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Femtosecond x rays from laser-plasma accelerators S. Corde, K. Ta Phuoc, G. Lambert, R. Fitour, V. Malka, and A. Rousse Laboratoire d’Optique Appliquée, ENSTA ParisTech - CNRS UMR7639 - Ećole Polytechnique, Chemin de la Hunière, 91761 Palaiseau, France A. Beck and E. Lefebvre CEA, DAM, DIF, 91297 Arpajon, 3 France 1 0 Relativistic interaction of short-pulse lasers with underdense plasmas has recently led 2 to the emergence of a novel generation of femtosecond x-ray sources. Based on radiation from electrons accelerated in plasma, these sources have the common properties n tobecompactandtodelivercollimated,incoherentandfemtosecondradiation. Inthis a article we review, within a unified formalism, the betatron radiation of trapped and J acceleratedelectronsintheso-calledbubbleregime,thesynchrotronradiationoflaser- 2 acceleratedelectronsinusualmeter-scaleundulators,thenonlinearThomsonscattering 2 fromrelativisticelectronsoscillatinginanintenselaserfield,andtheThomsonbackscat- tered radiation of a laser beam by laser-accelerated electrons. The underlying physics ] ispresentedusingidealmodels,therelevantparametersaredefined,andanalyticalex- h pressions providing the features of the sources are given. Numerical simulations and a p summaryofrecentexperimentalresultsonthedifferentmechanismsarealsopresented. - Each section ends with the foreseen development of each scheme. Finally, one of the m mostpromisingapplicationsoflaser-plasmaacceleratorsisdiscussed: therealizationof s acompactfree-electronlaserinthex-rayrangeofthespectrum. Intheconclusion,the a relevant parameters characterizing each sources are summarized. Considering typical l laser-plasma interaction parameters obtained with currently available lasers, examples p ofthesourcefeaturesaregiven. Thesourcesarethencomparedtoeachotherinorder . s todefinetheirfieldofapplications. c i s y CONTENTS E. Scalingsandperspectives 25 h p [ I. Introduction 2 V. PlasmaAcceleratorandConventionalUndulator: SynchrotronRadiation 26 1 II. GeneralFormalism: RadiationfromRelativistic A. Electronmotion 26 v Electrons 3 B. Radiationproperties 27 6 A. Radiationfeatures 3 C. Numericalresults 28 6 B. Tworegimesofradiation: undulatorandwiggler 4 D. Experimentalresults 28 0 C. Qualitativeanalysisoftheradiationspectrum 4 E. Perspectives 29 D. Durationanddivergenceoftheradiation 6 5 E. Analyticalformulasforthetotalradiatedenergyand VI. ElectromagneticWaveUndulator: NonlinearThomson . 1 thenumberofemittedphotons 7 ScatteringandThomsonBackscattering 29 0 F. Radiationfromanidealelectronbunch 7 A. NonlinearThomsonscattering 30 3 G. Radiationreaction 8 1. Electronorbitinanintenselaserpulse 30 1 H. Realelectronbunch: longitudinalandtransverse 2. Radiationproperties 31 : emittance 9 3. Numericalresults 32 v 4. Experimentalresults 33 i III. ElectronAccelerationinPlasma 10 X 5. Perspectives 34 A. Ponderomotiveforceandplasmawaves 10 B. Thomsonbackscattering 35 r B. Thecavitatedwakefieldorbubbleregime 11 a C. Experimentalproductionofrelativisticelectron 1. Electronorbitinacounterpropagatinglaser pulse 35 bunches 11 2. Radiationproperties 36 IV. PlasmaAcceleratorandPlasmaUndulator: Betatron 3. Numericalresults 36 Radiation 12 4. Experimentalresults 37 A. Electronorbitinanioncavity 13 5. Perspectives 38 B. Radiationproperties 15 1. Withoutacceleration 15 VII. CoherentRadiation: TowardaCompactX-Ray 2. Withacceleration 16 Free-ElectronLaser 39 C. Numericalresults 17 A. TheFELamplifier 40 1. Test-particlesimulation 17 1. Principleofthefree-electronlaserprocess 40 2. ParticleInCellsimulation 18 2. Requiredconditionsontheelectronbeam D. Experimentalresults 20 parameters 42 2 3. Seedingorself-amplifiedspontaneousemission In parallel, alternative and complementary methods configurations 44 basedonlaser-producedplasmashavebeendevelopedto B. Free-electronlaserfromalaser-plasmaaccelerator 45 produce ultrashort compact radiation sources covering a 1. Withaconventionalundulator 45 wide spectral range from the extreme ultraviolet (XUV) 2. Withanelectromagneticwaveundulator 47 to the gamma rays. While several laser-based source 3. Withaplasmaundulator 48 schemes were proposed in the early 1970s, this field of VIII. Conclusion 48 research has seen rapid development when lasers have been able to produce intense femtosecond pulses (Perry Acknowledgments 51 and Mourou, 1994; Strickland and Mourou, 1985). At References 51 laser intensities on the order of 1014 W/cm2, XUV radiation, in the few tens of electronvolts (eV) energy range, can be produced using the mechanism of high-order har- I. INTRODUCTION monics generation from gas targets (Brabec and Krausz, 2000; Corkum, 1993; Krausz and Ivanov, 2009; Protopa- X-ray radiation has been, ever since its discovery over paset al.,1997)orbyXUVlaseramplificationinalaser- a century ago, one of the most effective tools to explore produced plasma (Daido, 2002). These sources can de- thepropertiesofmatterforabroadrangeofscientificre- liver, in most recent configurations, up to a microjoule search. Successive generations of radiation sources have of radiation within a beam of a few milliradians diver- been developed providing radiation with always higher gence. At laser intensities on the order of 1016 W/cm2, brightness, shorter wavelength and shorter pulse dura- x-ray sources from laser solid target interaction can pro- tion (Koch, 1983). Despite remarkable progress on x-ray duce a short pulse of Kα line emission, emitted within generation methods, there is still a need for light sources 4π steradians (Kieffer et al., 1993; Murnane et al., 1991; deliveringfemtosecondpulsesofbrighthigh-energyx-ray Rousseet al.,1994). Discoveredmorethanadecadeago, and gamma-ray radiation, emitted from source size of these sources have been widely developed and have led theorderofamicron(Pfeiferet al.,2006;Service,2002). to the first structural dynamics experiments at the fem- Indeed, the intense activity on the production of such tosecond time scale (Cavalleri et al., 2001; Rischel et al., radiation is motivated by countless applications in fun- 1997; Rose-Petruck et al., 1999; Rousse et al., 2001a; damental science, industry or medicine.