Femtosecond x rays from laser-plasma accelerators
S. Corde, K. Ta Phuoc, A. Beck, G. Lambert, R. Fitour, E. Lefebvre, V. Malka, A. Rousse
FFemtosecond x rays from laser-plasma accelerators
S. Corde, K. Ta Phuoc, G. Lambert, R. Fitour, V. Malka, and A. Rousse
Laboratoire d’Optique Appliqu´ee,ENSTA ParisTech - CNRS UMR7639 - ´Ecole Polytechnique,Chemin de la Huni`ere,91761 Palaiseau,France
A. Beck and E. Lefebvre
CEA, DAM, DIF, 91297 Arpajon,France
Relativistic interaction of short-pulse lasers with underdense plasmas has recently ledto the emergence of a novel generation of femtosecond x-ray sources. Based on radi-ation from electrons accelerated in plasma, these sources have the common propertiesto be compact and to deliver collimated, incoherent and femtosecond radiation. In thisarticle we review, within a unified formalism, the betatron radiation of trapped andaccelerated electrons in the so-called bubble regime, the synchrotron radiation of laser-accelerated electrons in usual meter-scale undulators, the nonlinear Thomson scatteringfrom relativistic electrons oscillating in an intense laser field, and the Thomson backscat-tered radiation of a laser beam by laser-accelerated electrons. The underlying physicsis presented using ideal models, the relevant parameters are defined, and analytical ex-pressions providing the features of the sources are given. Numerical simulations and asummary of recent experimental results on the different mechanisms are also presented.Each section ends with the foreseen development of each scheme. Finally, one of themost promising applications of laser-plasma accelerators is discussed: the realization ofa compact free-electron laser in the x-ray range of the spectrum. In the conclusion, therelevant parameters characterizing each sources are summarized. Considering typicallaser-plasma interaction parameters obtained with currently available lasers, examplesof the source features are given. The sources are then compared to each other in orderto define their field of applications.
CONTENTS
I. Introduction 2II. General Formalism: Radiation from RelativisticElectrons 3A. Radiation features 3B. Two regimes of radiation: undulator and wiggler 4C. Qualitative analysis of the radiation spectrum 4D. Duration and divergence of the radiation 6E. Analytical formulas for the total radiated energy andthe number of emitted photons 7F. Radiation from an ideal electron bunch 7G. Radiation reaction 8H. Real electron bunch: longitudinal and transverseemittance 9III. Electron Acceleration in Plasma 10A. Ponderomotive force and plasma waves 10B. The cavitated wakefield or bubble regime 11C. Experimental production of relativistic electronbunches 11IV. Plasma Accelerator and Plasma Undulator: BetatronRadiation 12A. Electron orbit in an ion cavity 13B. Radiation properties 151. Without acceleration 152. With acceleration 16C. Numerical results 171. Test-particle simulation 172. Particle In Cell simulation 18D. Experimental results 20 E. Scalings and perspectives 25V. Plasma Accelerator and Conventional Undulator:Synchrotron Radiation 26A. Electron motion 26B. Radiation properties 27C. Numerical results 28D. Experimental results 28E. Perspectives 29VI. Electromagnetic Wave Undulator: Nonlinear ThomsonScattering and Thomson Backscattering 29A. Nonlinear Thomson scattering 301. Electron orbit in an intense laser pulse 302. Radiation properties 313. Numerical results 324. Experimental results 335. Perspectives 34B. Thomson backscattering 351. Electron orbit in a counterpropagating laserpulse 352. Radiation properties 363. Numerical results 364. Experimental results 375. Perspectives 38VII. Coherent Radiation: Toward a Compact X-RayFree-Electron Laser 39A. The FEL amplifier 401. Principle of the free-electron laser process 402. Required conditions on the electron beamparameters 42 a r X i v : . [ phy s i c s . p l a s m - ph ] J a n
3. Seeding or self-amplified spontaneous emissionconfigurations 44B. Free-electron laser from a laser-plasma accelerator 451. With a conventional undulator 452. With an electromagnetic wave undulator 473. With a plasma undulator 48VIII. Conclusion 48Acknowledgments 51References 51
I. INTRODUCTION
