Birefringence in thermally anisotropic relativistic plasmas and its impact on laser-plasma interactions
BBirefringence in thermally anisotropic relativistic plasmasand its impact on laser-plasma interactions
A. Arefiev, D. J. Stark, T. Toncian, and M. Murakami Department of Mechanical and Aerospace Engineering, University of California at SanDiego,La Jolla, CA 92093, USA Los Alamos National Laboratory, Los Alamos, New Mexico 87545,USA Institute for Radiation Physics, Helmholtz-Zentrum Dresden-Rossendorf e.V.,01328 Dresden, Germany Institute of Laser Engineering, Osaka University, Osaka 565-0871,Japan (Dated: 16 June 2020)
One of the paradigm-shifting phenomena triggered in laser-plasma interac-tions at relativistic intensities is the so-called relativistic transparency. Asthe electrons become heated by the laser to relativistic energies, the plasmabecomes transparent to the laser light even though the plasma density issufficiently high to reflect the laser pulse in the non-relativistic case. Thispaper highlights the impact that relativistic transparency can have on laser-matter interactions by focusing on a collective phenomenon that is associatedwith the onset of relativistic transparency: plasma birefringence in thermallyanisotropic relativistic plasmas. The optical properties of such a system be-come dependent on the polarization of light, and this can serve as the basisfor plasma-based optical devices or novel diagnostic capabilities.
I. INTRODUCTION
The advent of ultra-high intensity lasers has precipitated a corresponding effort to bettercharacterize the fundamental physics at play in high-amplitude laser-matter interactions.A laser pulse of high intensity irradiating a solid material quickly turns it into a plasma andthen continues to heat the plasma electrons. As the electron motion becomes relativisticdue to the heating induced by the laser pulse, the very optical properties of the plasmachange. The plasma can become transparent to the laser light even though its density issufficiently high to reflect the laser pulse in the case of non-relativistic electron energies .This fundamentally alters the nature of the interaction and has far-reaching repercussionson the subsequent evolution of the laser-plasma system.The goal of this paper is to highlight a novel collective phenomenon that is associatedwith the onset of relativistic transparency: plasma birefringence in anisotropic momen-tum distributions . Here the optical properties of a thermally anisotropic plasma becomedependent on the polarization of the laser light. An accurate description of the polariza-tion dependent dispersion in relativistic plasmas finds its application in the generation ofplasmonic devices , which have garnered much attention due to their tunability anddamage-resistant nature. The threshold for relativistic transparency in laser-plasma inter-actions has long been a topic of study because of its impact on the nature ofenergy transfer and particle dynamics; in particular, ion acceleration studies often operateat densities in the relativistic transparency regime and rely on precise characterizationsof this threshold . Understanding the role of polarization on these thresholds is critical forunderstanding the underlying physics at play in these systems, and it can also potentiallyserve as the basis for developing distribution function diagnostics in laboratory studies.First experimental validations of polarization rotation due to anisotropy of plasma heatedby a linearly polarized relativistic intensity laser pulse can be found in Refs. [12] and [24],and here we delineate in detail the origins and applications of this effect. Section II gives theanalytical treatment of the electromagnetic dispersion relation in a relativistic plasma withmomentum anisotropy, Section III provides particle-in-cell (PIC) simulations demonstratingthe birefrigent properties of these plasmas, and we close with a discussion in Section IV. a r X i v : . [ phy s i c s . p l a s m - ph ] J un II. ANALYTICAL TREATMENT OF THE OPTICAL PROPERTIES OF RELATIVISTICPLASMAS
