_{1}$ and $\boldsymbol{\epsilon}_{2}$ as the unit vectors perpendicular to the direction of observation and E as the resultant radiated electric field vector from all the charged particles, the Stokes parameters are given as: $s_{0}=|\boldsymbol{\epsilon}_{1}\cdot\textbf{E}|^{2}+|\boldsymbol{\epsilon}_{2}\cdot\textbf{E}|^{2}$; $s_{1}=|\boldsymbol{\epsilon}_{1}\cdot\textbf{E}|^{2}-|\boldsymbol{\epsilon}_{2}\cdot\textbf{E}|^{2}$; $s_{2}=2\textrm{Re}[(\boldsymbol{\epsilon}_{1}\cdot\textbf{E})^{*}(\boldsymbol{\epsilon}_{2}\cdot\textbf{E})]$; $s_{3}=2\textrm{Im}[(\boldsymbol{\epsilon}_{1}\cdot\textbf{E})^{*}(\boldsymbol{\epsilon}_{2}\cdot\textbf{E})]$ Jackson (1998); Born et al. (1970). The total linear and circular polarizations read as $\langle P_{\textrm{lin}}\rangle=\langle\sqrt{s_{1}^{2}+s_{2}^{2}}/s_{0}\rangle$, and $\langle P_{\textrm{circ}}\rangle=\langle s_{3}/s_{0}\rangle$ respectively, where $\langle s_{i}/s_{0}\rangle=\iint s_{i}d\textrm{A}d\omega/\iint s_{0}d\textrm{A}d\omega$ with A as the area of the detector and $i=(1,2,3)$ 333It may be noted that these Stokes parameters are defined for a monochromatic wave and their applicability to an ensemble of plasma particles considered here is facilitated by the random phase approximation; see Rybicki and Lightman (1985).. Fig.3 (a-b) show the frequency and space averaged radiated energy from Fig.1. The spectrum confirms the emission to be a synchrotron process. Since the $e_{b}^{+}$ and $e_{b}^{-}$ population have an energy difference and for a charged particle with Lorentz factor $\gamma$, the total energy radiated scales as $\mathcal{E}_{\textrm{rad}}\propto\gamma^{4}$ Jackson (1998), the CP radiation flux (caused by $\mathbf{B_{0}}$) due to beam $e_{b}^{+}$ will exceed those from beam $e_{b}^{-}$. This can also be confirmed by comparing Figs.3 (c) and (d) where regions of high degree of circular polarization show small amount of linear polarization. The frequency averaged degree of circular and linear polarizations with peak values of $\langle P_{\textrm{circ}}\rangle_{\omega}\approx 0.4(40\%)$ and $\langle P_{\textrm{lin}}\rangle_{\omega}\approx 0.6(60\%)$ are shown in Figs.3(c) and (d), respectively. The total fractional linear and circularly polarized radiation fluxes (integrated over the whole area in Figs.3 (c) and (d)) are $\langle P_{\textrm{circ}}\rangle\approx 0.18(18\%)$ and $\langle P_{\textrm{lin}}\rangle\approx 0.49(49\%)$, respectively. In all cases, polarized light is found to be $\sim(67-69)\%$, making the fraction of unpolarized light to be $\sim(33-31)\%$. Fig.4 shows the spectrum of the $\langle P_{\textrm{circ}}\rangle$ and $\langle P_{\textrm{lin}}\rangle$ peaks at $\omega\approx 100\omega_{\textrm{p}}$ for $n_{\textrm{b0}}=0.1n_{0}$, which corresponds to a wavelength of $\sim\textrm{3}\,\mu\textrm{m}$ (far infra-red) for $n_{0}=10^{16}\textrm{cm}^{-3}$. This frequency emission is promising to explain the observation of optical circular polarization in GRBs. The spectrum broadens for higher $n_{\textrm{b0}}$ due to increased radiated energy, arising due to strong WI/CFI magnetic field.