X-ray radiation has been, ever since its discovery overa century ago, one of the most effective tools to explorethe properties of matter for a broad range of scientific re-search. Successive generations of radiation sources havebeen developed providing radiation with always higherbrightness, shorter wavelength and shorter pulse dura-tion (Koch, 1983). Despite remarkable progress on x-raygeneration methods, there is still a need for light sourcesdelivering femtosecond pulses of bright high-energy x-rayand gamma-ray radiation, emitted from source size ofthe order of a micron (Pfeifer et al. , 2006; Service, 2002).Indeed, the intense activity on the production of suchradiation is motivated by countless applications in fun-damental science, industry or medicine.(Bloembergen,1999; Martin et al. , 1992; Rousse et al. , 2001b; Zewail,1997). For example, in the studies of structural dynam-ics of matter, the ultimate time scale of the vibrationalperiod of atoms is a few tens of femtoseconds. Fun-damental processes such as dissociation, isomerization,phonons, and charge transfer evolve at this time scale.High-energy radiation is used to radiograph dense ob-jects that are opaque for low-energy x rays, while micronsource size allows one to obtain high-resolution imagesand makes possible phase contrast imaging to see what isinvisible with absorption radiography. Several techniquesare being developed to produce femtosecond x rays. Inthe accelerator community, large-scale free-electron laserfacilities can now deliver the brightest x-ray beams ever,with unprecedented novel possibilities (Barty et al. , 2008;Brock, 2007; Chapman et al. , 2006; Fritz et al. , 2007;Gaffney and Chapman, 2007; Marchesini et al. , 2008;Neutze et al. , 2000). The slicing technique, combininga conventional accelerator with a femtosecond laser toisolate short electron slices, allows synchrotrons to pro-duce radiation pulses with duration of the order of 100 fs(Schoenlein et al. , 2000). High-energy radiation can bedelivered by radioactive sources, x-ray tube, and Comp-ton scattering sources based on a conventional accelera-tor. However, even if widely used these high-energy radi-ation sources have limitations in terms of storage, pulseduration, spectrum tunability, energy range, and sourcesize. In parallel, alternative and complementary methodsbased on laser-produced plasmas have been developed toproduce ultrashort compact radiation sources covering awide spectral range from the extreme ultraviolet (XUV)to the gamma rays. While several laser-based sourceschemes were proposed in the early 1970s, this field ofresearch has seen rapid development when lasers havebeen able to produce intense femtosecond pulses (Perryand Mourou, 1994; Strickland and Mourou, 1985). Atlaser intensities on the order of 10 W/cm , XUV radi-ation, in the few tens of electronvolts (eV) energy range,can be produced using the mechanism of high-order har-monics generation from gas targets (Brabec and Krausz,2000; Corkum, 1993; Krausz and Ivanov, 2009; Protopa-pas et al. , 1997) or by XUV laser amplification in a laser-produced plasma (Daido, 2002). These sources can de-liver, in most recent configurations, up to a microjouleof radiation within a beam of a few milliradians diver-gence. At laser intensities on the order of 10 W/cm ,x-ray sources from laser solid target interaction can pro-duce a short pulse of K α line emission, emitted within4 π steradians (Kieffer et al. , 1993; Murnane et al. , 1991;Rousse et al. , 1994). Discovered more than a decade ago,these sources have been widely developed and have ledto the first structural dynamics experiments at the fem-tosecond time scale (Cavalleri et al. , 2001; Rischel et al. ,1997; Rose-Petruck et al. , 1999; Rousse et al. , 2001a;Siders et al. , 1999; Sokolowski-Tinten et al. , 2003, 2001).With recent developments, laser systems can deliver fo-cused intensities above 10 W/cm and the laser-plasmainteraction has entered the relativistic regime (Mourou et al. , 2006; Umstadter, 2003). At this light intensity,relativistic effects become significant. Electrons can beaccelerated within the laser field or in the wakefield ofthe laser up to relativistic energies (Esarey et al. , 2009,1996b; Everett et al. , 1994; Joshi et al. , 1984; Malka et al. ,2002; Modena et al. , 1995; Patel, 2007; Tajima and Daw-son, 1979; Umstadter et al. , 1996a). In particular, laserwakefield acceleration has led to the production of high-quality femtosecond relativistic electron bunches (Faure et al. , 2004, 2006; Geddes et al. , 2004; Mangles et al. ,2004) created and accelerated up to the gigaelectron-volt level (Hafz et al. , 2008; Kneip et al. , 2009; Leemans et al. , 2006) within only a few millimeters or centimetersplasma. Using these relativistic electrons, several novelx-ray source schemes have been proposed over the pastdecades to produce collimated and femtosecond radiationin a spectrum ranging from the soft x rays to gammarays. Most of