Electron heating by an irradiating laser pulse can fundamentally alter the optical prop-erties of the plasma. In this section, we show that not only an otherwise opaque plasmacan become transparent to an electromagnetic wave due to relativistic electron motion, butit can also become birefringent. While the effect of relativistically induced transparency iswell-known, the relativistically induced birefringence, where the optical properties of themedium become dependent on the polarization of light, is a newly discovered effect .We begin by reviewing key properties of light propagation through a warm classical non-relativistic plasma. In the case of a plasma with an electron density n e and an electrontemperature T e , the dispersion relation for a linear electromagnetic wave is given by : ω = ω pe + c (cid:32) T e m e c ω pe ω (cid:33) k , (1)in which ω is the wave frequency, k is its wave vector, and ω pe = (cid:112) πn e e /m e (2)is the plasma frequency. Here c is the speed of light and m e and e are the electron massand charge respectively. It follows directly from Eq. (1) that the electromagnetic wave canpropagate only if the electron density is below the classical critical density defined as n ∗ ≡ m e ω πe . (3)The critical density is independent of the electron temperature, meaning that electronheating is inconsequential when it comes to transparency of a non-relativistic plasma.However, if the plasma electrons are heated to relativistic energies, then the relationbetween the electron momentum p and velocity v changes from p = m e v to p = γm e v ,where γ = (cid:112) p /m e c is the relativistic factor. This can effectively be interpreted asan increase in the electron mass. One can then expect, based on Eq. (3), that the criticaldensity would increase by a factor of γ as well compared to the classical critical density n ∗ : n crit ≈ γn ∗ . (4)This means that a plasma with an otherwise opaque electron density, n ∗ < n e < n crit ≈ γn ∗ , would become transparent to an electromagnetic wave if the electrons are heated torelativistic energies ∼ γm e c . The effect is often referred to as relativistic transparency. Should the polarization matter? (a) (b) ! !∆ $ $ FIG. 1. Two cases of the laser electric field E collinear (a) and orthogonal (b) to the electronmomentum p . This transparency is caused by the reduction in the electron current, because “heavier”electrons are less efficient in generating current in response to an electromagnetic wave.While the presented argument captures a well-known effect, it overlooks how the electroncurrent is actually driven by the wave. In order to illustrate an important subtlety, let usconsider how the velocity of a relativistic electron changes in response to a low-amplitudeelectromagnetic wave. The two cases of interest are shown in Fig. 1. If the electric field iscollinear to the electron momentum p , then a change of the momentum by ∆ p leads to achange in the electron velocity that can be estimated as∆ vc ≈ γ ∆ pp , (5)where γ = (cid:112) p /m e c . On the other hand, if the electric field is orthogonal to theelectron momentum p , then a change of the momentum by ∆ p leads to a change in theelectron velocity by ∆ vc ≈ ∆ pp . (6)Evidently, the velocity change is greatly reduced if the electric field is collinear with themomentum of the relativistic electron. This is because the velocity of the electron is alreadyclose to the speed of light in the direction of the intended change, so any potential changecan only be relatively small.These estimates indicate that the electron current should have a strong dependence onthe relative orientation of laser electric field E and particle momentum p . However, ifthe electron momentum distribution is isotropic, then the anisotropic effect from individualelectrons would be negated in the total electron current. On the other hand, an anisotropicdistribution should enable this effect to manifest itself on the macroscopic level as a birefrin-gent response to an electromagnetic wave, as the electrons would have a preferred directionto their momentum vectors.In order to quantitatively examine optical properties of an anisotropic relativistic plasma,we consider a simplified setup where the plasma is irradiated by a low-amplitude electro-magnetic wave. The unperturbed plasma is assumed to be uniform and anisotropic inmomentum space. Our goal is to derive a dispersion relation for