Dec 11, 2008 - pared to the Lorentz force, and is not expected to be a major determinant ... in Lorentz-. Abraham-Dirac LAD form 8 , and variety of its approxi-.
PHYSICAL REVIEW E 78, 066403 共2008兲
Plasma acceleration and cooling by strong laser field due to the action of radiation reaction force V. I. Berezhiani Andronikashvili Institute of Physics, Tamarashvili 6, Tbilisi 0177, Georgia
S. M. Mahajan Institute for Fusion Studies, The University of Texas at Austin, Austin, Texas 78712, USA
Z. Yoshida Graduate School of Frontier Sciences, The University of Tokyo, Kashiwa 277-8561, Chiba, Japan 共Received 7 April 2008; published 11 December 2008兲 It is shown that for super intense laser pulses propagating in a hot plasma, the action of the radiation reaction force 共appropriately incorporated into the equations of motion兲 causes strong bulk plasma motion with the kinetic energy raised even to relativistic values; the increase in bulk energy is accompanied by a corresponding cooling 共intense cooling兲 of the plasma. The effects are demonstrated through explicit analytical calculations. DOI: 10.1103/PhysRevE.78.066403
PACS number共s兲: 52.27.Ny, 52.27.Ep, 52.30.Ex, 52.38.Kd
I. INTRODUCTION
The bulk acceleration of a plasma to relativistic velocities represents one of the central issues of modern plasma physics and astrophysics. In most literature, devoted to the dynamics of plasmas embedded in the field of electromagnetic 共EM兲 radiation, the radiation reaction force 共RRF兲 acting on the plasma particles is neglected; it is ordered small compared to the Lorentz force, and is not expected to be a major determinant of plasma dynamics. There are two distinct cases when this assumption may not be justified: 共1兲 Several astrophysical situations in which the spatiotemporal scales of plasma motion are sufficiently large, and 共2兲 when the intensity of radiation is relativistically strong. The modern petawatt lasers systems are already capable of producing ultrashort pulses with the focal intensities I = 1021 – 1022 W / cm2 关1兴. Pulses of even higher intensities exceeding I = 1024 W / cm2 关2兴 and lasting from a few femtoseconds to picoseconds 共in the energy range 1 J – 1 kJ兲 are likely to be available soon. The power radiated by charged particles, accelerated in the EM field, increases with energy. Since the scattered radiation carries away the energy and momentum, the recoil force felt by the radiating charge 共i.e., the radiation reaction, or the radiation “damping”兲 could become significant. Recently Zhidkov et al. 关3兴 studied the effect of radiation reaction on the interaction of an ultraintense laser pulse with an overdense plasma slab via a relativistic particle-in-cell simulation. Applying the Landau and Lifshitz expression for the force it was shown that the effects of radiation reaction become significant if the laser intensity exceeds 1022 W / cm2 共corresponding to the normalized laser amplitude eE0 / mec ⬃ 50, see also 关4兴兲. For higher intensities considerable modification of the scattered radiation spectrum is expected, in particular, a burst of incoherent x rays could be emitted. The radiation pressure could also be important in astrophysical conditions. In fact, the acceleration of plasma by radiation pressure force has been considered as a possible mechanism for producing relativistic outflows 共jets兲 from very luminous radiation sources, such as the active galactic nuclei 共AGNs兲 or compact galactic objects 共see 关5兴 and ref1539-3755/2008/78共6兲/066403共8兲
erences therein兲. In astrophysical community it is customary to use the term “the radiation pressure” as a pressure arising on the plasma particles through Compton scattering of incoherent photon fluxes generated by those objects. Through scattering of external photons, individual particles in a plasmas lose energy simultaneous with momentum transfer to the plasma. The bulk flow can be either accelerated or decelerated 共i.e., radiative drag兲. The radiative drag force is derived by resorting to a phenomenological, test-particle approach. In this approach the energy-momentum conservation 共in the Thompson or the Compton-Klein-Nishina regime兲 is invoked to treat the particle-photon interaction with subsequent integration of the obtained force over the distribution function. It is interesting to remark that Landau and Lifshitz 关6兴 共see also 关7兴兲 demonstrated that the radiation drag force, acting on an electron which scatters photons, can be derived 共in the Thompson regime兲 not only through the energymomentum considerations but also by averaging the RRF. Thus, the test particle