these schemes are based on the wigglingof relativistic electrons accelerated in a laser wakefield.In this article, the physics of these sources is reviewed,and the opportunities offered by these relativistic elec-trons to generate ultrashort x-ray radiation (Catravas et al. , 2001; Fritzler et al. , 2003; Gr¨uner et al. , 2007;Hartemann et al. , 2007; Jaroszynski et al. , 2006; Lee-mans et al. , 2005; Malka et al. , 2008; Nakajima, 2008)are highlighted. These sources can deliver x rays orgamma rays as short as a few femtoseconds, as they in-herit the temporal profile of the laser-plasma electronbunch, whose few-femtosecond duration was recently ex-perimentally demonstrated (Lundh et al. , 2011).The aim of this article is to review the novel x-raysources based on relativistic laser and underdense plasmainteraction and to highlight their similitude by using acommon formalism for their description. The paper isorganized as follows. In Sec. II, the general formalism ofradiation from an accelerated relativistic electron is pre-sented, which provides a framework for the descriptionof the sources discussed throughout the paper. Fromthis formalism, the relevant parameters describing theproperties of the radiation, such as its spectrum, diver-gence, number of emitted photons and duration, can beextracted. As the x-ray sources presented here are basedon electrons accelerated by laser wakefields [ i.e. , by thelaser wakefield accelerator (LWFA)], a description of themost efficient laser-based electron accelerator to date isgiven in Sec. III.In Secs. IV, V and VI, different methods for the pro-duction of incoherent x rays from relativistic electrons arereviewed; the objective is to define the relevant regimesto accelerate and wiggle electrons in such a way that theyemit x rays. In Sec. IV, betatron radiation is described.In that case, a plasma cavity created in the wake of anintense laser pulse acts as both an electron acceleratorand a wiggler (referred to as a plasma undulator). Thescheme presented in Sec. V relies on the use of laser wake-field accelerated electrons, transported and wiggled in ameter-scale periodic arrangement of permanent magnets(conventional undulator). This method is the closest tosynchrotron technology. In Sec. VI, the nonlinear Thom-son scattering and the Thomson backscattering sourcesare reviewed. In nonlinear Thomson scattering, electronsare directly accelerated and wiggled in an intense laserfield. For the Thomson backscattering case, the plasmais used to accelerate electrons which are then wiggled ina counterpropagating electromagnetic wave (EM undula-tor).In Sec. VII, an introduction to the topic of the free-electron laser is given and the conditions for realizingsuch an ultrahigh-brightness and coherent source of ra-diation with wavelength down to the angstrom (hard xrays) from laser-accelerated electrons are discussed fordifferent types of undulators (plasma, conventional, andEM).Other radiation sources based on the laser-plasma in-teraction have been developed and could provide pho-tons in the keV range. For example, high-order harmon-ics from gas (Brabec and Krausz, 2000; Corkum, 1993;Krausz and Ivanov, 2009; Protopapas et al. , 1997) or solidtargets (Dromey et al. , 2006; Tarasevitch et al. , 2000;Teubner and Gibbon, 2009; Thaury et al. , 2007), sources based on the flying mirror concept (Bulanov et al. , 2003;Esirkepov et al. , 2009; Kando et al. , 2007; Mourou et al. ,2006), or the K α source (Kieffer et al. , 1993; Murnane et al. , 1991; Rousse et al. , 1994) can produce femtosec-ond XUV or x-ray radiation. However, they are not basedon the same physical principle of acceleration and wig-gling of relativistic electrons, and will therefore not bereviewed here. II. GENERAL FORMALISM: RADIATION FROMRELATIVISTIC ELECTRONS
In this section, the radiation from relativistic electronsis introduced. A qualitative understanding of the phe-nomenon is highlighted and analytical results for the rel-evant general parameters determining the radiation fea-tures are given. The formalism described here is generaland will be common to all the sources presented through-out the article. We follow the approach of Jackson (2001)and provide the basic results necessary for the remainderof the paper. The interested reader is referred to Wiede-mann (2007a) for a complete and detailed description ofsynchrotron radiation from bending magnets, undulator,and wiggler insertion devices (e.g. for the angular distri-bution of individual undulator harmonics, for polariza-tion or spatial and temporal coherence properties).