the low-amplitude wavefollowing a standard approach.A linearized kinetic equation for the plasma electrons, ∂f∂t + v ∂f∂ r − | e | (cid:18) E + 1 c [ v × B ] (cid:19) ∂F∂ p = 0 , (7)has the following form in Fourier representation: i ( k µ v µ − ω ) f − | e | (cid:18) E s + 1 c [ v × B ] s (cid:19) ∂F∂p s = 0 , (8)where f is the perturbation to the distribution function F induced by electric and magneticfields E and B . Here ω and k are the corresponding frequency and wave-vector of theperturbation and v and p are the electron velocity and momentum, respectively. Using thedefinition of the electron current, j α ≡ − (cid:90) | e | v α f d p, (9)and taking into account that B = cω [ k × E ] , (10)one can find that j α = (cid:90) ie v α E β k µ v µ − ω (cid:20) δ sβ (cid:18) − k µ v µ ω (cid:19) + k s v β ω (cid:21) ∂F∂p s d p. (11)A general expression for the dielectric tensor is then ε αβ = δ αβ + 4 πe ω (cid:90) ∂F∂p β v α d p − πe ω (cid:90) v α v β k s k µ v µ − ω ∂F∂p s d p. (12)In order to provide a striking example of relativistically induced birefringence, we considera plasma that consists of two counter-streaming relativistic flows: F = 12 [ n δ ( p − p ) + n δ ( p + p )] . (13)where p = m e u / (cid:112) − u /c is the electron momentum associated with the flow velocity u . The corresponding dielectric tensor that follows from Eq. (12) is ε αβ = δ αβ (cid:32) − γ ω p ω (cid:33) + 1 γ ω p ω u α u β c (cid:0) ω − k c (cid:1) (cid:0) ω + ( k µ u µ ) (cid:1) ( k µ u µ − ω ) ( k µ u µ + ω ) + 1 γ ω p ω k µ u µ ( k α u β + k β u α )( k µ u µ − ω ) ( k µ u µ + ω ) . (14)If the counter-streaming flows are non-relativistic, then a simplified expression follows fromEq. (14) by setting γ = 1.We simplify the analysis by assuming that the wave propagation is transverse to thecounter-streaming flows, such that ( k · u ) = 0. Let us also assume without any loss ofgenerality that both k and u are in the ( x, y )-plane and that u = u e y and k = k e x . Thewave dispersion relations are determined from a general conditiondet (cid:20) k α k β c ω − k c ω + ε αβ (cid:21) = 0 . (15)For the dielectric tensor given by Eq. (14), this equation leads to (cid:34) − γ ω p ω (cid:35) (cid:34) − γ ω p ω − (cid:32) γ ω p ω u c (cid:33) k c ω (cid:35)(cid:34) − γ ω p ω − k c ω (cid:35) = 0 . (16)The three dispersion relations that follow are ω = ω p / γ , (17) ω = ω p (cid:14) γ + k c (cid:32) γ ω p ω u c (cid:33) , (18) ω = ω p / γ + k c . (19)The corresponding wave polarizations are E = E e x , and B = 0 , (20) E = E e y , and B = E kcω e z , (21) E = E e z , and B = − E kcω e y , (22)where e x , e y , and e z are unit vectors.The key result is that all three waves have different dispersion relations. The transverselypolarized electromagnetic waves described by Eqs. (18) and (19) have different phase veloc-ities and dramatically different cutoff densities: n ( y ) crit = γ n ∗ , (23) n ( z ) crit = γn ∗ , (24)where the upper index indicates the polarization of the electric field in the correspondingwave. Equation (18) is implicit, but it has a convenient form for determining the cutoffdensity.In agreement with the qualitative estimates given earlier in this section, the wave whoseelectric field is collinear with the counter-streaming electron motion drives electron currentless efficiently. As a result, the corresponding cutoff density (23) is significantly higher thanin the case where the electric field driving the current is orthogonal to the electron motionin the flow. The plasma with two counter-streaming flows is clearly birefringent, as the twotransversely polarized waves have different cutoff densities and different phase and groupvelocities. III. PIC SIMULATIONS DEMONSTRATING BIREFRINGENCE IN RELATIVISTICALLYTRANSPARENT PLASMAS