approach 共based on semiclassical treatment of recoil arising from photon scattering兲 is equivalent to the classical treatment 共including RRF in the equation of motion兲 as long as the photon energy in the particle rest frame ប Ⰶ mc2, and the strength of EM fields is kept below the Schwinger limit. Though this statement sounds trivial, the discovery that the inclusion of radiation reaction in the equation of motion of a charged particle 关in LorentzAbraham-Dirac 共LAD兲 form 关8兴, and variety of its approximations and/or modifications 关6,9兴兴 could, in certain cases, lead to particle energy gain 共see, for instance, 关4,10,11兴兲 was labeled “surprising” and, in fact, counterintuitive. Indeed, the accelerated charged particles radiate the EM field, and consequently their motion should dampen. According to Zel’dovich 关12兴 a systematic acceleration of charged particles, in the direction of EM wave propagation, is the main consequence of inclusion of RRF force in the equation of motion. The origin of this effect is related to the fact that the charged particle, absorbing energy from the wave, also absorbs a corresponding momentum. The energy is not accumulated by the particle but is reemitted. However, the reemitted energy is not “directed” exactly forward; the momentum lost by the particle is less than the momentum
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©2008 The American Physical Society
PHYSICAL REVIEW E 78, 066403 共2008兲
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gain. Keitel et al. 关13兴 conducted careful analytical and numerical analysis of the electron motion in the field of a strong laser pulse, and confirmed that the damping of electron motion takes place in the laser field polarization direction, while in the propagation direction electrons are accelerated due to the absorption and reabsorption of emitted radiation. In this paper the radiative acceleration to relativistic velocities of a hot, optically thin, underdense plasma, exposed to a relativistic strong laser pulse, is investigated. We use the plasma dynamics model, recently developed in 关14兴 in which the RRF is consistently included in the relativistic fluid equations. By taking moments of the relativistic kinetic equation 共RRF in the Landau-Lifshitz form兲 the authors derived a manifestly covariant relativistic hydrodynamic equations for a multispecies relativistic plasma in an external EM field. It was found that the derived equations, in the limit of a plasma embedded in the field of incoherent photon fluxes, can be reduced to the equations for plasma bulk motion, obtained 共in astrophysical context兲 in previous literature by adopting the test particle approach 关5兴. It has been shown by O’Dell 关15兴 that a hot 共relativistic兲 plasma, immersed in the photon fluxes from astrophysical objects, feels a much larger radiation pressure than a cold 共nonrelativistic兲 gas, and tends to drive itself away from the radiation source with momentum derived largely from the anisotropic loss of its own internal energy. This “Compton rocket” is, however, always accompanied by “Compton cooling” of the plasma 共see Phinney in Ref. 关5兴兲. Similar effect could be operative in laser plasma as well. Laser sources generate highly coherent but anisotropic photon fluxes. We will soon demonstrate that ultrastrong laser pulses, impinging on a hot plasma, can indeed strongly accelerate as well as cool the bulk plasma via the action of RRF. It turns out that, under certain simplified assumptions, the effect can be demonstrated analytically.
II. BASIC EQUATIONS
We use the set of relativistic hydrodynamic equations derived in 关14兴 for optically thin multispecies plasma. Throughout this paper, we adopt the notations and conventions used in 关14兴 excepting the metric that is chosen to be g␣ = 共1 , −1 , −1 , −1兲. The fluid equations, valid for each species, can be written in a manifestly covariant form
T␣ ␣ − qF␣nU = Frad , x
共1兲
where the greek indices go from 0 to 3; ␣ = / x␣ = 共c−1 / t , 兲; T␣ is the energy-momentum tensor of plasma species with charge q and mass M, and U␣ = 共␥ , ␥V / c兲 is the local four velocity with ␥ = 共1 − V2 / c2兲−1/2 共U␣U␣ = 1兲; the rest-frame particle density n satisfies the continuity equation nU␣ / x␣ = 0. Note that we do not label the fluid species by an additional index for brevity. The EM field tensor can be formally written as F␣ = 关E , B兴 and it satisfies the Maxwell equations F␣ = −共4 / c兲J␣, ⑀␣␥␦F␥␦ = 0, where J␣ = 共c , J兲, and J are, respectively, the total charge and the