A. Radiation features
Relativistic electrons can produce bright x-ray beamsif their motion is appropriately driven. For all the laser-based x-ray sources discussed in this article, the radia-tion mechanism is the emission from accelerated relativis-tic electrons. The features of this relativistically movingcharge radiation are directly linked to the electron tra-jectories. Obtained from the Li´enard-Wiechert field, thegeneral expression that gives the radiation emitted byan electron, in the direction of observation (cid:126)n , as a func-tion of its position, velocity, and acceleration along thetrajectory is written (Jackson, 2001) d Idωd
Ω = e π (cid:15) c × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) + ∞−∞ e iω [ t − (cid:126)n.(cid:126)r ( t ) /c ] (cid:126)n × (cid:104) ( (cid:126)n − (cid:126)β ) × ˙ (cid:126)β (cid:105) (1 − (cid:126)β.(cid:126)n ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (1)This equation represents the energy radiated within aspectral band dω centered on the frequency ω and a solidangle d Ω centered on the direction of observation (cid:126)n . Here (cid:126)r ( t ) is the electron position at time t , (cid:126)β is the velocityof the electron normalized to the speed of light c , and˙ (cid:126)β = d(cid:126)β/dt is the usual acceleration divided by c . Westress that this expression assumes an observer placed ata distance far from the electron so that the unit vector (cid:126)n is constant along the trajectory. The expression (1)for the radiated energy shows an important number ofgeneric features:1. When ˙ (cid:126)β = 0, no radiation is emitted by the elec-tron. This means that the acceleration is responsi-ble for the emission of electromagnetic waves fromcharged particles.2. According to the term (1 − (cid:126)β.(cid:126)n ) − , the radiated en-ergy is maximum when (cid:126)β . (cid:126)n →
1. This conditionis satisfied when β (cid:39) (cid:126)β (cid:107) (cid:126)n . Thus, a rela-tivistic electron ( β (cid:39)
1) will radiate orders of mag-nitude higher than a nonrelativistic electron, andits radiation will be directed along the direction ofits velocity. This is simply the consequence of theLorentz transformation: for an electron emittingan isotropic radiation in its rest frame, the Lorentztransformation implies that the radiation is highlycollimated in the small cone of typical opening an-gle of ∆ θ = 1 /γ around the electron velocity vector,when observed in the laboratory frame (see Fig. 1).In the following, we consider ultrarelativistic elec-trons, γ (cid:29)
1, and all angles which will be definedare supposed to be small so that tan θ (cid:39) sin θ (cid:39) θ .3. The term ( (cid:126)n − (cid:126)β ) × ˙ (cid:126)β , together with the relation˙ (cid:126)β (cid:107) ∝ (cid:126)F (cid:107) /γ and ˙ (cid:126)β ⊥ ∝ (cid:126)F ⊥ /γ between applied forceand acceleration (respectively for a force longitudi-nal or transverse with respect to the velocity (cid:126)β ),indicate that applying a transverse force (cid:126)F ⊥ (cid:126)β ismore efficient than a longitudinal force. The termalso shows that the radiated energy increases withthe square of the acceleration ˙ (cid:126)β . More precisely, P ∝ F (cid:107) and P ∝ γ F ⊥ , where P is the radiatedpower. Thus, it is much more efficient to use atransverse force in order to obtain high radiatedenergy.4. The phase term e iω [ t − (cid:126)n.(cid:126)r ( t ) /c ] can be locally ap-proximated by e iω (1 − β ) t . The integration over timewill give a nonzero result only when the integrand,excluding the exponential, varies approximately atthe same frequency as the phase term which oscil-lates at ω ϕ = ω (1 − β ). Given that the velocity (cid:126)β of the electron varies at the frequency ω e − , thecondition ω ϕ ∼ ω e − is required to have a nonzeroresult. The electron will radiate at the higher fre-quency ω = ω e − / (1 − β ) (cid:39) γ ω e − . Thus, the usualDoppler upshift is directly extracted from this gen-eral formula. This indicates the possibility to pro-duce x-ray beams ( ω X ∼ s − ) by wiggling arelativistic directional electron beam at a frequencyfar below the x-ray range: ω e − (cid:39) ω X / (2 γ ). This analysis underlines the directions for the produc-tion of x rays from relativistic electrons: the goal forx-ray generation from relativistic electrons is to force arelativistic electron beam to oscillate transversally. Thistransverse motion will be responsible for the radiation.This is the principle of synchrotron facilities, where a pe-riodic static magnetic field, created by a succession ofmagnets, is used to induce a transverse motion to theelectrons. The laser-based sources presented here relyon this principle. In the next sections, the propertiesof moving charge radiation in two different regimes arereviewed. The different laser-based x-ray sources whichwill be discussed can work in both regimes depending onthe interaction parameters. B. Two regimes of radiation: undulator and wiggler