The counter-streaming electron distribution that we have chosen provides an easy exampleto analyze, but it is extremely unstable. The distribution would quickly start to evolveafter being initialized, so that the anisotropy and accompanying birefringence would bevery difficult to probe with a laser pulse. In this context, Eqs. (23) and (24) should beviewed as an upper estimate for the degree of birefringence in a plasma whose characteristicelectron relativistic factor is γ . However, one could envision setups in which the plasma iscontinuously pumped by laser or particle beams and thus anisotropy can be maintained forlonger periods of time.In order to further explore propagation of electromagnetic waves through a plasma withan anisotropic distribution, we have performed fully self-consistent one-dimensional PICsimulations using a fully-relativistic code EPOCH . In these simulations a 50 µ m thickplasma slab with n e = 0 . n ∗ is irradiated by a 150 fs long circularly polarized laser pulsewhose maximum electric field amplitude is E ≡ V/m. The laser wavelength is 1 µ m.There are 100 cells per wavelength, each with 2000 macro-particles representing electronsand 2000 macro-particles representing ions. The ions are treated as immobile to preventplasma expansion; this does not alter the effect of birefringence. At the same time, this en-sures that the electron density remains unchanged, so that any changes of optical propertiesof the plasma slab are only due to changes in the electron momentum distribution.In the first simulation, the plasma is cold in all directions and thus the electron momentumdistribution is isotropic. The polarization of the laser pulse transmitted by the plasma isshown in the upper panel of Fig. 2, where the dots represent the electric field componentson the grid used by the PIC code. The maximum amplitude and the polarization of thetransmitted laser pulse are the same as those of the incoming laser pulse. In the secondsimulation, the plasma consists of two counter-streaming electron flows aligned along the y -axis. The corresponding distribution is given by Eq. (13), where p ≈ . m e c and thecorresponding flow velocity is u = 0 . c . This highly unstable electron distribution is given100 fs to relax to a slowly evolving distribution before it is irradiated by the laser pulse.The polarization of the laser pulse transmitted by the plasma is shown in the lower panelof Fig. 2. In contrast to the previous case, the plasma changes the laser polarization toelliptical due to the phase velocity discrepancy between the y and z -polarizations. Themaximum amplitude of the electric field also changes, as it becomes higher than that in theincoming laser pulse (shown with a solid curve).We can therefore conclude that the anisotropy that results from a highly unstable two-stream electron distribution produces the analytically predicted optical changes in a mixedpolarization laser pulse. This is in spite of the plasma isotropizing due to the unstableplasma distribution. The anisotropy in the simulation is sufficiently long-lived to be probedby a 150 fs long laser pulse. It takes over 300 fs for the laser pulse to pass fully through theconsidered plasma slab.It might appear that a relativistic plasma flow represents a stable case of a birefringentrelativistic plasma. This is however not the case, because the flow can be eliminated by con-sidering wave propagation in a frame of reference moving with the flow velocity. Since thereis no birefringence without the flow and since all inertial frames of reference are equivalent,two electromagnetic waves with different polarizations propagating in the same directionthrough a plasma flow would have the same phase and group velocities. In other words,the optical properties are independent of the polarization. The same result regarding theabsence of birefringence in a flowing plasma can be confirmed using a standard perturbativeanalysis given in Appendix A.The key conclusion then is that a relativistically induced birefringence requires ananisotropy of the irradiated electron momentum distribution that excludes an overallelectron flow. However, the anisotropy is not required to be very severe in order to producespectacular results. The final example of this section is designed to illustrate just that.In this 1D PIC simulation, electrons are initialized with an anisotropic distribution func-tion F = nN ( α, β ) exp − α (cid:115) p z + β ( p x + p y ) m e c , (25)where N ( α, β ) = (cid:90) exp − α (cid:115) p z + β ( p x + p y ) m e c d p (26)is a normalization factor. Here 1 /α is an effective temperature normalized to m e c and β (cid:54) = 1introduces anisotropy into the distribution. We set α = 2 . β = 0 .