current density of plasma and ⑀␣␥␦ is the antisymmetric tensor. Equation 共1兲, expressing the conservation of momentum and energy, has, in addition to the conventional terms 共left␣ ; the latter represents the hand side兲, an extra term Frad 4-force density related to the radiation reaction 共emission and/or scattering兲, and is necessary to correct the momentum energy balance. Note that the momentum change due to collisions is ignored in Eq. 共1兲. This assumption can be justified by the fact that with increase of EM radiation intensity, the role of collisional processes decreases and they can be ignored in comparison to the radiation reaction that scales up with the intensity of the electromagnetic wave. The energy momentum tensor T␣ is assumed to be that of an ideal isotropic fluid: T␣ = wU␣U − g␣ p, where w = E + p is the enthalpy per unit volume, and E共p兲 is the proper internal energy density 共pressure兲 of the fluid. For a Maxwellian ¯兲 while p = nT 关16兴 with G plasma, w = nc2MG共z ¯兲 / K3共z ¯兲 where K2 and K3 are, respectively, the modi= K3共z fied Bessel functions of the second and third order. The argument ¯z = Mc2 / T, and T is the temperature measured in the ¯兲 defines the rest frame of a fluid element. The function G共z “effective” temperature-dependent mass of the particles, and ¯ for ¯z has the following limiting expressions: G ⬇ 1 + 5 / 2z ␣ Ⰷ 1, and G ⬇ 4 /¯z for ¯z Ⰶ 1. The expression for Frad has been derived in 关14兴, and is ␣ = Frad
2q3 F␣ ␥ ¯ ␣U ¯兲/z ¯兴T T + n兵关1 + 2G共z  3M 2c4 x␥  ¯ ␥U U U␣其, ¯兲/z ¯兴T − 关1 + 6G共z  ␥
共2兲
where = 8q4 / 3M 2c4 is the Thomson cross section, and ¯T␣ = 1 共−F␣␥F + 1 g␣F␥␦F 兲 is the energy momentum ten␥␦ ␥ 4 4 sors of the EM field. Equations 共1兲 and 共2兲 along with the continuity and Maxwell’s equations form the closed set describing consistently the bulk motion of a multispecies relativistic plasma in the presence of strong EM radiation. It is convenient to introduce the following dimensionless variable: x⬘␣ = 共 / c兲x␣ ¯ = 1 / T⬘兲, F⬘␣ 关i.e., t⬘ = t, r⬘ = 共 / c兲r兴, T⬘ = T / Mc2 共z ␣ = 共q / Mc兲F 关i.e., E⬘ = 共q / Mc兲E, B⬘ = 共q / Mc兲B兴, n⬘ = n / N0, p⬘ = p / N0Mc2, V⬘ = V / c, and ␥ = 冑1 − V⬘2 关U⬘␣ = 共␥ , ␥V⬘兲兴, T⬘␣ = T␣ / 共N0Mc2兲, ¯T⬘␣ = 4共q / Mc兲2¯T␣, J⬘␣ = J␣ / 共qN0c兲. In this notation, N0 is the equilibrium density of the plasma and is the characteristic frequency of the EM field. Suppressing the superscripts, we arrive at the dimensionless equations
T␣ ␣ − F␣nU = Frad , x ␣ = Frad
共3兲
F␣ ␥ ¯ ␣U ¯兲/z ¯兴T T + n兵关1 + 2G共z  x␥ 
¯ ␥U U U␣其, ¯兲/z ¯兴T − 关1 + 6G共z  ␥
共4兲
where = 0 and 0 = 2q2 / 3Mc3 关for electrons 共M = me , q = −兩e兩兲 0 = 0.6⫻ 10−23 s兴.
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In order to explicitly display the content of these equations, it helps to go to the more familiar notation of space and time components. In what follows we work in terms of three vectors sacrificing manifest covariance. We begin by writing the EM field tensor in the matrix form
冋
册
S ¯T␣ ⬅ W, , S, − ik
共5兲
W = ¯T00 = 21 共E2 + B2兲
is the EM field energy density, S where = 关E ⫻ B兴 is the pointing vector and ik is a stress tensor given by the expression ik = 关− 21 ␦ik共E2 + B2兲 + EiEk + BiBk兴 共i , k = 1 , 2 , 3兲. Simple algebra leads to the following identity:
冋
册
2 2 ¯T␣U = ␥ 共E + B 兲/2 − V · 共E ⫻ B兲 ,  AL1 + V共B2 − E2兲/2
冉
1 共B2 − E2兲 2␥2
冊冋 册
␥ . ␥V
共7兲
The remaining terms, appearing explicitly in Eqs. 共3兲 and 共4兲, are T␥␥F␣ and T␣. One can show that T␥␥F␣ = ␥nGUdF␣ / dt − pF␣ 关here d / dt = / t + 共V · 兲兴. Using the Maxwell equation F␣ = −⍀L2 J␣ 关where ⍀L = 共4q2N0 / M兲1/2 / is the normalized plasma frequency兴 and applying the useful relation F␣U = ␥共E · V , E + 关V ⫻ B兴兲, we get ␣
冤
V
dE dt
F T␥ ␥ = ␥2nG dE x dB + V⫻ dt dt
冋
while for T␣ we have the identity
册冥
+ p⍀L2
冋册 J
dS = ⍀L2 共 − V · J兲 − 共2G兲␥−1共B2 − E2兲 dt
冋 冉
1 2 2G dE dB + V⫻ ⍀LJ − V共B2 − E2兲 + G␥ ¯z dt dt ␥¯z
共13兲
¯G兲/z ¯N兴 + const. S = ln关␥K2 exp共z
共14兲
where Deriving Eq. 共13兲 we used the relations U␣J␣ = ␥共 − V · J兲 and identity 关17兴
冉 冊
1 dS ␥d N d . G− = dt N dt ¯z␥ ¯z dt
共15兲
One can see that without the RRF 共 = 0兲 the plasma dynamics is isentropic with a corresponding relativistic adiabatic equation of state. With radiation reaction, the entropy is no longer a constant along the streamline. In dimensionless form the Maxwell equations are ⫻ E = −tB, · B = 0 and
共8兲
⫻B=
E − ⍀L2 J. t
N + · 共NV兲 = 0, t
共16兲 共17兲
共10兲
冊册
¯兲V兵关E + 共V ⫻ B兲兴 − 共V · E兲 其. − ␥ 共1 + 6G/z 2
共11兲
Here N = ␥n is the density in the laboratory frame and pressure is p = N /¯z␥ 共p = NT / ␥ in dimensional units兲. Equation 共10兲 is the vector fluid equation of a relativistic plasma generalized to include the radiation reaction force R.