We consider ultrarelativistic electrons with a velocityalong the direction (cid:126)e z executing transverse oscillations inthe (cid:126)e x direction. Two regimes can be distinguished.The undulator regime corresponds to the situationwhere an electron radiates in the same direction at alltimes along its motion, as shown in Fig. 1. This occurswhen the maximal angle of the trajectory ψ is smallerthan the opening angle of the radiation cone ∆ θ = 1 /γ .The wiggler regime differs from the undulator by thefact that the different sections of the trajectory radiatein different directions. Thus, emissions from the differ-ent sections are spatially decoupled. This occurs when ψ (cid:29) /γ . The fundamental dimensionless parameterseparating these two regimes is K = γψ . The radiationproduced in these two regimes have different qualitativeand quantitative properties in terms of spectrum, diver-gence, and radiated energy and number of emitted pho-tons. C. Qualitative analysis of the radiation spectrum
The shape of the radiation spectrum can be determinedusing simple qualitative arguments. In most of the casesdiscussed throughout the article the electron trajectorycan be approximated by a simple transverse sinusoidaloscillation of period λ u at a constant velocity β and con-stant γ . The orbit can be written as x ( z ) = x sin( k u z ) = ψk u sin( k u z ) = Kγk u sin( k u z ) , (2)where k u = 2 π/λ u is the wave-vector norm, x is thetransverse amplitude of motion, and ψ is the maximumangle between the electron velocity and the longitudinaldirection (cid:126)e z . Since the electron energy is constant, anincrease of the transverse velocity leads to a decrease ofthe longitudinal velocity. This can be explicitly derived ZY ZYXX Ψ e - orbit e - o r b i t UndulatorWiggler Δϑ ∼ γΔϑ ∼ γ K<<1K>>1
FIG. 1 Illustration of the undulator and wiggler limits, atthe top and the bottom, respectively. The lobes represent thedirection of the instantaneously emitted radiation. ψ is themaximum angle between the electron velocity and the prop-agation axis (cid:126)e z and ∆ θ is the opening angle of the radiationcone. When ψ (cid:28) ∆ θ (undulator), the electron always radi-ates in the same direction along the trajectory, whereas when ψ (cid:29) ∆ θ (wiggler), the electron radiates toward different di-rections in each portion of the trajectory. from the assumed trajectory (2), β z (cid:39) β (cid:18) − K γ cos ( k u z ) (cid:19) , (3) β z (cid:39) β (cid:0) − K γ (cid:1) (cid:39) − γ (cid:0) K (cid:1) . (4)With the trajectory of the electron periodic, the emit-ted radiation is also periodic since each time the electronis in the same acceleration state, the radiated amplitudeis identical. The period of the radiated field can be cal-culated to obtain the fundamental frequency of the radi-ation spectrum. The radiation emitted in the direction (cid:126)n , forming an angle θ with the (cid:126)e z direction, is consid-ered, as represented in Fig. 2. The field amplitude (cid:126)A radiated in the direction (cid:126)n by the electron at z = 0 and t = 0 propagates at the speed of light c . At z = λ u and t = λ u / ( β z c ), the electron radiates an amplitude (cid:126)A = (cid:126)A . The distance separating both amplitudes ( (cid:126)A and (cid:126)A ) corresponds to the spatial period λ of the radi-ated field and is given by λ = λ u β z − λ u cos θ (cid:39) λ u γ (1 + K γ θ ) . (5)The radiation spectrum consists necessarily in the funda-mental frequency ω = 2 πc/λ and its harmonics. To knowif harmonics of the fundamental are effectively present inthe spectrum, it is instructive to look at the electron mo-tion in the average rest frame, moving at the velocity β z in the (cid:126)e z direction with respect to the laboratory frame. Z e - o r b i t λ YX λ u λ u / β z ϑ A A FIG. 2 Schematic for the calculation of the spatial period λ ofthe radiation emitted toward the direction of observation (cid:126)n ,forming an angle θ with the (cid:126)e z direction. At the two positionsmarked by a blue point, the electron radiates the same fieldamplitude. The distance between these two amplitudes at agiven time corresponds to λ . If K (cid:28)