(Bloembergen, Siders et al., 1999; Sokolowski-Tinten et al., 2003, 2001). 1999; Martin et al., 1992; Rousse et al., 2001b; Zewail, With recent developments, laser systems can deliver fo- 1997). For example, in the studies of structural dynam- cusedintensitiesabove1018 W/cm2 andthelaser-plasma ics of matter, the ultimate time scale of the vibrational interaction has entered the relativistic regime (Mourou period of atoms is a few tens of femtoseconds. Fun- et al., 2006; Umstadter, 2003). At this light intensity, damental processes such as dissociation, isomerization, relativistic effects become significant. Electrons can be phonons, and charge transfer evolve at this time scale. accelerated within the laser field or in the wakefield of High-energy radiation is used to radiograph dense ob- the laser up to relativistic energies (Esarey et al., 2009, jectsthatareopaqueforlow-energyxrays,whilemicron 1996b;Everettetal.,1994;Joshietal.,1984;Malkaetal., source size allows one to obtain high-resolution images 2002;Modenaet al.,1995;Patel,2007;TajimaandDaw- andmakespossiblephasecontrastimagingtoseewhatis son, 1979; Umstadter et al., 1996a). In particular, laser invisiblewithabsorptionradiography. Severaltechniques wakefield acceleration has led to the production of high- are being developed to produce femtosecond x rays. In quality femtosecond relativistic electron bunches (Faure the accelerator community, large-scale free-electron laser et al., 2004, 2006; Geddes et al., 2004; Mangles et al., facilities can now deliver the brightest x-ray beams ever, 2004) created and accelerated up to the gigaelectron- withunprecedentednovelpossibilities(Bartyetal.,2008; volt level (Hafz et al., 2008; Kneip et al., 2009; Leemans Brock, 2007; Chapman et al., 2006; Fritz et al., 2007; et al., 2006) within only a fewmillimeters or centimeters Gaffney and Chapman, 2007; Marchesini et al., 2008; plasma. Using these relativistic electrons, several novel Neutze et al., 2000). The slicing technique, combining x-ray source schemes have been proposed over the past a conventional accelerator with a femtosecond laser to decadestoproducecollimatedandfemtosecondradiation isolate short electron slices, allows synchrotrons to pro- in a spectrum ranging from the soft x rays to gamma duceradiationpulseswithdurationoftheorderof100fs rays. Most of these schemes are based on the wiggling (Schoenlein et al., 2000). High-energy radiation can be of relativistic electrons accelerated in a laser wakefield. delivered by radioactive sources, x-ray tube, and Comp- In this article, the physics of these sources is reviewed, ton scattering sources based on a conventional accelera- and the opportunities offered by these relativistic elec- tor. However,evenifwidelyusedthesehigh-energyradi- trons to generate ultrashort x-ray radiation (Catravas ation sources have limitations in terms of storage, pulse et al., 2001; Fritzler et al., 2003; Gru¨ner et al., 2007; duration, spectrum tunability, energy range, and source Hartemann et al., 2007; Jaroszynski et al., 2006; Lee- size. mans et al., 2005; Malka et al., 2008; Nakajima, 2008) 3 are highlighted. These sources can deliver x rays or based on the flying mirror concept (Bulanov et al., 2003; gamma rays as short as a few femtoseconds, as they in- Esirkepovet al.,2009;Kandoet al.,2007;Mourouet al., herit the temporal profile of the laser-plasma electron 2006), or the K source (Kieffer et al., 1993; Murnane α bunch, whose few-femtosecond duration was recently ex- et al., 1991; Rousse et al., 1994) can produce femtosec- perimentally demonstrated (Lundh et al., 2011). ondXUVorx-rayradiation. However,theyarenotbased on the same physical principle of acceleration and wig- The aim of this article is to review the novel x-ray gling of relativistic electrons, and will therefore not be sourcesbasedonrelativisticlaserandunderdenseplasma reviewed here. interaction and to highlight their similitude by using a common formalism for their description. The paper is II. GENERAL FORMALISM: RADIATION FROM organized as follows. In Sec. II, the general formalism of RELATIVISTIC ELECTRONS radiation from an accelerated relativistic electron is presented, which provides a framework for the description Inthissection,theradiationfromrelativisticelectrons of the sources discussed throughout the paper. From is introduced. A qualitative understanding of the phe- this formalism, the relevant parameters describing the nomenonishighlightedandanalyticalresultsfortherel- properties of the radiation, such as its spectrum, diver- evant general parameters determining the radiation fea- gence, number of emitted photons and duration, can be tures are given. The formalism described here is general extracted. As the x-ray sources presented here are based andwillbecommontoallthesourcespresentedthrough- on electrons accelerated by laser wakefields [i.e., by the outthearticle. WefollowtheapproachofJackson(2001) laser wakefield accelerator (LWFA)], a description of the andprovidethebasicresultsnecessaryfortheremainder most efficient laser-based electron accelerator to date is of the paper. The interested reader is referred to Wiede- given in Sec. III. mann (2007a) for a complete and detailed description of In Secs. IV, V and VI, different methods for the pro- synchrotron radiation from bending magnets, undulator, ductionofincoherentxraysfromrelativisticelectronsare and wiggler insertion devices (e.g. for the angular distri- reviewed; the objective is to define the relevant regimes bution of individual undulator harmonics, for polariza- toaccelerateandwiggleelectronsinsuchawaythatthey tion or spatial and temporal coherence properties). emit x rays. In Sec. IV, betatron radiation is described. In that case, a plasma cavity created in the wake of an intense laser pulse acts as both an electron accelerator A. Radiation features and a wiggler (referred to as a plasma undulator). The schemepresentedinSec. Vreliesontheuseoflaserwake- Relativistic electrons can produce bright x-ray beams field accelerated electrons, transported and wiggled in a if their motion is appropriately driven. For all