共18兲
represent the closed system of Maxwell and relativistic fluid equations. The RRF force explicitly depends on the plasma temperature. It is interesting to remark that in the cold plasma limit 共T → 0兲, ¯z → ⬁, and G → 1, the first two terms on the righthand side of Eq. 共11兲 vanish and R reduces to the Landau and Lifshitz 关6兴 expression for the force acting on a single particle: R = ␥
¯兲关共E ⫻ B兲 + B ⫻ 共B ⫻ V兲 + E共V · E兲兴 + 共1 + 2G/z 2
− 共4G兲␥兵关E + 共V ⫻ B兲兴2 − 共V · E兲2其,
共9兲
1 d 共G␥V兲 + p = E + 共V ⫻ B兲 + R, dt N
2
共12兲
that, in three vector notation, becomes 共S is the entropy per particle兲
· E = − ⍀L2 ,
The preceding preparation is enough to spell out Eq. 共3兲 into space and time components; the spatial component reads as
R=
T␣ ␣ = U␣Frad , x
These equations along with Eqs. 共10兲 and 共13兲, and the continuity equation
d T␣ p = n␥ 共GU␣兲 − . dt x x␣
where
U␣
共6兲
where AL1 = 共E ⫻ B兲 + B ⫻ 共B ⫻ V兲 + E共V · E兲. By similar manipulations, we derive ¯T␥UU␥ = ␥2AL2 + 21 共B2 − E2兲, where AL2 = 关E + 共V ⫻ B兲兴2 − 共V · E兲2, and U␣¯T␥UU␥ = ␥2 AL2 +
The temporal 共zero兲 component of Eq. 共3兲 can be readily written, and we do not display it here. We will, instead, complete the fluid system by deriving the 共equivalent兲 equation for entropy by projecting Eq. 共3兲 “along” the four velocity U␣. Multiplying 共3兲 by U␣, and using the obvious identity F␣UU␣ = 0, we get
冋 冉
dE dB + V⫻ dt dt
冊册
+ 关共E ⫻ B兲 + B ⫻ 共B ⫻ V兲
+ E共V · E兲兴 − ␥2V兵关E + 共V ⫻ B兲兴2 − 共V · E兲2其. 共19兲 For high temperatures, however, the Landau-Lifshitz expression is modified by the temperature-dependent factors, and new terms appear 共11兲. ¯ → 0兲 the effective For ultrarelativistic temperature T Ⰷ 1 共z mass of the fluid element enhanced to G ⯝ 4T. Consequently
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the Lorentz force on a fluid element becomes weaker in contradistinction to the RRF at ultrarelativistic temperature,
R = ␥−1T⍀L2 J + 4T␥
冋 冉
dE dB + V⫻ dt dt
冊册
− 8T2V共B2 − E2兲 + 8T2关共E ⫻ B兲 + B ⫻ 共B ⫻ V兲 + E共V · E兲兴 − 24␥2T2V兵关E + 共V ⫻ B兲兴2 − 共V · E兲2其 共20兲 that becomes stronger because it scales as ⬃T2 关see last three terms on the right-hand side of Eq. 共20兲兴. At sufficiently high temperatures T共Ⰷ1兲, RRF can match or even beat the Lorentz force. The temperature enhancement of the radiation drag is consistent with the results of O’Dell 关15兴 共see also 关5兴兲, who demonstrated that a hot 共relativistic兲 plasma embedded in the photon fluxes from the astrophysical objects feels a much larger radiation pressure than a cold 共nonrelativistic兲 gas. The main equations derived in Refs. 关15,5兴 by test particle approach can be recovered by averaging Eqs. 共3兲 and 共4兲 over the random phases of incoherent “photons” 关14兴. In the averaging procedure the Lorentz force vanishes, and the high frequency components in U␣ are ignored. Strictly speaking in such an averaging procedure, one neglects the effects related to the high-frequency pressure 共ponderomotive pressure兲 arising, for instance, by averaging the magnetic part of the Lorentz force 关⬃共V ⫻ B兲兴. However, the corresponding force is a gradient force which vanishes for the uniform spatiotemporal structure of the radiation fluxes, i.e., the case that is mostly considered in astrophysical situations. For a monochromatic EM pulse, the dynamics of plasma acceleration is ruled by the Lorentz force, while in the expression of RRF all terms can be equally important. In most cases of interest RRF can be ignored since it is considerably smaller than the Lorentz force. Indeed, Eq. 共20兲 contains a small parameter = 0 which 共for electrons兲 can be estimated to be = 1.2⫻ 10−8 / 关m兴, where 关m兴 is the vacuum wavelength in m. Thus, for available laser sources ⯝ 10−8 – 10−9 and consequently the action of RRF can be ignored even for the laser pulses with intensities as high as I = 1020 W / cm2 共note that such intensities have already been achieved in several laboratories worldwide兲. It is only at higher intensities that the RRF becomes comparable to the Lorentz force. A feature RRF worth emphasizing is that it is a nonpotential force and the plasma particles will be left with residual energies after interaction with the entire pulse. In this case, the bulk plasma is set in motion. If the plasma is relativistically hot 共or is heated by the laser pulse itself兲 the RRF is stronger and it could provide a mechanism for the plasma bulk energy to grow at the expense of its internal energy and intensive plasma cooling will be the consequence. To extract all the above-mentioned physics, one will have to resort to numerical methods to solve the complex system of Eqs. 共10兲–共19兲. However, under certain simplified assumption, the system is amenable to analytical solutions.