1, the longitudinal velocity reduction is negli-gible: β z (cid:39) β and γ z = 1 / (cid:113) − β z (cid:39) γ . The motioncontains only the fundamental component. Indeed, themotion reduces to a simple dipole in the average restframe. The spectrum consists in a single peak at thefundamental frequency ω which depends on the angle ofobservation θ . As K →
1, radiation also appears at har-monics.If K (cid:29)
1, the longitudinal velocity reduction is sig-nificant: γ z = γ/ (cid:113) K . In the average rest frame,the trajectory is a figure-eight motion. It can containmany harmonics of the fundamental. In the laboratoryframe, this can be explained by the fact that an ob-server receives short bursts of light of duration τ . In-deed, the instantaneous radiation is contained within acone of opening angle ∆ θ = 1 /γ centered on (cid:126)β and pointstoward an observer positioned in the direction (cid:126)n duringa time ∆ t (see Fig. 3), corresponding to the variationof (cid:126)β by an angle ∆ θ = 1 /γ . Locally, a portion of thetrajectory can be approximated by a portion of a cir-cle of radius ρ , such that the direction of the velocity (cid:126)β changes by an angle ∆ θ when the electron travels adistance d e = 2 πρ × (∆ θ/ π ) = ρ/γ , which correspondsto a time ∆ t = t e = d e / ( βc ). During the time ∆ t , theradiation has covered a distance d γ = 2 ρ sin(1 / γ ) cor-responding to a propagation time of t γ = d γ /c . Theradiation burst duration τ as observed by an observerreads τ = t e − t γ (cid:39) ρ γ c . (6)The temporal profile (see the inset of Fig. 3) of the radia-tion emitted in the wiggler regime has been qualitativelyobtained. The Fourier transform of this typical profilegives a precise representation of the radiation spectrum.With the time profile a succession of bursts of duration τ , the spectrum will contain harmonics up to the criticalfrequency ω c ∼ /τ ∼ γ cρ . (7) e - orbit Δϑ ∼ γ Observer time P λ /c τ β z t t + Δ t FIG. 3 In the wiggler limit, the radiation cone points towardthe observer during a time ∆ t , which corresponds to a du-ration τ for the emitted radiation. This is repeated at eachperiod: the observer receives bursts of radiation separatedby a time λ/c . The inset gives the temporal profile of theradiation power seen by the observer. Note that the spectrum that arises from a complete cal-culation of the radiation emitted by a relativistic chargedparticle in instantaneously circular motion (Jackson,2001) is in agreement with the above estimation. Suchcalculation yields the synchrotron spectrum, which iswritten in terms of radiated energy per unit frequencyand per unit time, dPdω = P γ ω c S ( ω/ω c ) ,S ( x ) = 9 √ π x (cid:90) ∞ x K / ( ξ ) dξ,P γ = e cγ π(cid:15) ρ = 2 e ω c π(cid:15) cγ ,ω c = 32 γ cρ , (8)where we introduced the exact definition of the criticalfrequency ω c of the synchrotron spectrum which will beused throughout the review, P γ = (cid:82) dP/dω is the radi-ated power, and K / is the modified Bessel function ofthe second kind.The expression for the radius of curvature ρ can beobtained for an arbitrary trajectory. Its value can becalculated at each point of the trajectory, correspondingto a particular direction of observation. In the particularcase of a sinusoidal trajectory, the radius of curvaturereads ρ ( z ) = ρ [1 + ψ cos ( k u z )] / / | sin( k u z ) | and itis minimum when the transverse position is extremum( x = ± x ) and its value at this point reads ρ = λ u πψ = γ λ u πK , (9)leading to the following expression for the critical fre- quency: ω c = 32 Kγ πc/λ u (10)= 34 Kω { K (cid:28) ,θ =0 } . (11)Note that for a nonplanar and nonsinusoidal trajectory,the spectrum extends up to a critical frequency deter-mined by the minimal radius of curvature of the trajec-tory.The parameter K can be considered as the number ofdecoupled sections of the trajectory. With each sectionradiating toward a different direction, the radiation isspatially decoupled and leads to bursts of duration τ ineach direction and to a broad spectrum with