the laser- meter-scale periodic arrangement of permanent magnets based x-ray sources discussed in this article, the radia- (conventional undulator). This method is the closest to tionmechanismistheemissionfromacceleratedrelativis- synchrotrontechnology. InSec. VI,thenonlinearThom- tic electrons. The features of this relativistically moving son scattering and the Thomson backscattering sources charge radiation are directly linked to the electron tra- arereviewed. InnonlinearThomsonscattering,electrons jectories. Obtained from the Liénard-Wiechert field, the are directly accelerated and wiggled in an intense laser general expression that gives the radiation emitted by field. For the Thomson backscattering case, the plasma an electron, in the direction of observation (cid:126)n, as a func- is used to accelerate electrons which are then wiggled in tion of its position, velocity, and acceleration along the acounterpropagatingelectromagneticwave(EMundula- trajectory is written (Jackson, 2001) tor). In Sec. VII, an introduction to the topic of the free- d2I e2 = electron laser is given and the conditions for realizing dωdΩ 16π3(cid:15) c 0 sduiacthioannwuiltthrawhiagvhe-lbenriggthhtndeoswsnantdo ctohheearenngtstsrooumrc(ehoafrdrax- (cid:12)(cid:12)(cid:12)(cid:90) +∞ (cid:126)n×(cid:104)((cid:126)n−β(cid:126))×β(cid:126)˙(cid:105) (cid:12)(cid:12)(cid:12)2 (1) ×(cid:12) eiω[t−(cid:126)n.(cid:126)r(t)/c] dt(cid:12) . rays) from laser-accelerated electrons are discussed for (cid:12)(cid:12) −∞ (1−β(cid:126).(cid:126)n)2 (cid:12)(cid:12) different types of undulators (plasma, conventional, and (cid:12) (cid:12) EM). This equation represents the energy radiated within a Other radiation sources based on the laser-plasma in- spectralbanddω centeredonthefrequencyω andasolid teraction have been developed and could provide pho- angledΩcenteredonthedirectionofobservation(cid:126)n. Here tons in the keV range. For example, high-order harmon- (cid:126)r(t) is the electron position at time t, β(cid:126) is the velocity ics from gas (Brabec and Krausz, 2000; Corkum, 1993; KrauszandIvanov,2009;Protopapasetal.,1997)orsolid of the electron normalized to the speed of light c, and targets (Dromey et al., 2006; Tarasevitch et al., 2000; β(cid:126)˙ = dβ(cid:126)/dt is the usual acceleration divided by c. We TeubnerandGibbon,2009;Thauryet al.,2007), sources stressthatthisexpressionassumesanobserverplacedat 4 a distance far from the electron so that the unit vector This analysis underlines the directions for the produc- (cid:126)n is constant along the trajectory. The expression (1) tion of x rays from relativistic electrons: the goal for for the radiated energy shows an important number of x-ray generation from relativistic electrons is to force a generic features: relativistic electron beam to oscillate transversally. This transverse motion will be responsible for the radiation. 1. When β(cid:126)˙ = 0, no radiation is emitted by the elec- Thisistheprincipleofsynchrotronfacilities,whereape- riodic static magnetic field, created by a succession of tron. This means that the acceleration is responsi- magnets, is used to induce a transverse motion to the ble for the emission of electromagnetic waves from electrons. The laser-based sources presented here rely charged particles. on this principle. In the next sections, the properties 2. Accordingtotheterm(1−β(cid:126).(cid:126)n)−2,theradiateden- of moving charge radiation in two different regimes are ergy is maximum when β(cid:126).(cid:126)n → 1. This condition reviewed. The different laser-based x-ray sources which is satisfied when β (cid:39) 1 and β(cid:126) (cid:107)(cid:126)n. Thus, a rela- willbediscussed canworkinbothregimes depending on tivistic electron (β (cid:39)1) will radiate orders of mag- the interaction parameters. nitude higher than a nonrelativistic electron, and its radiation will be directed along the direction of its velocity. This is simply the consequence of the B. Two regimes of radiation: undulator and wiggler Lorentz transformation: for an electron emitting anisotropicradiationinitsrestframe, theLorentz We consider ultrarelativistic electrons with a velocity transformation implies that the radiation is highly alongthedirection(cid:126)ez executingtransverseoscillationsin collimated in the small cone of typical opening an- the(cid:126)ex direction. Two regimes can be distinguished. gleof∆θ =1/γaroundtheelectronvelocityvector, The undulator regime corresponds to the situation whenobservedinthelaboratoryframe(seeFig. 1). where an electron radiates in the same direction at all In the following, we consider ultrarelativistic elec- times along its motion, as shown in Fig. 1. This occurs trons, γ (cid:29) 1, and all angles which will be defined when the maximal angle of the trajectory ψ is smaller are supposed to be small so that tanθ (cid:39)sinθ (cid:39)θ. than the opening angle of the radiation cone ∆θ = 1/γ. The wiggler regime differs from the undulator by the 3. The term ((cid:126)n−β(cid:126))×β(cid:126)˙, together with the relation fact that the different sections of the trajectory radiate β(cid:126)˙ ∝F(cid:126) /γ3 and β(cid:126)˙ ∝F(cid:126) /γ between applied force in different directions. Thus, emissions from the differ- (cid:107) (cid:107) ⊥ ⊥ ent sections are spatially decoupled. This occurs when andacceleration(respectivelyforaforcelongitudi- ψ (cid:29) 1/γ. The fundamental dimensionless parameter nal or transverse with respect to the velocity β(cid:126)), separating these two regimes is K = γψ. The radiation indicate that applying a transverse force F(cid:126) ⊥β(cid:126) is produced in these two regimes have different qualitative more efficient than a longitudinal force. The term and quantitative properties in terms of spectrum, diver- also shows that the radiated energy increases with gence, and radiated energy and number of emitted pho- the square of the acceleration β(cid:126)˙. More precisely, tons. P ∝ F2 and P ∝ γ2F2, where P is the radiated (cid:107) ⊥ power. Thus, it is much more efficient to use a transverse force in order to obtain high radiated C. Qualitative analysis of the radiation spectrum energy. Theshapeoftheradiationspectrumcanbedetermined 4. The phase term eiω[t−(cid:126)n.