III. TRANSPARENT PLASMAS
The simplest nontrivial example is the EM pulse propagation in an underdense unmagnetized electron-positron plasma. We have chosen this system not only for its relative simplicity, but also due to the fact that in the ultrarelativistic limit 共due to the strong laser field or due to the relativistic temperatures兲, the electron-ion plasma also behaves similarly to the electron-positron one. The electron-positron plasma is further relevant because the electron-ion collisions give rise to copious pair creation via Bremsstrahlung photons, or by the trident processes 关18兴. We consider a transversely polarized 1D pulse E = 共E⬜ , 0兲 propagating along z axis in a highly transparent plasma 共⍀L Ⰶ 1兲. The group velocity of such a pulse is vg = 共1 − 1 / ⍀L2 兲1/2 ⯝ 1. We assume that the EM pulse profile is “given” and does not change in the course of propagation. The equilibrium state of the plasma is characterized by an overall charge neutrality Ne0 = N p0 = N0, where Ne0 and N p0 are the unperturbed number densities of the electrons and positrons. In most mechanisms for creating e-p plasmas, the pairs appear simultaneously and due to the symmetry of the problem it is natural to assume that Te0 = T p0 = T0, where Te0 and T p0 are the respective equilibrium temperatures. The longitudinal motion of plasma is driven by the ponderomotive pressure, and RRF. Being the same for the electrons and positrons, these forces do not cause charge separation. Because of the symmetry between the electron and positron fluids, their temperatures, being initially equal, will also remain equal during the evolution of the system. It follows, then, that Ne = N p = N and Te = T p = T, and consequently Ez = = Jz = 0. Assuming that all variables depend on = 共t − z兲 and taking 2 共1 − Vz兲 and into account the relations B = 关zˆ ⫻ E⬜兴, AL1 = zˆ E⬜ 2 2 AL2 = E⬜共1 − Vz兲 共where zˆ is the unit vector directed along z axis兲, Eq. 共10兲 can be reduced to the following equation for the transverse momentum of the fluid dE⬜ d 共GP⬜兲 = E⬜ + G 共 ␥ − P z兲 d d 2 ¯兲E⬜ 共␥ − Pz兲P⬜ − 共1 + 6G/z
共21兲
Here P = ␥V is the momentum of the fluid. The evolution of the longitudinal component of the momentum is governed by 共 ␥ − P z兲
冉 冊
␥ d N d 共GPz兲 − d N d ␥¯z
冉
= P⬜ · E⬜ + G
冊
dE⬜ 2 共␥ − Pz兲 + E⬜ 共 ␥ − P z兲 d
¯兲 − 共1 + 6G/z ¯兲Pz共␥ − Pz兲兴, ⫻关共1 + 2G/z
共22兲
and of the entropy 共13兲, by
冋
冉 冊册
d ␥ d N dS = ¯z G− d d N d ␥¯z
2 = − 4GE⬜ 共␥ − Pz兲. 共23兲
Note that in the system of equations 共21兲–共23兲 the charge and the mass of the particle species cannot be seen explicitly since they are hidden in normalizations employed in the first
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section. It is now convenient to attribute this system to the positron fluid so that the normalizations are with respect to q = 兩e兩 and M = me. To write the corresponding set of electron equations, we need to make the trivial replacement in 共21兲–共23兲: E⬜ ⇒ −E⬜ and J⬜ ⇒ −J⬜. One can easily see that e = the longitudinal momentum of electrons Pze = Pz while P⬜ 2 2 e −P⬜ and consequently ␥e = ␥ = 共1 + Pz + P⬜兲1/2 and Vz = Vz. Taking the scalar product of Eq. 共21兲 with P⬜, subtracting the result from Eq. 共22兲 and applying Eq. 共23兲, we find
冉
冊
1 d 2 −2 ¯兲. G 共1 + 6G/z = E⬜ d G共␥ − Pz兲
共24兲
Equation 共24兲 can be readily integrated. For the initial conditions 共at = 0, the plasma is at rest兲, P⬜ = 0 = Pz, while T = T0 关G = G0共T0兲兴, the solution is G共␥ − Pz兲 − K共兲 = 0,
2 共GP⬜兲2 = 共1 + 2K2兲E⬜ 共兲.