In order to check the validity of our simulation results, we use the well-known synchrotron radiation formalism Jackson (1998) and calculated the degree of polarization analytically. The Fourier spectrum of the radiation field is, \textbf{E}(\textbf{r},\omega)=2\,e\,\omega\big{[}(1+\gamma^{2}\theta^{2})\,K_{2/3}(\textrm{x})\hat{\textbf{e}}_{1}/(\sqrt{3}\gamma^{2}\omega_{0})\pm i\theta{\sqrt{1+\gamma^{2}\theta^{2}}}\,K_{1/3}(\textrm{x})\hat{\textbf{e}}_{2}/(\sqrt{3}\gamma\omega_{0})\big{]}\textrm{e}^{i\omega r}/(c\,r), where $e\equiv(e_{b}^{-},e_{b}^{+})$, $+$ is for $e_{b}^{+}$ and $-$ for $e_{b}^{-}$, $x=\omega/(3\gamma^{3}\omega_{0})(1+\gamma^{2}\theta^{2})^{3/2}$, $\omega_{0}$ is the cyclotron frequency, $r$ the distance from the charge to the observation point, and $K_{1/3}(x)$ and $K_{2/3}(x)$ are the modified Bessel functions. Thus, the radiation will be circularly polarized for $\theta\neq 0$. Numerically integrating the Stokes parameters $s_{3}$ and $s_{0}$ for the radiated field given before over the entire solid angle and the frequency range, we obtain the total circularly polarized radiation flux, $P_{\textrm{cflux}}=7.2(e/c)^{2}\gamma^{4}$ and the total radiated energy $\mathcal{E}_{\textrm{rad}}=8.556(e/c)^{2}\gamma^{4}$ for a single particle. For the pair beam with distribution function from PIC simulations, $f(\gamma)_{\alpha}$, where $\alpha=(e_{b}^{-},e_{b}^{+})$, the resultant degree of circular polarization $\langle P_{\textrm{circ}}\rangle=(\langle P_{\mathrm{cflux}}\rangle_{e_{b}^{-}}-\langle P_{\mathrm{cflux}}\rangle_{e_{b}^{+}})/(\langle\mathcal{E}_{\mathrm{rad}}\rangle_{e_{b}^{-}}+\langle\mathcal{E}_{\mathrm{rad}}\rangle_{e_{b}^{+}})$, where $\langle P_{\mathrm{cflux}}\rangle_{\alpha}=\int P_{\textrm{cflux}}f(\gamma)_{\alpha}d\gamma/\int f(\gamma)_{\alpha}d\gamma$, yields

[TABLE]