(cid:126)r(t)/c] can be locally ap- using simple qualitative arguments. In most of the cases proximatedbyeiω(1−β)t. Theintegrationovertime discussed throughout the article the electron trajectory will give a nonzero result only when the integrand, can be approximated by a simple transverse sinusoidal excluding the exponential, varies approximately at oscillationofperiodλ ataconstantvelocityβ andcon- u the same frequency as the phase term which oscil- stant γ. The orbit can be written as lates at ω = ω(1−β). Given that the velocity ϕ β(cid:126) of the electron varies at the frequency ω , the ψ K e− x(z)=x sin(k z)= sin(k z)= sin(k z), (2) condition ωϕ ∼ ωe− is required to have a nonzero 0 u ku u γku u result. The electron will radiate at the higher fre- quencyω =ω /(1−β)(cid:39)2γ2ω . Thus,theusual where k = 2π/λ is the wave-vector norm, x is the e− e− u u 0 Dopplerupshiftisdirectlyextractedfromthisgen- transverse amplitude of motion, and ψ is the maximum eral formula. This indicates the possibility to pro- angle between the electron velocity and the longitudinal duce x-ray beams (ω ∼ 1018 s−1) by wiggling a direction (cid:126)e . Since the electron energy is constant, an X z relativisticdirectionalelectronbeamatafrequency increase of the transverse velocity leads to a decrease of far below the x-ray range: ω (cid:39)ω /(2γ2). the longitudinal velocity. This can be explicitly derived e− X 5 X e- orbit Δϑ ∼ 1/γ Undulator λu/βz K<<1 λ X Y Z A1 A2 ϑ Y Z λ u e- orbit Δϑ ∼ 1/γ Wiggler X Ψ K>>1 FIG.2 Schematicforthecalculationofthespatialperiodλof the radiation emitted toward the direction of observation (cid:126)n, Z forminganangleθwiththe(cid:126)e direction. Atthetwopositions Y z marked by a blue point, the electron radiates the same field amplitude. The distance between these two amplitudes at a e- orbit given time corresponds to λ. FIG. 1 Illustration of the undulator and wiggler limits, at If K (cid:28) 1, the longitudinal velocity reduction is negli- (cid:113) thetopandthebottom,respectively. Thelobesrepresentthe 2 gible: β (cid:39) β and γ = 1/ 1−β (cid:39) γ. The motion direction of the instantaneously emitted radiation. ψ is the z z z contains only the fundamental component. Indeed, the maximum angle between the electron velocity and the propagation axis(cid:126)e and ∆θ is the opening angle of the radiation motion reduces to a simple dipole in the average rest z cone. When ψ (cid:28) ∆θ (undulator), the electron always radi- frame. The spectrum consists in a single peak at the atesinthesamedirectionalongthetrajectory,whereaswhen fundamental frequency ω which depends on the angle of ψ (cid:29) ∆θ (wiggler), the electron radiates toward different di- observation θ. As K →1, radiation also appears at har- rections in each portion of the trajectory. monics. If K (cid:29) 1, the longitudinal velocity reduction is sig- (cid:113) from the assumed trajectory (2), nificant: γz = γ/ 1+ 12K2. In the average rest frame, the trajectory is a figure-eight motion. It can contain (cid:18) K2 (cid:19) many harmonics of the fundamental. In the laboratory β (cid:39)β 1− cos2(k z) , (3) z 2γ2 u frame, this can be explained by the fact that an ob- (cid:0) K2(cid:1) 1 (cid:0) K2(cid:1) server receives short bursts of light of duration τ. In- βz (cid:39)β 1− 4γ2 (cid:39)1− 2γ2 1+ 2 . (4) deed, the instantaneous radiation is contained within a coneofopeningangle∆θ =1/γ centeredonβ(cid:126) andpoints With the trajectory of the electron periodic, the emit- toward an observer positioned in the direction (cid:126)n during tedradiationisalsoperiodicsinceeachtimetheelectron a time ∆t (see Fig. 3), corresponding to the variation is in the same acceleration state, the radiated amplitude of β(cid:126) by an angle ∆θ = 1/γ. Locally, a portion of the is identical. The period of the radiated field can be cal- trajectory can be approximated by a portion of a cir- culated to obtain the fundamental frequency of the radi- cle of radius ρ, such that the direction of the velocity ation spectrum. The radiation emitted in the direction β(cid:126) changes by an angle ∆θ when the electron travels a (cid:126)n, forming an angle θ with the (cid:126)ez direction, is consid- distance de = 2πρ×(∆θ/2π) = ρ/γ, which corresponds ered, as represented in Fig. 2. The field amplitude A(cid:126)1 to a time ∆t = te = de/(βc). During the time ∆t, the radiated in the direction (cid:126)n by the electron at z = 0 and radiation has covered a distance d = 2ρsin(1/2γ) cor- γ t = 0 propagates at the speed of light c. At z = λu responding to a propagation time of tγ = dγ/c. The and t = λu/(βzc), the electron radiates an amplitude radiation burst duration τ as observed by an observer A(cid:126) = A(cid:126) . The distance separating both amplitudes (A(cid:126) reads 2 1 1 and A(cid:126) ) corresponds to the spatial period λ of the radi- 13ρ 2 τ =t −t (cid:39) . (6) ated field and is given by e γ 24γ3c λ λ K2 Thetemporalprofile(seetheinsetofFig. 3)oftheradia- λ= βu −λucosθ (cid:39) 2γu2(1+ 2 +γ2θ2). (5) tionemittedinthewigglerregimehasbeenqualitatively z obtained. The Fourier transform of this typical profile Theradiationspectrumconsistsnecessarilyinthefunda- gives a precise representation of the radiation spectrum. mentalfrequencyω =2πc/λanditsharmonics. Toknow With the time profile a succession of bursts of duration ifharmonicsofthefundamentalareeffectivelypresentin τ, the spectrum will contain harmonics up to the critical thespectrum,itisinstructivetolookattheelectronmo- frequency tion in the average rest frame, moving at the velocity βz ω ∼1/τ ∼γ3c. (7) in the(cid:126)ez direction with respect to the laboratory frame. c ρ 6 P quency: λ/c 3 ω = Kγ22πc/λ (10) Observer c 2 u Δϑ ∼ 1/γ τ 3 = Kω . (11) 4 {K(cid:28)1,θ=0} time Note that for a nonplanar and nonsinusoidal trajectory, β z the spectrum extends up to a critical frequency determined by the minimal radius of curvature of the trajec- e- orbit t + Δt tory. t The parameter K can be considered as the number of decoupled sections of the trajectory. With each section radiating toward a different