1, so that effectivelythe x and y directions are “hotter” than the z direction. We sampled this distribution 15000times and show the scatterplot (see Fig. 3b) of the points in ( p y , p z ) space to illustrate thegreater momentum spread in the y -direction.A plasma with such an initial electron momentum distribution is irradiated by a linearlypolarized laser pulse propagating along the x -axis. The laser electric field is polarized at a45 ◦ angle in the transverse ( y, z )-plane, such that max( E y ) = max( E z ) = E = 4 . × V/m. The incoming pulse is 250 fs long with a wavelength of 1 µ m. Note that the noisein the incoming pulse shown in Fig. 3 is physical and it is caused by the fields emitted bythe plasma itself as its electron distribution evolves. The electron density profile is shownin Fig. 3a, where the slab is 150 µ m thick and the peak electron density is 3 . n ∗ . There are184 cells per micron with 200 electron and 100 ion macro-particles per cell. The ions areagain treated as immobile.A laser pulse transmitted by the plasma is shown in Fig. 3a. Remarkably, we now havetwo linearly polarized laser pulses instead of just one. The leading pulse is polarized alongthe y -axis, whereas the trailing pulse is polarized along the z axis. This is because theplasma is more transparent for the y -polarized part of the incoming pulse, so its groupvelocity is faster than for the z -polarized part of the pulse. This difference causes the twopulses to eventually split. In our previous example the propagation was not long enoughfor this effect to accumulate, but the corresponding phase velocity discrepancy resulted ina single elliptically polarized pulse. IV. SUMMARY AND DISCUSSION
In this work we have examined a fundamental physical phenomena resulting from the so-called relativistically induced transparency. It is shown that the relativistic transparencyimpacts the laser propagation by causing the irradiated plasma to become birefringentwhen there is a thermally anisotropic electron momentum distribution. This birefringencecan lead to such easily observable features as a change of polarization and pulse splitting,where a single pulse splits into two staggered pulses with orthogonal linear polarizations.The ability of a plasma to affect such optical changes on a pulse without damaging anyequipment provides an attractive option for future optical devices.The presented 1D PIC simulations are intended to illustrate the striking features of thisbirefringence. While the simulations do capture the evolving plasma distributions fromthe inherent instabilities, 3D simulations would provide the most robust description of thisinstability-driven isotropization timescale. These were performed in Ref. 11, in which theanistropy was shown to persist long enough to be probed by the selected laser pulses. Withthis context, the 1D simulations here complement this prior work and can provide bothan upper limit to the birefringence and also an example of what can occur in continuouslypumped systems in which anisotropy is better maintained.With the recent experimental demonstration of this birefringent phenomenon , char-acterizing momentum anisotropy in relativistic plasmas also becomes possible. Across thespectrum of high-intensity laser-plasma interactions, many variations of plasma distributionfunctions are generated. These distributions largely influence the plasma applications underconsideration, so proper diagnosis of said distributions would provide invaluable feedbackfor both modeling and experimental efforts . ACKNOWLEDGMENTS
This research was supported by AFOSR (Grant No. FA9550-17-1-0382). Simulationswere performed by EPOCH (developed under UK EPSRC Grant Nos. EP/G054950/1,EP/G055165/1, and EP/G056803/1) using HPC resources provided by TACC at the Uni-versity of Texas.
