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Radiation reaction effects on electron nonlinear dynamics and ion acceleration in laser–solid interaction

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Radiation reaction effects on electron nonlinear dynamics and ion acceleration in laser–solid interaction
    a  r   X   i  v  :   1   0   1   1 .   5   6   3   5  v   1   [  p   h  y  s   i  c  s .  p   l  a  s  m  -  p   h   ]   2   5   N  o  v   2   0   1   0 Radiation Reaction E ff  ects on Electron Nonlinear Dynamicsand Ion Acceleration in Laser-solid Interaction M. Tamburini a, ∗ , F. Pegoraro a , A. Di Piazza b , C. H. Keitel b , T. V. Liseykina c , A. Macchi a,d a  Dipartimento di Fisica “E. Fermi”, Universit`a di Pisa, Largo Bruno Pontecorvo 3, I-56127 Pisa, Italy b  Max-Planck-Institut f¨ ur Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany c  Institute of Computer Technologies, SD-RAS, Novosibirsk, Russia and Institute of Physics, University of Rostock, Germany d   Istituto Nazionale di Ottica, CNR, research unit “A. Gozzini”, Pisa, Italy Abstract Radiation Reaction (RR) e ff  ects in the interaction of an ultra-intense laser pulse with a thin plasma foil are investigated analyticallyand by two-dimensional (2D3P) Particle-In-Cell (PIC) simulations. It is found that the radiation reaction force leads to a significantelectron cooling and to an increased spatial bunching of both electrons and ions. A fully relativistic kinetic equation including RRe ff  ects is discussed and it is shown that RR leads to a contraction of the available phase space volume. The results of our PICsimulations are in qualitative agreement with the predictions of the kinetic theory. Keywords:  Radiation Reaction, Ion Acceleration, Laser-Plasma Interaction, Radiation Pressure 1. Introduction Current laser systems may deliver intensities up to10 22 Wcm − 2 [1] and intensities up to 10 26 Wcm − 2 are expectedat the Extreme Light Infrastructure (ELI). In such ultrahigh-intensity regime and for typical laser wavelength  λ  ∼  0 . 8  µ mthe motion of electrons in the laser field is ultra-relativistic andRadiation Reaction (RR) e ff  ects may become important. TheRR force describes the back-action of the radiation emitted byan accelerated electron on the electron itself and accounts forthe loss of the electron energy and momentum due to the emis-sion of such radiation. Apart from the need of including RRe ff  ects in the dynamics of laser-plasma interactions in the ultra-relativistic regime, the latter also o ff  ers for the first time theopportunity to detect RR e ff  ects experimentally [2, 3]. In this paper we present an approach to a kinetic descrip-tion of laser-plasma interactions where RR e ff  ects are includedvia the Landau-Lifshitz (LL) force [4]. Some properties of thekinetic equation with RR are discussed and in particular it isproved that the RR force leads to a  contraction  of the phasespace volume. Then, PIC simulations are used to study RR ef-fects on the acceleration of a thin plasma foil in the regime of Radiation Pressure dominance [5]. Numerical simulations [5] suggested that Radiation Pressure Acceleration (RPA) becomesthe dominant mechanism of ion acceleration at intensities ex-ceeding 10 23 Wcm − 2 . Such RPA regime is attractive becauseof the foreseen high e ffi ciency, the quasi-monoenergetic fea-tures expected in the ion energy spectrum and the possibilityto achieve a potentially “unlimited” acceleration [6]. PreviousParticle-In-Cell (PIC) simulations [7] showed signatures of RR ∗ Corresponding author  Email address:  (M. Tamburini) e ff  ects at intensities exceeding 5 × 10 22 Wcm − 2 and increasingnonlinearly with the laser intensity. More recent simulationsstudies of RPA both for thick targets [8, 9] and ultrathin tar- gets [10] suggested that the inclusion of the RR force cools theelectrons and may improve the quality of the ion spectrum.Our approach to the inclusion of RR e ff  ects in a PIC codehas been discussed in detail in Ref.