direction, the radiation is FIG.3 Inthewigglerlimit,theradiationconepointstoward spatially decoupled and leads to bursts of duration τ in the observer during a time ∆t, which corresponds to a du- each direction and to a broad spectrum with harmonics ration τ for the emitted radiation. This is repeated at each of the fundamental up to ω . period: the observer receives bursts of radiation separated c by a time λ/c. The inset gives the temporal profile of the radiation power seen by the observer. An infinite periodical motion has been considered so far, leading to harmonics that are spectrally infinitely thin. ForafinitenumberofoscillationperiodsN,Fourier Note that the spectrum that arises from a complete cal- transformpropertiesimplythattheharmonicofnumber culationoftheradiationemittedbyarelativisticcharged n and of wavelength λn =λ/n has a width given by particle in instantaneously circular motion (Jackson, ∆λ 1 2001) is in agreement with the above estimation. Such n = . (12) λ nN calculation yields the synchrotron spectrum, which is n written in terms of radiated energy per unit frequency This corresponds to the harmonic bandwidth in a given and per unit time, direction characterized by the angle θ. Since the wavelength λ depends on the angle θ [see Eq. (5)], the inte- n dP P = γ S(ω/ω ), grationoftheradiationoverasmallapertureoffinitedi- c dω ω √c mensionbroadenseachharmonic. Whenintegratingover 9 3 (cid:90) ∞ thetotalangulardistribution,harmonicsoverlapandthe S(x)= x K (ξ)dξ, 8π 5/3 radiation spectrum becomes continuous, but keeps the x e2cγ4 2e2ω2 same extension (up to ωc). In the undulator limit, in Pγ = 6π(cid:15) ρ2 = 27π(cid:15) ccγ2, which only the fundamental wavelength is present, the 0 0 bandwidth is highly degraded when integrating over the 3 c ω = γ3 , (8) total angular distribution. c 2 ρ where we introduced the exact definition of the critical D. Duration and divergence of the radiation frequency ω of the synchrotron spectrum which will be c (cid:82) used throughout the review, P = dP/dω is the radi- γ Thepulsedurationandthedivergenceoftheradiation ated power, and K is the modified Bessel function of 5/3 emitted by a single electron can be deduced from the the second kind. previous analysis. If N is the number of periods of the The expression for the radius of curvature ρ can be trajectory, the radiation consists in N periods of length obtained for an arbitrary trajectory. Its value can be λ and the total duration of the pulse is τ =Nλ/c. calculated at each point of the trajectory, corresponding Foranundulator,K (cid:28)1,thedirectionro|Nfet=h1evectorβ(cid:126) toaparticulardirectionofobservation. Intheparticular varies along the trajectory by an angle ψ negligible com- case of a sinusoidal trajectory, the radius of curvature paredto∆θ =1/γ. Hence,theradiationfromallsections reads ρ(z) = ρ [1 + ψ2cos2(k z)]3/2/|sin(k z)| and it 0 u u ofthetrajectoryoverlapandthetypicalopeningangleof is minimum when the transverse position is extremum theradiationissimplyθ =1/γ 1. Forawiggler,K (cid:29)1, r (x=±x ) and its value at this point reads 0 λ λ u u ρ = =γ , (9) 0 2πψ 2πK 1 ForK<1,therootmeansquareangleoftheangulardistribution oftheradiatedenergyatthefundamentalfrequencyis(cid:104)θ2(cid:105)1/2= leading to the following expression for the critical fre- 1/γz,whichsimplifiesto1/γ forK(cid:28)1(Jackson,2001). 7 thedivergenceincreasesinthedirectionofthetransverse N electrons contained in the bunch, assuming that they e oscillation(cid:126)e ,butremainsidenticalintheorthogonaldi- all have exactly the same energy and the same initial x rection(cid:126)e . In the direction of the motion(cid:126)e , the typical momentum(zeroemittance). Theradiationfromseveral y x openingangleoftheradiationisθ =ψ =K/γ2,which electronsisobtainedbysummingthecontributionofeach Xr is greater than ∆θ = 1/γ, while the typical opening an- electron before taking the squared norm, gle of the radiation in the direction(cid:126)e is θ =1/γ. For y Yr the general case of a transverse motion occurring in the d2I e2 = (cid:126)ex and(cid:126)ey directions, the angular profile can take various dωdΩ 16π3(cid:15)0c strhaajpeecstodrye.pending on the exact three-dimensional (3D) (cid:12)(cid:12)(cid:12)(cid:88)Ne (cid:90) +∞ (cid:126)n×(cid:104)((cid:126)n−β(cid:126)j)×β(cid:126)˙j(cid:105) (cid:12)(cid:12)(cid:12)2 ×(cid:12) eiω(t−(cid:126)n.(cid:126)rj(t)/c) dt(cid:12) . (cid:12)(cid:12)(cid:12)j=1 −∞ (1−β(cid:126)j.(cid:126)n)2 (cid:12)(cid:12)(cid:12) E. Analytical formulas for the total radiated energy and the (17) number of emitted photons This formula expresses the coherent addition of the ra- Analytical calculations provide simple expressions for diation field of each electron. It can be considerably the radiated energy and the number of emitted photons simplified by considering that all electrons follow simi- perperiod. Usingtheexpressionoftheradiatedpowerby lar trajectories linked to each other by a spatiotemporal anelectronP(t)=(e2/6π(cid:15)0c)γ2[(dp(cid:126)ˆ/dt)2−(dγ/dt)2](p(cid:126)îs translation (tj,R(cid:126)j) of a reference trajectory (cid:126)r(t): themomentumnormalizedtomc),theaveragedradiated powerPγ andthetotalradiatedenergyperperiodIγ can (cid:126)rj(t)=R(cid:126)j +(cid:126)r(t−tj), bederivedforboththeundulatorandthewigglercasefor (cid:12) (cid:12)2 anarbitrarytrajectory. Inthecaseofaplanarsinusoidal d2I =(cid:12)(cid:12)(cid:12)(cid:88)Ne eiω(tj−(cid:126)n.R(cid:126)j/c)(cid:12)(cid:12)(cid:12) e2 trajectory, the result is dωdΩ (cid:12) (cid:12) 16π3(cid:15) c (cid:12)j=1 (cid:12) 0 Pγ = π3e(cid:15)20c γ2λK2u2, (13) ×(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) +∞eiω(t−(cid:126)n.(cid:126)r(t)/c)(cid:126)n×(cid:104)((cid:126)n−β(cid:126))×β(cid:126)˙(cid:105)dt(cid:12)(cid:12)(cid:12)(cid:12)2. I = πe2 γ2K2. (14) (cid:12)(cid:12) −∞ (1−β(cid:126).