Interestingly one can use the preceding equation to eliminate the transverse momentum in Eqs. 共27兲 and 共28兲 to obtain Pz =
␥=
K=
共25兲 =
1 2 −2 兰0 E⬜ G 共1
¯兲d⬘ + + 6G/z
. G−1 0
共26兲
After straightforward algebra, Eq. 共25兲 yields the important relations Pz =
2 兲 − K2 G2共1 + P⬜ , 2GK
2 兲 + K2 G2共1 + P⬜ . ␥= 2GK
Now we can simplify Eqs. 共21兲 and 共23兲 further. Using these relations and Eq. 共25兲, the equation for the transverse momentum can be written as
冉
冊
d GP⬜ E⬜ − E⬜ = . d K K
d 2 2 ¯兲, ¯兲K兴 = − 4E⬜ 关F共z K F共z d
共30兲
¯兲exp关z ¯G共z ¯兲兴 K2共z . ¯G共z z ¯兲
2 共兲 G2 + K2 + 共1 + 2K2兲E⬜ 2GK
共34兲
1 2 兰0 E⬜ 共⬘兲d⬘ + 1
冉
共E20 + 1兲−1 共E20d
if 0 ⬍ ⬍ d
−1
+ 1兲 = const
if ⬎ d
冊
.
共35兲
1 + E20 − K2 1 1 = 共1 + E20兲共1 + E20兲 − 2 2K 2共1 + E20兲 共36兲
while the residual momentum acquired by the plasma after the pulse is gone is calculated to be 1 1 . Pz共res兲 = 共1 + E20d兲 − 2 2共1 + E20d兲
共37兲
Similarly, 1 1 ␥ = 共1 + E20兲共1 + E20兲 + 2 2共1 + E20兲
if 0 ⬍ ⬍ d 共38兲
and 1 1 ␥共res兲 = 共1 + E20d兲 + 2 2共1 + E20d兲
if ⬎ d . 共39兲
The expression for the plasma density variation 共␦N = N − 1兲 is
␦N =
¯−1兲 via where the function F depends on temperature T共=z ¯兲 = F共z
Pz =
共29兲
To find the equation for the evolution of the plasma temperature, we will invoke the entropy equation 共23兲. Integrating the continuity equation, and assuming that N共 = 0兲 = 1 we obtain N = ␥ / 共␥ − Pz兲 = ␥G / K共兲. Using this expression, and Eqs. 共14兲 and 共25兲, we find that the temperature evolves as
共33兲
Since Ⰶ 1 and K 艋 1, we can neglect terms 2K2 in 共33兲 and 共34兲. The plasma momentum in the body of the pulse 共i.e., 0 ⬍ ⬍ d兲 is found to be
共27兲
共28兲
2 共兲 G2 − K2 + 共1 + 2K2兲E⬜ , 2GK
giving Pz and ␥ entirely in terms of the field intensity of the EM wave. In the cold plasma limit T → 0 共G = G0 = 1兲, Eq. 共26兲 yields
where K共兲 =
共32兲
1 1 ␥ − 1 = 共1 + E20兲2共1 + E20兲 − 2 2 K共兲
if 0 ⬍ ⬍ d 共40兲
共31兲 and
In what follows we integrate the basic system of ordinary differential equations 共26兲–共31兲. For simplicity the incident pulse will be taken to be circularly polarized; the electric field of the pulse is given by E⬜ = 共xˆ sin + yˆ cos 兲E⬜共兲, where E⬜共兲 is the slowly varying envelope 关共E⬜兲 / E⬜ Ⰶ 1兴 and xˆ and yˆ are the unit vectors. Since, for this choice 2 2 = E⬜ 共兲 does not contain high harmonics, K of the pulse, E⬜ and T are also slowly varying. Equation 共29兲, then, yields
␦N共res兲 = 21 共1 + E20d兲2 − 21
if ⬎ d .
共41兲
In the absence of RRF we recover the well-known results 关19兴. For instance, if = 0, we have Pz = E20 / 2, and Pz共res兲 = 0, which implies that the plasma particles are temporally accelerated only inside the region occupied by the pulse; the result is a consequence of the fact that the magnetic part of the Lorentz force 共that is responsible for the longitudinal
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acceleration of the plasma兲 is a potential force. The action of RRF manifest itself not only by the enhancement of acceleration but also as the residual momentum that the whole plasma acquires. The pulse leaves in its wake an accelerated plasma which follows the pulse; its momentum, energy, and density are given by Eqs. 共37兲, 共39兲, and 共41兲. The strength of the acceleration caused by RRF is determined by the parameter E20 and the duration of the pulse. The generated plasma flow becomes relativistic 关Pz共res兲 ⬎ 1兴 when the pulse duration d ⬎ acc, where the “acceleration” time is given by
acc =
1
It turns out that the magnitude of the zero temperature results are quite insensitive to small but finite temperature 共T Ⰶ 1兲. However, a new qualitative feature emerges for finite temperature plasmas; the acceleration, now, is accompanied by a cooling of the plasma. Indeed, in the nonrelativistic limit ¯z Ⰷ 1 共T Ⰶ 1兲 we can use the asymptotic form of the ¯兲 ⯝ 共 / 2z ¯兲e−z¯ and G ⯝ 1 modified Bessel function K2共z ¯兲 in Eq. 共31兲, and approximate F ⯝ 共 / 2兲1/2T3/2. On + 5 / 共2z substituting F into Eq. 共30兲, and applying the relation 2 , we get for the plasma temperature d共K−1兲 / d = E⬜ if 0 ⬍ ⬍ d ,
T0 Td = 2 共E0d + 1兲2
if ⬎ d
4T0 2 6T0兰0E⬜共⬘兲d⬘ +
1
,
d 2 2 2 2 关T K共兲兴 = − 4E⬜ KT; d
+ 1兲5/6
if ⬎ d .