[11] where one-dimensional(1D)simulationsofRPA havebeenalso reported. Inthepresentpaper we report both additional 1D simulations and first two-dimensional (2D) simulations using parameters similar to thoseof Ref.[12] where, in particular, the impact of a Rayleigh-Taylor-like instability on a thin foil acceleration was studied.Inclassical electrodynamics,thee ff  ectofRR canbeincludedby means of the LL force [4] F  R  =  −  4 π 3 r  e λ   ·  γ    ∂∂ t  + v ·∇  E + v ×   ∂∂ t  + v ·∇  B  −  E + v × B  × B +  v · E  E  +  γ  2  E + v × B  2 −  v · E  2  v  ,  (1)where  v  is the electron velocity,  γ   is the relativistic factor, r  e  ≡  e 2 / mc 2 ≈  2 . 8  ×  10 − 9  µ m is the classical electron radius, λ  =  2 π c /ω  is the laser wavelength and we use dimensionlessquantities as in the PIC code: time, space and momentum arenormalized in units of   ω − 1 ,  c ω − 1 and  mc , respectively. Conse-quently, EM fields are normalized in units of   m ω c / | e |  and den-sities in units of the critical density  n c  =  m ω 2 / 4 π e 2 .The LL approach holds in the classical framework and quan-tum e ff  ects are neglected. As pointed out in [11], the first termof the LL force Eq.(1) i.e. the one containing the  derivatives of the electric and magnetic fields, should be neglected becauseits e ff  ect is smaller than quantum e ff  ects such as the spin force. Preprint submitted to Nuclear Instruments and Methods Section A November 29, 2010  However, in Sec.(2) we show the e ff  ect of each term of the LLforce Eq.(1) on the rate of change of the phase space volume. 2. The kinetic equation with Radiation Reaction In this section, a fully relativistic kinetic equation that in-cludes the RR e ff  ects is discussed. We show a few basic prop-erties ofthe kineticequationpointingoutthepeculiaritiesoftheRR force whose main new feature is that it  does not   conservethe phase-space volume.Generalized kinetic equations for non-conservative forcesand in particular for the RR force have been known since latesixties forthe Lorentz-Abraham-Dirac(LAD)equation[13, 14] and late seventies for the LL equation [15]. Recently, the gen-eralized kinetic equation with the LL force included has beenused to study the RR e ff  ects on thermal electrons in a magnet-ically confined plasma [16] and to develop a set of closed fluidequations with RR [17–19]. In this paper, we give the kinetic equation in a non-manifestly covariant form, see [15, 16] for the kinetic equation in a manifestly Lorentz-covariant form.The relativistic distribution function  f   =  f  ( q , p , t  ) evolvesaccording to the collisionless transport equation ∂  f  ∂ t  + ∇ q  · (  f   v ) + ∇ p  · (  f   F )  =  0 ,  (2)where  q  are the spatial coordinates,  v  =  p /γ   is the three-dimensional velocity,  γ   =   1 + p 2 is the relativistic factor and F  =  F  L  +  F  R  is the mean force due to external and collectivefields ( F  L  ≡ − ( E + v × B ) is the Lorentz force and  F  R  is givenin Eq.(1)). Physically, Eq.(2) implies the conservation of the number of particles.The new key feature compared to the usual Vlasov equa-tion is that for the RR force  F  R  we have  ∇ p  ·  F  R    0. UsingLagrangian coordinates  q ( t  ) ,  p ( t  ), Eq.(2) can be recast in theequivalent form d  ln  f dt  =  −∇ p  · F .  (3)According to Eq.(3), ∇ p · F  providesthe percentageof variationofthedistributionfunction  f   withinthecharacteristictimescale ω − 1 . IntegratingEq.(3) alongits characteristics,we findthatthedistribution function  f   remains positive as required.Introducingtheentropydensityin thephasespace  s ( q , p , t  )  = −  f  ( q , p , t  )ln  f  ( q , p , t  ), from Eq.(2) we get the equation for theevolution of the entropy density ∂ s ∂ t  + ∇ q  · ( s v ) + ∇ p  · ( s F )  =  f   ∇ p  · F .  (4)Integrating Eq.(4) in the phase space, we get the rate of varia-tion of the total entropy  S  ( t  ) dS  ( t  ) dt  =    d  3 qd  3  p f   ∇ p  · F .  (5)The Lorentz force  F  L  ≡ − ( E + v × B ) gives ∇ p  · F  L  =  0 identi-cally thus ∇ p  · F  =  ∇ p  · F  R . Moreover, the distribution function  f  ( q , p , t  ) is always non-negative  f   ≥  0 thus the sign of   dS  / dt   isgiven by ∇ p  · F  R  solely.From the LL force Eq.(1) we get [20] ∇ p  · F  R  =  −  4 π 3 r  e λ  ∇ q  · E − v ·  ∇ q  × B −  ∂ E ∂ t   + 2  E 2 + B 2 γ   + 4 γ   v × E  2 +  v × B  2 − 2 v ·  E × B  . (6)In a plasma, the kinetic equation is coupled with the Maxwellequations for the self-consistent fields ∇ q  · E =  ρ ρ c =  1 n c   j = e , i  Z   j    d  3  p f   j ( q , p , t  ) (7) ∇ q  × B −  ∂ E ∂ t  =  j  j c =  1 n c c   j = e , i  Z   j    d  3  p  v  f   j ( q , p , t  ) ,  (8)where  ρ c  ≡ | e | n c ,  j c  ≡ | e | n c c ,    d  3 qd  3  p f   j ( q , p , t  )  =  N   j  is thetotal number of particles for each species (  j  =  e  electrons,  j  =  i ions) and  Z   j  is the charge of the particle species in units of   | e | (for electrons  Z  e  =  − 1). For a plasma, Eq.(6) can be recast as ∇ p  · F  R = −  4 π 3 r  e λ   ρ ρ c − v ·  j  j c  + 2  E 2 + B 2 γ   + 4 γ   v × E  2 +  v × B  2 − 2 v ·  E × B  .  (9)The terms of Eq.(9) proportionalto the charge density  ρ  and tothe current density  j  come from the first term of the LL forceEq.(1) i.e. the term containing the derivatives of the fields. Ingeneral, these terms can give either a positive or negative con-tribution to  ∇ p  ·  F  R . The second term of Eq.(9) i.e. the termproportional to ( E 2 +  B 2 ) has always a negative sign, its ef-fect decreaseswith increasingelectronenergyandit is typicallynegligible. The third term of Eq.(9) comes from the stronglyanisotropic “friction” term of the LL force i.e. the term propor-tional to  γ  2 in Eq.(1) (see [11] for a detailed discussion of this term) and dominates in the ultra-relativistic limit  γ   ≫  1.It is possible to prove [20] the following statement: for any  v such that | v | ∈  [0 , 1[ then  ( v × E ) 2 + ( v × B ) 2 − 2 v · ( E × B )  +  E 2 + B 2 2 γ  2   ≥  0 , (10)therefore according with Eqs.(5, 9), the terms of the LL forceEq.(1) that  do not   depend on the derivativesof the fields alwayslead to a  contraction  of the available phase space volume. In afew special cases, the e ff  ect of the terms of the LL force Eq.(1)that depend on the derivatives of the fields (i.e. the terms pro-portional to  ρ  and  j  in Eq.(9)) might lead to an expansion of thephase space volume. Anyway, their e ff  ect should be negligiblecompared to quantum e ff  ects as discussed in [11].We show explicitly the contraction of the phase space in thespecial case of a small bunch of electrons interacting with aplane wave where collective fields are assumed to be negligi-ble compared with the plane wave fields. Assuming an initialdistribution  f   =  g ( q ) δ 3 ( p − p 0 ), from Eqs.(5, 9) we have dS  ( t  ) dt  = −  4 π 3 r  e λ     d  3 q g ( q )  2  E 2 + B 2 γ  ( p 0 )  + 4 γ  ( p 0 ) ··  ( v 0  × E ) 2 + ( v 0  × B ) 2 − 2 v 0 · ( E × B )  ,  (11)2  where  v 0  =  p 0 /γ  ( p 0 ). If the electron bunch counter-propagateswith the plane wave ([ v 0  ·  ( E × B )]  <  0) or propagates in thetransverse direction ([ v 0 · ( E × B )]  =  0), from Eq.(11) it is clearthat RR leads to a contraction of the phase space. In particular,in thecase ofcounter-propagation(using | E |  =  | B | ,  E · B  =  0)wehave ∇ p · F  R  =  − (4 π r  e / 3 λ )4 E 2   2 γ  ( p 0 ) | v 0 | (1 + | v 0 | ) + 1 /γ  ( p 0 )  .On the other hand, if the bunch propagates in the same direc-tion of the plane wave ( v 0  parallel to  E  ×  B ), then the con-tribution of the friction term (proportional to  γ   in Eq.