Eq.(1) indicates that a finite circular polarization arises due to the difference in average kinetic energies between the beam species. For the purpose of comparison of PIC simulation results and Eq.(1), we performed two sets of PIC simulations: First, with $n_{\textrm{b0}}/n_{0}=[0.1,0.5,1.0]$, keeping the beam initial $\gamma_{\textrm{b0}}=50$ and then with $\gamma_{\textrm{b0}}=[10,20,30,40,50]$, keeping the $n_{\textrm{b0}}/n_{0}=1.0$. The $\langle\gamma^{4}\rangle_{\alpha}$ of the beam species was calculated for the duration $(750-1500)\omega_{\textrm{p}}^{-1}$ using the distribution obtained from the simulation data. The $\langle P_{\textrm{circ}}\rangle$ followed the scaling of Eq.1 as shown in Fig.5 (a-b). The $\langle P_{\textrm{circ}}\rangle$ decreases while the $\langle P_{\textrm{lin}}\rangle$ (insets) increases with $n_{\textrm{b0}}/n_{0}$. Since asymmetric energy dissipation in $e^{+}_{b}$ and $e^{-}_{b}$ species is larger at low density (as seen in Fig.2), the CP flux is also higher at lower densities. At higher density the energy difference between beam species is lower, yielding strong linear polarization as expected from synchrotron emission Rybicki and Lightman (1985); Medvedev and Loeb (1999). Contrastingly, $\langle P_{\textrm{circ}}\rangle$ increases ($\langle P_{\textrm{lin}}\rangle$ decreases) with $\gamma$ (peaks around $\gamma_{0}=30$ ) and decreases afterwards as shown in Fig. 5(b) for density ratio $n_{b0}/n_{0}=1$. The decrease can be associated with the effect of plasma hosing and modulation instabilities in plasma wakefield regime which may further reduce and saturate the $\langle P_{\mathrm{circ}}\rangle$ for high beam densities. Panels (c) and (d) show the dependence on the external magnetic field $B_{0}$ for different initial $n_{\textrm{b0}}$. Both components of polarizations increase with $B_{0}$ and saturate when the WI/CFI generated magnetic field equals the external magnetic field. The WI/CFI fields add a degree of randomness to the motion of charged particles reducing $\langle P_{\mathrm{circ}}\rangle$ and $\langle P_{\mathrm{lin}}\rangle$. For high $B_{0}$, the motion of the beam particles is dominated by $B_{0}$ and $\langle P_{\mathrm{circ}}\rangle$ arises only due to the energy difference between the beam $e_{b}^{-}$ and $e_{b}^{+}$ as seen in Fig.5 (c-d). In case of an $e_{b}^{-},e_{b}^{+}$ ambient plasma, both the species of the ambient plasma contribute equally to neutralize the current filaments of the beam species. As a result, there is no asymmetry in the kinetic energies between the beam species. Then $\langle P_{\textrm{circ}}\rangle=0$ for such a case and is independent of $n_{\textrm{b0}}/n_{0}$, as the $\langle P_{\textrm{circ}}\rangle$ due to the beam $e_{b}^{-}$ is canceled by that due to the beam $e_{b}^{+}$; see also Ref. Sinha et al. (2015) where generation of circular polarization in a different physical configuration is attributed to the topological changes in pitch-angle distribution of electrons instead of the asymmetric energy dissipation discussed here.

In the PIC simulations, we have not included the electron-positron annihilation and radiation reaction effects. The annihilation rate in a plasma, in the ultra-relativistic limit, can be written as $\dot{n}_{+}=(2\pi c\,n_{+}n_{-}r_{e}^{2}/\gamma_{0})\left[\mathrm{ln}\,2\gamma_{0}-1\right]$ sec*-1*, where $n_{+},n_{-}$ are the electron and positron densities respectively, $r_{e}$ is the classical electron radius, and $\gamma_{0}$ is the Lorentz factor of the pair beam Svensson (1982). For PIC simulation parameters, $n_{+}=n_{-}=10^{15}$cm*-3*, and $\gamma_{0}=50$, it yields $\dot{n}_{+}\sim 1$ fs*-1*, justifying the assumption of ignoring the pair annihilation in our PIC simulations. Also, the magnetic field due to the WI doesn’t grow to large values to necessitate the inclusion of RR force in our PIC simulations 444We carried out PIC simulations by including the Landau-Lifshitz radiation reaction force and found results to be unchanged in agreement with the results of Ref. D’Angelo et al. (2015). Also we have ignored that plasma collisions due to lower plasma density employed in PIC simulations.

In conclusion, we have shown that a pair beam ($e_{b}^{-},e_{b}^{+}$) propagating through a magnetized electron-proton plasma emits radiation by the synchrotron mechanism. The asymmetry in the kinetic energies of the beam species arising due to the energy dissipation into generating the transverse magnetic field do not allow the left and right circularly polarized radiation fluxes from the beam species to cancel, resulting in a finite circularly polarized radiation flux. The degree of linear and circular polarization depend on the initial $n_{b0}$ and $\gamma_{0}$ of the fireball, the initial magnetization and the composition of the ambient plasma. These results can be readily tested in laboratory astrophysics experiments. In addition, the origin of circular polarization due to asymmetric energy dissipation between beam $e_{b}^{-}$ and $e_{b}^{+}$ species opens the scope to explain the recently observed circular polarization in the optical frequency range from GRBs Wiersema et al. (2014).

Acknowledgements.

The authors would like to thank Dr. K. Hatsagortsyan for useful discussions. U. Sinha would like to thank the SMILEI development team for technical support.