共48兲
共4T兲2 + K2 + E20共兲 . 2共4T兲K
共49兲
Note that at the initial stage of interaction 共small 兲, although K ⬃ 4T0 Ⰷ 1 is large, the inequality 共2K2 Ⰶ 1兲 still holds and cannot be violated for any known physical conditions because ⯝ 10−8 – 10−9 is so exceedingly small for laser pulses. From Eqs. 共45兲–共49兲 we can extract detailed expressions for the relativistic factor, 1 1 ␥ = 共6E20T0 + 1兲1/6 + 2 2 2共6E0T0 + 1兲1/6 + 共6E20T0 + 1兲11/6
共44兲
共45兲
共46兲
the latter can be readily integrated 关using the relation 2 / 2兴 for the temperature T, d共K−1兲 / d = 3E⬜ T0 2 共6E0T0
+ 1兲5/6
E20 32T20
共50兲
if 0 ⬍ ⬍ d, and
while the equation for temperature 共30兲 reduces to
T=
␥=
共43兲
implying that the cooling begins immediately with the pulse interaction and the temperature keeps its minimum value 共Td兲 after the pulse is gone. It should be emphasized that in the relatively simple scenario of the “given” pulse, cooling is inhomogeneous in the sense that it is limited to the region that the pulse occupies. In a self-consistent treatment 共i.e., when the pulse undergoes structural changes and/or modification, or depletion兲, however, plasma cooling will be more profoundly inhomogeneous. When the plasma has ultrarelativistic temperatures 共T Ⰷ 1兲, the temperature function F ⯝ 27T2, and Eq. 共26兲 simplifies to K=
T0 2 共6E0T0d
It follows from 共47兲 and 共48兲 that even in this ultrahot limit the plasma temperature begins to fall via RRF as soon as the plasma begins to interact with the laser pulse; the temperature remains at the lowest value achieved after the pulse is long gone. In this ultrarelativistic limit, Eq. 共34兲 for ␥ factor can be approximated by 共2K2 Ⰶ 1兲
共42兲
. E20
T0 T= 2 共E0 + 1兲2
Td =
if 0 ⬍ ⬍ d ,
共47兲
1 1 ␥共res兲 = 共6E20T0d + 1兲1/6 + 共51兲 2 2 2共6E0T0d + 1兲1/6 if ⬎ d. Evidently the high temperature not only enhances the bulk plasma acceleration, it also makes it faster; the time of acceleration acc = 1 / 共6E20T0兲 is reduced by the factor T0共Ⰷ1兲. We would like to reiterate that the acceleration of the plasma is augmented by its intensive cooling 共47兲 and 共48兲. The physical origin of the bulk motion can be traced to the fact that the hot particles in the fluid cell radiate most of their energy in the direction of the radiation source, and part of the energy of relativistic thermal motion is converted into the bulk motion. One must beware that cooling reduces the temperature, and at some stage, the relativistic temperature approximation will be violated especially if the duration of the pulse far exceeds the acceleration time d Ⰷ 6E20T0. For long pulses, therefore, we must numerically solve the problem to obtain quantitative results. Qualitative features of the system are not likely to be much affected since we have already shown that the acceleration goes on 共but at a slower ate兲 even for temperatures that are nonrelativistic. Let us now go back to the cold plasma and estimate the acceleration and/or cooling efficiency that depends on the following dimensionless parameters , E0, T0, and . The strength of the radiation field is related to the intensity by I 2 典 were 具¯典 means averaging over the radiation = 共c / 4兲具E⬜ high frequency periods. In terms of intensity, the dimensionless strength of the field E0共=eE0 / mc in units兲 is given by
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E20 = 3.65 ⫻ 10−19I关W/cm2兴 ⫻ 2关m兴
共52兲
implying that the acceleration time 共42兲, the time required for plasma flow to acquire relativistic velocities, can be estimated as
acc关fs兴 =
1.2 ⫻ 1026 1.2 ⫻ 105 = , I关W/cm2兴 I21
共53兲
where I21 = I 关W / cm2兴 / 1021. Note that acc does not depend on the frequency of the EM field. At currently achievable intensities I = 1021 W / cm2, E0 ⬇ 20 for = 1 m leading to a acc = 100 ps. Since acc is much longer than the typical durations of such laser pulses, radiative acceleration will be minimal and not quite observable. However, for the future superintensity pulses, acc could be considerably smaller than d. Indeed, for the laser field intensity I = 1023 W / cm2 共E0 ⬇ 190 for = 1 m兲 we have acc ⬃ 1 ps, while for I = 1024 W / cm2 共E0 ⬇ 600兲, acc will be several tens of femtoseconds. Then, plasma will be strongly accelerated acquiring relativistic bulk motion whose magnitude may be estimated to be ␥res = d / acc Ⰷ 1. Here we would like to emphasize that the validity of the classical treatment of the problem must be checked carefully for super-intense fields. Classical treatment of the problem is valid provided the energy of the photon involved in the process of interaction is smaller than the electron energy, i.e., L = បem / ␥emec2 ⬍ 1 共for ultrarelativistic case ␥e = E0兲. Since the characteristic frequency of the photons emitted by accelerated electrons in the field of a circularly polarized laser field is em ⯝ E30, the factor L ⯝ 0.1 is for I = 1023 W / cm2 approaching L ⯝ 1 when the intensities rise to I = 1024 W / cm2. Thus, for I 艌 1024 W / cm2, classical treatment becomes questionable. Recently Bulanov et al. 