(9)) be-comes comparable with the contribution of the second term(proportional to ( E 2 +  B 2 ) in Eq.(9)) and we have  ∇ p  ·  F  R  = − (4 π r  e / 3 λ )  4 E 2 / (1 + | v 0 | ) 2 γ  3 ( p 0 )   which still leads to a con-traction of the phase space but with a rate  γ  4 smaller than thecase of counter-propagation. This reinforcesthe evidenceofthestrongly anisotropic featuresoftheLLforceEq.(1)(see[11]for further details).The physical interpretation of the above properties is that theRR force acts as a cooling mechanism for the system: part of the energy and momentum are radiated away and the spread inboth momentum and coordinate space may be reduced. Thisgeneral prediction is confirmed by our PIC simulations (seeSec.3) where we foundthat RR e ff  ects lead to bothan increasedbunching in space and to a noticeable cooling of hot electrons.Finally, it is worthwhile mentioning that Eq.(2) is more gen-eral than the Vlasov equation but the PIC approach is still validi.e. the PIC approach provides a solution for Eq.(2) and it notlimited to the Vlasov equation [20]. 3. PIC simulations Suitable approximations to the LL force and our approachto its inclusion in a PIC code are described in Ref.[11]. Thenumerical approach is based on the widely used Boris particlepusher and it can be implemented in codes of any dimensional-ity. Inclusion of RR e ff  ects via this method in PIC simulationsleads to only a  ∼  10% increase in CPU time, which may beessential to perform large-scale simulations with limited com-puting power. 3.1. 1D simulations We firstreportone-dimensional(1D3P)PIC simulationswithlaser and plasma parameters similar to Ref.[5]. Previous 1Dsimulations in this regimehave been reportedin Ref.[11] wherea detailed comparison with other work is also made. In thepresent paper we review the basic observations in the 1D caseand we include results at intensities higher than those investi-gated in Ref.[11].The target is a plasma foil of protons with uniform initialdensity  n 0  =  100 n c  and thickness  ℓ   =  1 λ  where  λ  =  0 . 8  µ m isthe laser wavelength and  T   =  λ/ c  ≈  2 . 67fs is the laser period.In these simulations, the laser pulse front reaches the edge of the plasma foil at  t   =  0, the profile of the laser field amplitudehas a “trapezoidal” shape in time with one-cycle, sin 2 -functionrise and fall and a five cycles constant plateau. We consideredthree intensities  I   =  2 . 33 × 10 23 Wcm − 2 ,  I   =  5 . 5 × 10 23 Wcm − 2 and  I   =  10 24 Wcm − 2 for both Circular (CP) and Linear (LP)polarization of the laser pulse.In the CP case, we found that RR e ff  ects on the ion spectrumare negligible even at intensities of   I   =  10 24 Wcm − 2 as shownin Fig.1. For CP, electrons pile up and the numerical densitygrows exceeding thousand of times the critical density  n c . Thelaser pulse does not penetrate deeply into the target (i.e. thee ff  ectiveskindepthis a verysmall fractionofthe foil thickness)and electrons move in a field much weaker than the vacuumfield.In Ref.[5] it was expected that RR e ff  ects in the radiation-pressure dominated acceleration of the thin foil would havebeen weak because in this regime the whole foil becomesquickly relativistic, hence in the foil frame the laser wavelength λ ′ increasesandthe typicalstrengthof theRR parameter ∼  r  e /λ [seeEq.(1)]decreases. ThepresentcaseofaccelerationwithCPpulses appears to confirm this picture. The weakness of RR ef-fects may also be explained on the basis of the LL equation foran electron moving into a plane wave [21]. As electrons movein the forward direction coherently with the foil (while rotatingin the transverse plane in the CP field) and the amplitude of thereflected wave is weak when the foil is strongly relativistic, thesituation is similar to an electron co-propagatingwith the planewave at a velocityclose to  c , forwhichthe LLforcealmost van-ishes [11]. The relativistic motion of the foil also prevents theonset of Self-Induced Transparency by increasing the opticalthickness parameter  ζ   =  π n 0 ℓ/ n c λ  in the foil frame (see [22]and references therein) . For smaller target thickness, break-through of the laser pulse occurs and RR e ff  ects are greatly en-hanced also for CP [11].It may be worth noticing that, at the highest intensity consid-ered  I   =  10 24 Wcm − 2 , in principle one would expect the classi-cal approach to RR to break down due to the onset of quantumelectrodynamics (QED) e ff  ects, as discussed in Ref.