(cid:126)n)2 (cid:12)(cid:12) γ 3(cid:15) λ (cid:12) (cid:12) 0 u (18) Toobtainanestimationofthenumberofemittedpho- tons N , a mean energy of photons must be determined. γ The radiated energy per unit frequency and unit solid For K (cid:28) 1, the spectrum is quasimonochromatic in the angle from an electron bunch is equal to the radiated forward direction and the mean energy of photons, af- energyfromasingleelectronfollowingthetrajectory(cid:126)r(t) ter integrating over the angular distribution, is equal to multiplied by the coherence factor (cid:126)ωθ=0/2. The number of emitted photons reads (cid:12) (cid:12)2 Nγ = 23παK2, (15) c(ω)=(cid:12)(cid:12)(cid:12)(cid:88)Ne eiω(tj−(cid:126)n.R(cid:126)j/c)(cid:12)(cid:12)(cid:12) . (19) (cid:12) (cid:12) (cid:12)j=1 (cid:12) where α=e2/(4π(cid:15)0(cid:126)c) is the fine structure constant. ForK (cid:29)1,thespectrumissynchrotronlikewiththecrit- Thevalueofc(ω)dependsontheelectrondistributionin ical frequency ωc. Usin√g the fact that for a synchrotron the (t ,R(cid:126) ) space. For a uniform distribution, c(ω) = 0, j j spectrum (cid:104)(cid:126)ω(cid:105) = (8/15 3)(cid:126)ωc, the following estimation whereas for a random distribution, c(ω) = N on aver- e is derived: √ age. If the distribution is microbunched at the wave- 5 3π length λ = 2πc/ω , the summation is coherent for the N = αK. (16) b b γ 6 frequencyωb anditsharmonics,c(nωb)=Ne2 forn∈N∗. In large accelerators or in laser-plasma accelerators, electrons are randomly distributed inside the bunch at F. Radiation from an ideal electron bunch the x-ray wavelength scale, and the radiation is incoher- ently summed, Theradiationfromasingleelectronhasbeendiscussed so far. We now take into account the fact that there are c(ω)=N , (20) e d2I d2I 2 Herethetypicalopeningangleisdefinedasthemaximumdeflec- dωdΩ(cid:12)(cid:12)Ne =NedωdΩ(cid:12)(cid:12)Ne=1. (21) tion angle of the electron trajectory ψ, such that the full width oftheangulardistributionoftheradiatedenergyinthedirection The spectrum shape and the radiation divergence θr re- (cid:126)ex is2K/γ. main unchanged for an electron bunch. The temporal 8 profile of the radiation is given by the convolution be- wherethefirsttermistheLorentzforcewithF theex- µν tween the electron bunch temporal profile and the ra- ternal electromagnetic field tensor, the last two terms diation profile from a single electron. Since the typical that define the RR self-force correspond, respectively, electron bunch length is in the micron range, whereas to the Schott term and the radiation damping term, the radiation length from a single electron l =Nλ p = mu is the electron quadrimomentum and τ = is in the nanometer range for x rays, the durra|Nteio=n1 of the 2rµ /(3c) =µ e2/(6π(cid:15) mc3) = 6.26×10−24 s, with r 0the e 0 e radiationfromanelectronbunchisinmostcasesapprox- classical electron radius. The Schott term τ d2p = 0 τ µ imately equal to the bunch duration, τ =τ . −d G accounts for the change of energy momentum r|Ne b τ µ G of the external field, while the radiation damping µ Inaddition,theelectronexperiencestheradiationfrom term −τ0pµ(dτpνdτpν)/(m2c2) = −dτHµ accounts for other electrons in the case of a bunch, which can modify thechangeofenergy-momentumHµofthescatteredelec- its motion and its energy. This interaction of the bunch tromagnetic wave (Hartemann, 2002). withitsownradiationcanleadtoamicrobunchingofthe Several difficulties appear in the point electron model electron distribution within the bunch at the fundamen- described by the Dirac-Lorentz equation (22). First, talwavelengthoftheradiationanditsharmonics. Thisis it admits unphysical runaway solutions with exponen- the free-electron laser (FEL) process which produces co- tially increasing acceleration, which can be eliminated herent radiation. Here the coherence factor c(nω ) is N2 by requiring the Dirac-Rohrlich asymptotic condition b e insteadof Ne, indicating thattheFELradiatesordersof limτ→±∞dτpµ = 0. Second, physical solutions presents magnitude higher than conventional synchrotrons. How- acausal preacceleration, i.e., that electron momentum ever, the FEL effect requires stringent conditions on the changes before an external force is suddenly applied, on electron beam quality and an important number N of a time scale τ0. Equation (22) can be approximated oscillations. At the present status of laser-plasma accel- by evaluating the RR self-force with the solution of the erators, realizing a FEL represents a technological chal- zeroth-order equation dτpµ = −eFµνuν (Landau and lenge. In the following sections, the interaction between Lifshitz, 1994). This yields the Landau-Lifshitz equa- theelectronbunchanditsradiationwillnotbetakeninto tionwhichneitheradmitsrunawaysolutionsnorpresents accountbecauseintheseschemestheconditionsrequired acausal preacceleration behavior. fortheFELarenotfulfilled: theelectrondistributionre- ForthecaseofanelectronundulatingaccordingtoEq. mains random along the propagation and the radiation (2), it is important to define the range of parameters for is incoherent. In Sec. VII, the underlying physics of which radiation reaction comes into play and has to be thefree-electronlaserispresentedinmoredetailandthe includedinthedescriptionoftheelectronmotionandits possible realization of such high-brightness coherent ra- radiation. Forrelativisticelectrons,thedominanttermin diation using laser-plasma accelerators is discussed. theradiationreactioncomesfromtheenergymomentum transferred to the scattered electromagnetic wave [while forrestelectrons,theradiationreactiondescribesthedi- G. Radiation reaction rectexchangeofenergymomentumbetweentheexternal field and the scattered wave (Hartemann, 2002)]. The The radiation reaction (RR) corresponds to the ef- rate of energy loss ν for the electron can be estimated γ fect of the electromagnetic field scattered by an electron from mc2dγ/dt = −P , with P the average power ra- γ γ onitself,theso-calledself-interaction(Hartemann,2002; diated by the electron given by Eq. (13). It leads to Jackson, 2001; Landau and Lifshitz, 1994). RR effects γ(t)=γ /(1+ν t) with 0 γ can modify the electron trajectory and its energy and (cid:18) (cid:19)2 τ 2πc it is therefore important to define the range of param- ν = 0γ K2 , (23) γ 0 eters for which these effects come into play. The main 2 λu steps, followed by Dirac to derive the relativistically co- and γ is initial gamma factor of the electron (Esarey, 0 variant form of the self-force