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The propagation of electromagnetic waves in plasmas (Addison-Wesley Publishing Com-pany Inc., 1964). T. D. Arber, K. Bennett, C. S. Brady, A. Lawrence-Douglas, M. G. Ramsay, N. J. Sircombe, P. Gillies,R. G. Evans, H. Schmitz, A. R. Bell, and C. P. Ridgers, “Contemporary particle-in-cell approach tolaser-plasma modelling,” Plasma Physics and Controlled Fusion , 113001 (2015). A. Davies, D. Haberberger, J. Katz, S. Bucht, J. Palastro, W. Rozmus, and D. Froula, “Picosecondthermodynamics in underdense plasmas measured with thomson scattering,” Physical review letters ,155001 (2019). A. Davies, D. Haberberger, J. Katz, S. Bucht, J. Palastro, R. Follett, and D. Froula, “Investigation ofpicosecond thermodynamics in a laser-produced plasma using thomson scattering,” Plasma Physics andControlled Fusion , 014012 (2019). Appendix A: Dispersion relation in a relativistic plasma flow
We consider a cold plasma flow of both species with F = n δ ( p − p ) , (A1)0where p = m e u / (cid:112) − u /c is the electron momentum associated with the flow velocity u . The corresponding dielectric tensor readily follows from Eq. (12) and it is given by ε αβ = δ αβ (cid:32) − γ ω p ω (cid:33) + 1 γ ω p ω u α u β c ω − k c ( k µ u µ − ω ) + 1 γ ω p ω k α u β + k β u α k µ u µ − ω . (A2)One can derive wave dispersion relations for an arbitrarily directed wave vector k withrespect to u using the dielectric tensor given by Eq. (A2). Here we consider the case wherethe wave propagation is transverse to the plasma flow. Let us also assume without any lossof generality that both k and u are in the ( x, y )-plane and that u = u e y and k = k e x . Itfollows from Eq. (15) that the dispersion relations of propagating waves must satisfy thefollowing equation: (cid:34) − γ ω p ω (cid:35) (cid:34) − γ ω p ω − k c ω (cid:35) = 0 , (A3)where we explicitly take into account that γ = (1 − u /c ) − / . The dispersion relationsthat follow from Eq. (A3) are ω = ω p (cid:14) γ , (A4) ω = ω p / γ + k c . (A5)In a reference frame moving with velocity − u along the y -axis, the considered cold plasmaflow case reduces to a well-known problem of wave propagation in a cold plasma without aflow. The three modes in such a plasma are a plasma wave (˜ ω = ˜ ω p ) and two electromag-netic waves (˜ ω = ˜ ω p + ˜ k c ), where the tilde marks quantities in the frame of referencewithout the flow. Using the Lorentz transformation, we find that ˜ ω = γω and ˜ ω p = ω p /γ .It is now straightforward to establish the correspondence between the waves in a plasmawithout a flow and the modes described by Eqs. (A4) and (A5). The mode given byEq. (A4) corresponds to the plasma wave. In the presence of a flow, the plasma wave alsoinvolves transverse electric field oscillations. The modes whose dispersion relation is givenby Eq. (A5) correspond to the electromagnetic waves (note that ˜ ω − ˜ k c = ω − k c ).The mode whose electric field is polarized along the flow now also includes longitudinalelectric field oscillations.Evidently, there is no birefringence for the electromagnetic modes in a cold plasma witha flow. Both modes have the same dispersion relation, meaning that their group and phasevelocities are identical.1 -1 -0.5 0 0.5 1-1-0.500.51 -1 -0.5 0 0.5 1-1-0.500.51 -1 10 0.5-0.5-1-0.500.51 -1 10 0.5-0.5-1-0.500.51 (a)(b) ⁄ " " $ ⁄ " " $ ⁄" % " $ FIG. 2. Polarization of a laser pulse with an initial peak amplitude E transmitted by plasmaswith (bottom) and without (top) an anisotropy in electron momentum distribution. The solidcurve shown the peak amplitude in the incoming laser pulse. -1 -0.5 0 0.5 1 1.5 2 2.5 3 x (m) -4 -505 x , µm × E , V / m n e / n * incoming transmittedelectrondensity ! " = ! $ ! $ ! " (a) (b) ⁄" $ % & ⁄ " ' $ % & FIG. 3. (a) Laser pulse splitting due to plasma birefringence in a plasma with an anisotropicelectron momentum distribution in the plane transverse to the pulse propagation. (b) Initializedelectron distribution function for the target plasma in ( p y , p zz