[11]. How-ever, it can be shown by a direct analysis of the simulation datathat the thresholdconditionforQED e ff  ect is not violatedin theCP case. Figure 1: Ion energy spectra at  t   =  66 T   with (top) and without (bottom) RRfor Circular Polarization. The laser intensity  I   is 2 . 33 × 10 23 Wcm − 2 (yellow),5 . 5 × 10 23 Wcm − 2 (blue), 10 24 Wcm − 2 (red) and the target thickness is  ℓ   =  1 λ . 3  Figure 2: Ion energy spectra at  t   =  14 T   with (top) and without (bottom) RRfor Linear Polarization. The laser intensity  I   is 2 . 33 ×  10 23 Wcm − 2 (yellow),5 . 5 × 10 23 Wcm − 2 (blue), 10 24 Wcm − 2 (red) and the target thickness is  ℓ   =  1 λ . For linear polarization(LP), di ff  erentlyfrom the CP case, wefound that RR e ff  ects are important leading to a reduction of the maximum achievable ion energy and to some narrowing of the width of the ion spectrum as shown in Fig.2. This di ff  er-ent dynamics for LP is correlated with the strong longitudinaloscillatory motion driven by the oscillating component of the  j  ×  B  force which is suppressed in the CP case. This allows adeeper penetration of the laser pulse into the foil with a signifi-cantfractionofelectronsonthefrontsurfacemovinginastrongelectromagnetic field of the same order of vacuum fields [11].The relative reduction in the ion energy when RR is included isclose to the percentage of the laser pulse energy which is lostas high-energy radiation escaping from the plasma.The results for LP (Fig.2) are shown for the same intensityvalues of the CP case (Fig.1) for a direct comparison. How-ever, at least for the highest intensity case, the LP results mustbe taken with some caution as the condition for the validity of a classical approach tends to be significantly violated. In suchregime, an analysis based on quantumRR e ff  ects might be nec-essary [23, 24]. 3.2. 2D simulations We report preliminary two-dimensional (2D3P) PIC simula-tions with laser and plasma parameters similar to Ref.[12]. Tothebest ofourknowledge,thisis thefirst paperreportingresultsof two-dimensional PIC simulations with RR e ff  ects included.The target is a plasma slab of fully ionized deuterium(  Z  /  A  = 1 / 2) of width 40 λ , density  n 0  =  169 n c  and thickness  ℓ   =  0 . 5 λ .The size of the computational box is 95 λ ×  40 λ  with a spatialresolution  ∆  x  = ∆  y  =  λ/ 80 and 625 quasi-particles per cellcorresponding to a total of 8  ×  10 7 quasi-particles. The laserpulse is s-polarized with the electric field along the  z -axis. Itsnormalizedamplitude is  a 0  =  320 correspondingto an intensity  I   =  1 . 4 × 10 23 Wcm − 2 with a wavelength λ  =  1 . 0  µ mandperiod T   =  λ/ c  ≈  3 . 3fs. Thepulsehas a Gaussian transverseprofileof width 20 λ  FWHM and a sin 2 longitudinal profile of length 40 λ Figure 3: Plots of the 2D PIC simulations at  t   =  70 T  . The laser pulse is s-polarized with an intensity  I   =  1 . 4 ×  10 23 Wcm − 2 and the target thickness is ℓ   =  0 . 