associated with RR, are 2000; Huang and Ruth, 1998; Koga et al., 2005; Michel to solve for the self-quadripotential Asµ scattered by the et al., 2006; Telnov, 1997). Therefore, the radiation re- electronintermsofGreen’sfunctions,andthentocalcu- action can be neglected when the interaction duration latetheassociatedelectromagneticforceontheelectron, or equivalently the number of oscillations satisfy respec- Fµs = −e(∂µAsν −∂νAsµ)uν, where uν = dxν/dτ is the tively, τ (cid:28) νγ−1 and N (cid:28) NRR = λu/(2π2cτ0γ0K2). electron quadrivelocity with τ the electron proper time. With conventional undulators (see Sec. V), RR will al- This leads to the Dirac-Lorentz equation of motion for a ways be negligible; for ∼ 10 GeV electrons, λ ∼ 1 cm u pointlike electron, and K ∼1, the limiting number of period N is on the RR dp (cid:20)d2p p (cid:18)dp dpν(cid:19)(cid:21) orderof1.3×107. Forcurrentandshort-termlaser-based µ =−eF uν +τ µ − µ ν , betatron experiments (see Sec. IV), RR is negligible but dτ µν 0 dτ2 m2c2 dτ dτ in the long term or for electron beam driven plasma ac- (22) celerators [e.g., parameters of the Facility for Advanced 9 Accelerator Experimental Tests (FACET) (Hogan et al., For a relativistic electron beam traveling in the (cid:126)e di- z 2010)], ∼ 10 GeV electrons, λ ∼ 1 cm, and K ∼ 100 rection, an emittance is defined for each dimension: the u lead to N ∼ 1.3×103. For Thomson backscattering z one is called longitudinal and two others are trans- RR (seeSec. VI),GeVelectronscollidingwithalaserpulseof verse (x and y). For a uniform distribution with sharp strength parameter K ∼ 10 and wavelength λ = 0.8 µm boundary, the normalized emittance (cid:15) is defined as aN (λ = λ/2) leads to N ∼ 60. The radiation reaction the area occupied by electrons in the (a,p /mc) space u RR a canbeneglectedinThomsonbackscatteringforsub-GeV divided by π (a = x,y,z). But because realistic elec- electronbeamsandlaserpulsesofstrengthparameteron tronbeamshavediffuseboundaries,thenormalizedroot- the order of unity, for which N (cid:38)6×103. mean-square (rms) emittance is used and defined as RR In the realm of quantum electrodynamics (QED), the (cid:112) (cid:15) = (cid:104)∆a2(cid:105)(cid:104)∆p2(cid:105)−(cid:104)∆a∆p (cid:105)2/mc, (24) radiation reaction corresponds to the recoil experienced aN a a by an electron due to consecutive incoherent photon witha=x,y,z andwhere∆a=a−(cid:104)a(cid:105),∆p =p −(cid:104)p (cid:105). a a a emissions (Di Piazza et al., 2010). Quantum effects be- For transverse dimensions, it is convenient to use the come important when the electron energy loss associ- unnormalized emittance (cid:15) , which is related to the area a ated with the emission of a photon is on the order of occupiedbyelectronsinthe(a,a(cid:48))tracespace(a=x,y), the electron energy. Signatures of quantum effects can where a(cid:48) (cid:39)p /p is the transverse angle with respect to a z be observed before entering this quantum regime. In- the propagation axis z, deed, quantum fluctuations, which imply that different (cid:112) electrons emit a different number of photons (that carry (cid:15)a = (cid:104)∆a2(cid:105)(cid:104)∆a(cid:48)2(cid:105)−(cid:104)∆a∆a(cid:48)(cid:105)2, a=x,y. different energies) and hence lose a different amount of (25) energy, can lead to an observable increase of the elec- Itisgenerallyexpressedinπ.mm.mrad. Forcylindrically tron beam energy spread (Esarey, 2000). Experimen- symmetric beams, the emittance (cid:15)r (defined in the same tally, a study of the quantum regime is accessible in the way by putting a = r) can be used. The normalized framework of Compton scattering (Sec. VI). QED ef- emittance is related to the unnormalized one by (cid:15)N = fects, such as nonlinear Compton scattering (Bula et al., γβ(cid:15) and has the advantage of being conserved during 1996) and the production of electron-positron pairs from acceleration (in systems which preserve the emittance). light (Burke et al., 1997), were observed in the SLAC In a focal plane, where there is no correlation between E-144 experiment, where 46.6 GeV electron beams col- positionandangle,theemittanceissimplytheproductof lidedwithrelativisticlaserpulseswithintensitiesof1018 the rms transverse size σa by the rms angular dispersion W.cm−2. σa(cid:48): (cid:15)a =σaσa(cid:48), a=x,y,r. (26) H. Real electron bunch: longitudinal and transverse For incoherent radiation of electrons oscillating in un- emittance dulator or wiggler devices, the emittance and energy spread have the following effects. The electron beam Intheprevioussection,idealbuncheswithelectronsat is focused in the device, with a transverse size σ and the same energyand same momentum have been consid- a divergence θ satisfying (cid:15) = σθ. Because of the angu- ered in order to simply discuss the effect of summation lar spread, the radiation angular distributions from each of the radiation from each electron. However, in realis- singleelectronareslightlyshiftedfromoneanother,lead- ticbunches, electronshaveslightlydifferentenergiesand ing to a redshifted broadening of harmonic bandwidths momenta. More precisely, inside the bunch, electrons [an electron with direction θ with respect to the prop- that are at the same location can have different energies agation axis contributes on axis with higher wavelength andmomenta. Thisimplies,forexample,thatthebunch λ = λ (1+γ2θ2), see Eq. (5)]. The energy spread θ=0 z cannot be focused and compressed on an infinitely small effect is straightforward: electrons with different ener- point. Thislimitationisfundamentalandinherenttothe giesradiateatslightlydifferentwavelengths,leadingtoa bunch, it does not depend on practical realization. The broadening of harmonic bandwidths. These effects re- parameterwhichaccountsforthatiscalledtheemittance sult in a modified bandwidth given by (∆λ /λ )2 = n n and is related to the volume occupied by the electrons (1/(nN))2 +(2∆γ/γ)2 +(γ2(cid:15)2/σ2)2. Hence, the band- z in the 6D phase space (x,y,z,p ,p ,p ) at a given time width of an harmonic at a given direction comes from x y z (Humphries, 1990). The 6D phase volume is constant three different effects: the