共see Ref. 关4兴兲 demonstrated phenomenologically that inclusion of quantum effects in the RRF do not alter significantly the result obtained via the classical approximation up to laser intensities I = 1.38⫻ 1025 W / cm2 共E0 = 2500 for = 1 m兲. In general, rigorous treatment of quantum correction in the problem of RRF is challenging and is beyond the scope of the current paper. Our entire calculation has been based on the “given field” approximation. This approximation, surely, breaks down when a large part of the radiation energy is converted into the kinetic energy of the plasma, or is scattered away. We will now derive the conditions under which the “given field” approach may be valid. Assuming that the radiation losses due to scattering are the same order as the energy needed for plasma acceleration, we can make a reasonable estimate of depletion length for the pulse. Comparing the initial energy of the pulse with the plasma residual energy, 2 E⬜
4
cTL ⬇ mec2N␥resLD
共54兲
we find for the pulse depletion length LD, LD ⬇ D共cd兲, where
共55兲
D=
acc Nc 2 E . d N 0
共56兲
Here Nc is the critical density of the plasma. For strong acceleration we should have 共acc / d兲 Ⰶ 1 but for a transparent plasma, the critical density Nc Ⰷ N. With E20 Ⰷ 1, D can be considerably larger unity implying that the pulse depletion length can be much longer than the pulse width. Thus, if the plasma slab is shorter than the LD, the laser pump depletion may be ignored and the ignored, i.e., the “given field” approximation will give approximately correct results. By dwelling on a relatively simple hot optically thin, underdense electron-positron plasma, exposed to a relativistic strong laser pulse, we have analytically demonstrated that the radiation reaction force 共that becomes stronger with high tempearture兲 can cause an increase in the bulk kinetic energy of the fluid 共that may be boosted up to relativistic values兲 accompanied by a corresponding cooling or intense cooling as the case may be. When the super-intensity laser pulses 共I 艌 1024 W / cm2兲 become available in the future, it will be possible to create conditions in the laboratory such that the workings of the mechanism of the simultaneous gain of kinetic energy and loss of thermal energy 共conjectured to play a fundamental role in advancing understanding of a variety of astrophysical phenomena兲 could be explored; the predictions of this and similar works can, then, be experimentally tested. There are many ways by which this simple analytical work can be augmented. More detailed studies should be undertaken that include: 共1兲 fluids with different masses, 共2兲 denser plasmas, 共3兲 giving up the “given field” approximation 共a self-consistent calculation where the field intensity is also evolved兲, and 共4兲 multidimensional effects. We end this paper by giving the gist of our preliminary studies on an electron-ion plasma irradiated by super-strong laser pulses. We will continue assuming a radiation intensity of 共I 艌 1024 W / cm2兲. We find that the electrons are cooled and accelerated rapidly to relativistic speeds while ions lag behind owing to their larger inertia. The ensuing strong charge separation electrostatic fields cause a considerable slow down of the process of electron acceleration initially. However, at the subsequent stage, the strong charge separation field effectively accelerates the ions. When the ions catch up with the electrons the charge separation field becomes small, and the electrons are, once again, rapidly reaccelerated by the radiation field. The reported sequence of events takes place at acc. ACKNOWLEDGMENTS
We thank Professor K. Mima for his interest in this problem and useful discussions. V.I.B. wishes to acknowledge the hospitality of the Graduate School of Frontier Sciences of the University of Tokyo, where most of the work was carried out. S.M.M. acknowledges support form USDOE Contract No. DE-FG02-04ER-54742. The work of V.I.B. was partially supported by Georgian NSF grant projects GNSF 69/07 共Grant No. GNSF/ST06/4-057兲 and GNSF 195/07 共Grant No. GNSF/ST07/4-191兲.
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