5 λ . From top to bottom, ion  n i  and electron  n e  density distributions with(left column) and without (right column) RR, longitudinal  E   x  (first row) andtransverse  E   z  (second row) electric field, ion and electron spectrum with (red)and without (blue) RR. FWHM. In these simulations, the frontof the lase pulse reachesthe foil at  t   =  0.Comparing the results of our simulations with and withoutRR (see Fig.3, we report the results at  t   =  70 T  ) it is apparent4  that RR leads to both an increased electron and ion bunchingand to a strong cooling of electrons. These results are qualita-tively consistent with our expectations from the kinetic theorythat we have discussed in Sec.2 and in particular with the pre-diction of a contraction of the electrons available phase spacevolume.A qualitative understanding of these results can be achievedrecalling that the RR force Eq.(1) is mainly a stronglyanisotropic and non-linear friction-like force that reaches itsmaximum for electrons that counter-propagate with the laserpulse [11]. The backward motion of electrons is thus impededby RR, more electrons and consequently ions are pushed for-ward leading to an enhanced clumping that improves the e ffi -ciency of the RPA mechanism. In fact, the ion spectrum withRR shows a region between about three hundred and six hun-dredMeVwithasignificantincreaseinthenumberofionscom-pared to the case without RR (Fig.3). This picture is confirmedby both the enhancement of the longitudinal electric field  E   x and by the formation of denser bunches in the ion density com-pared to the case without RR (see Fig.3). However, for linearpolarization, hot electrons are always generated by the oscillat-ing component of the  j  ×  B  force. The generation of hot elec-trons providesa competingaccelerationmechanismto RPA andultimatelyleadstothegenerationofthefractionofionswiththehighest energy. The noticeable suppression of the  j × B  heatingmechanismdue to the RR forcethereforeleads to a lower maxi-mum cut-o ff  energyboth in the electron and in the ion spectrum(see Fig.3).These preliminary results for two-dimensional simulationswith RR e ff  ects includedsuggest that, in the LP case, the trendsfound in one-dimensional simulations hold qualitatively evenfor higher dimensions. More detailed studies and quantitativecomparisons between one-dimensional and two-dimensionalPIC simulations are left for forthcoming publications. 4. Conclusions We summarize our results as follows. Radiation Reactione ff  ects on the electron dynamics in the interaction of an ultra-intense laser pulse with a thin plasma foil were studied analyt-ically and by one-dimensional and two-dimensional PIC simu-lations. The details of the numerical implementation of the RRforce in our PIC code were described in Ref.[11].In one-dimensional simulations, we checked RR e ff  ects forthree di ff  erent intensities:  I   =  2 . 33  ×  10 23 Wcm − 2 ,  I   =  5 . 5  × 10 23 Wcm − 2 and  I   =  10 24 Wcm − 2 comparing the results forCircular and Linear Polarization of the laser pulse. For CP,we found that RR e ff  ects are not relevant even at intensityof   I   =  10 24 Wcm − 2 whenever the laser pulse does not break through the foil. In contrast, for LP we found that RR e ff  ectsare important reducing the ion energy significantly.In two-dimensional simulations, we found that RR reducesthe  j × B  heating mechanism leading to a lower maximum cut-o ff   energy both in the electron and in the ion spectrum. More-over, RR increases the spatial bunching of both electrons andions which are collected into denser clumps compared to thecase without RR. This might lead to a somewhat beneficial ef-fect with a longer and more e ffi cient radiation pressure acceler-ation phase whose signature would be an ion energy spectrumpeaking at an intermediate energy.A generalized relativistic kinetic equation including RR ef-fects has been discussed and we have shown that RR leads to acontraction of the available phase space volume. This predic-tion is in qualitative agreement with the results of our PIC sim-ulations where we observed both an increased spatial bunchingand a significant electron cooling as discussed above. 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