finite number of periods, the in time if only smooth external forces are applied and if energy spread, and the angular spread (which depends collisionsareneglected(thephasevolumeconservationis on the emittance). a consequence of the collisionless Boltzmann equation). Thetransverseemittanceisessentialfortransportcon- The emittance reflects the quality of the electron beam siderations and applications such as the free-electron because it quantitatively indicates if electrons have the laser. Concerning the FEL application, required condi- same coordinates, direction, and energy. tions on the transverse emittance and the energy spread 10 will be given in Sec. VII. A smaller transverse emittance plasma wave. After a brief description of the character- permits one to transport or focus the electron beam on isticsoftheaccelerationmechanism,recentexperimental a smaller focal spot size. progress is presented. We refer the interested reader to the recent article of Esarey et al. (2009) for a complete review of laser-plasma electron accelerators. III. ELECTRON ACCELERATION IN PLASMA The possibility to accelerate electrons in laser- A. Ponderomotive force and plasma waves producedplasmaswasoriginallyproposedbyTajimaand Dawson (1979). They suggested to use the intense elec- Theponderomotiveforceisaforceassociatedwiththe tricfieldofarelativisticplasmawave,createdinthewake intensity gradients in the laser pulse, that pushes both of an intense laser pulse, to accelerate electrons to rela- electronsandionsoutofthehigh-intensityregions. Ions, tivistic energies. The main advantage of plasmas relies being much heavier than electrons, still remain for short on their ability to sustain an accelerating gradient much interaction times whereas electrons are cast away. This larger (on the order of 100 GeV/m) than a conventional leads, in an underdense plasma, to the formation of a radio frequency accelerating module (on the order of 10 relativistic plasma wave whose fields can accelerate elec- MeV/m). Thismeansthatelectronscouldbeaccelerated trons. Here a short description of the ponderomotive up to 1 GeV in millimeter- or centimeter-scale plasmas forceandoftheexcitationofaplasmawave(Kruer,1988) (Claytonetal.,2010;Froulaetal.,2009;Hafzetal.,2008; is given. Kneipet al.,2009;Leemanset al.,2006)whileafewtens Because of the mass of the plasma ions, they can be of meters would be necessary to reach the same energy considered motionless for short interaction times. Con- in conventional accelerators. sidering a fluid description for the plasma electrons, the This acceleration method has experienced a remark- equation of motion for a fluid element submitted to the able development over the past decades, mainly thanks electromagnetic force reads 4 to the advent of high-intensity lasers and to a better un- ∂p(cid:126)ˆ ∂(cid:126)a derstandingofthephysicalmechanismsdrivingtheaccel- = +c∇(cid:126)(φ−γ), (27) ∂t ∂t eration. Different plasma accelerator schemes have been developed over the years, 3 leading to electron bunches wherep(cid:126)ˆ=p(cid:126)/mcisthenormalizedmomentumofanelec- with ever-increasing quality. The most efficient to date tron fluid element, γ is the relativistic factor of an elec- is the so-called bubble, blowout or cavitated wakefield tron fluid element, and φ = eV/mc2 and (cid:126)a = eA(cid:126)/mc regime (Lu et al., 2006a,b, 2007; Pukhov et al., 2004a; are, respectively, thenormalizedscalarpotentialandthe Pukhov and Meyer-ter Vehn, 2002). In that regime, de- normalizedvectorpotentialoftheelectromagneticfields. pending on the chosen parameters, electron bunches can (cid:126)a describes the high-frequency laser pulse, c∇(cid:126)φ is the nowbeproducedwithtunableenergyinthehundredsof Coulomb force associated with the charge distribution, MeVrange(Faureet al.,2006),lowdivergence(mrad),a and −c∇(cid:126)γ is the relativistic ponderomotive force which relatively high charge (∼100 pC), and a bunch duration expelselectronsawayfromthelaserpulse. Intheabsence oflessthan10fs(Davoineet al.,2008;Faureet al.,2004; ofCoulombandponderomotiveforces,theequationsim- Geddes et al., 2004; Glinec et al., 2007; Lundh et al., plifiestop(cid:126)ˆ=(cid:126)a,correspondingtothefastelectronoscilla- 2011; Mangles et al., 2004, 2006; Thomas et al., 2007; tioninthelaserpulse. Dependingontheamplitudeofthe van Tilborg et al., 2006; Tsung et al., 2006, 2004). Be- laser pulse normalized vector potential a , plasma elec- 0 cause the x-ray sources which will be reviewed are based trons oscillate with relativistic velocities |(cid:126)v|(cid:39)c (a >1) 0 on laser-plasma accelerators, this section is dedicated to or with velocities much smaller than c (a (cid:28) 1). An 0 a short description of wakefield acceleration in the cavi- inhomogenous laser intensity distribution leads to an in- tatedregime. Inparticulartwoimportantphysicalmech- homogenousγ distributionandtoaponderomotiveforce anisms are introduced: the ponderomotive force and the thatpushesplasmaelectronsfromthehighγ region(cor- respondingtohigha )tothelowγ region(lowa ). This 0 0 slowdriftmotionoftheplasmaelectronsleadstoacharge density distribution ρ responsible for a Coulomb force 3 For an overview of the historical development of the field, see c∇(cid:126)φ (by virtue of the Poisson equation (cid:52)φ=ρ/(cid:15) ). 0 Amiranoff et al. (1998, 1995); Bingham et al. (2004); Clayton On the other hand, a small charge density perturba- etal.(1993);Coverdaleetal.(1995);Esareyetal.(2009,1996b); tion in a plasma oscillates at a characteristic frequency, Everett et al. (1994); Gahn et al. (1999); Gordon et al. (1998); Joshi(2007);Joshietal.(1984);Kitagawaetal.(1992);Leemans et al. (2002); Malka et al. (2002); Mangles et al. (2005); Mod- ena et al. (1995); Moore et al. (1997); Najmudin et al. (2003); Nakajima et al. (1995); Patel (2007); Pukhov (2003); Santala 4 Equation (27) assumes that ∇(cid:126) ×(p(cid:126)ˆ−(cid:126)a) is initially zero, which et al. (2001); Tajima and Dawson (1979); Ting et al. (1997); is thecase in practicebecause both p(cid:126)ând(cid:126)a are zerobeforethe Umstadteret al.(1996a);andWagneret al.(1997). passageofthelaserpulseintheplasma.