harmonicsof the fundamental up to ω c .An infinite periodical motion has been considered sofar, leading to harmonics that are spectrally infinitelythin. For a finite number of oscillation periods N , Fouriertransform properties imply that the harmonic of number n and of wavelength λ n = λ/n has a width given by∆ λ n λ n = 1 nN . (12)This corresponds to the harmonic bandwidth in a givendirection characterized by the angle θ . Since the wave-length λ n depends on the angle θ [see Eq. (5)], the inte-gration of the radiation over a small aperture of finite di-mension broadens each harmonic. When integrating overthe total angular distribution, harmonics overlap and theradiation spectrum becomes continuous, but keeps thesame extension (up to ω c ). In the undulator limit, inwhich only the fundamental wavelength is present, thebandwidth is highly degraded when integrating over thetotal angular distribution. D. Duration and divergence of the radiation
The pulse duration and the divergence of the radiationemitted by a single electron can be deduced from theprevious analysis. If N is the number of periods of thetrajectory, the radiation consists in N periods of length λ and the total duration of the pulse is τ r | Ne =1 = N λ/c .For an undulator, K (cid:28)
1, the direction of the vector (cid:126)β varies along the trajectory by an angle ψ negligible com-pared to ∆ θ = 1 /γ . Hence, the radiation from all sectionsof the trajectory overlap and the typical opening angle ofthe radiation is simply θ r = 1 /γ . For a wiggler, K (cid:29) For
K <
1, the root mean square angle of the angular distributionof the radiated energy at the fundamental frequency is (cid:104) θ (cid:105) / =1 /γ z , which simplifies to 1 /γ for K (cid:28) the divergence increases in the direction of the transverseoscillation (cid:126)e x , but remains identical in the orthogonal di-rection (cid:126)e y . In the direction of the motion (cid:126)e x , the typicalopening angle of the radiation is θ Xr = ψ = K/γ , whichis greater than ∆ θ = 1 /γ , while the typical opening an-gle of the radiation in the direction (cid:126)e y is θ Y r = 1 /γ . Forthe general case of a transverse motion occurring in the (cid:126)e x and (cid:126)e y directions, the angular profile can take variousshapes depending on the exact three-dimensional (3D)trajectory. E. Analytical formulas for the total radiated energy and thenumber of emitted photons
Analytical calculations provide simple expressions forthe radiated energy and the number of emitted photonsper period. Using the expression of the radiated power byan electron P ( t ) = ( e / π(cid:15) c ) γ [( d ˆ (cid:126)p/dt ) − ( dγ/dt ) ] (ˆ (cid:126)p isthe momentum normalized to mc ), the averaged radiatedpower P γ and the total radiated energy per period I γ canbe derived for both the undulator and the wiggler case foran arbitrary trajectory. In the case of a planar sinusoidaltrajectory, the result is P γ = πe c (cid:15) γ K λ u , (13) I γ = πe (cid:15) γ K λ u . (14)To obtain an estimation of the number of emitted pho-tons N γ , a mean energy of photons must be determined.For K (cid:28)
1, the spectrum is quasimonochromatic in theforward direction and the mean energy of photons, af-ter integrating over the angular distribution, is equal to (cid:126) ω θ =0 /
2. The number of emitted photons reads N γ = 2 π αK , (15)where α = e / (4 π(cid:15) (cid:126) c ) is the fine structure constant.For K (cid:29)
1, the spectrum is synchrotronlike with the crit-ical frequency ω c . Using the fact that for a synchrotronspectrum (cid:104) (cid:126) ω (cid:105) = (8 / √ (cid:126) ω c , the following estimationis derived: N γ = 5 √ π αK. (16) F. Radiation from an ideal electron bunch
The radiation from a single electron has been discussedso far. We now take into account the fact that there are Here the typical opening angle is defined as the maximum deflec-tion angle of the electron trajectory ψ , such that the full widthof the angular distribution of the radiated energy in the direction (cid:126)e x is 2 K/γ . N e electrons contained in the bunch, assuming that theyall have exactly the same energy and the same initialmomentum (zero emittance). The radiation from severalelectrons is obtained by summing the contribution of eachelectron before taking the squared norm, d Idωd
Ω = e π (cid:15) c × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N e (cid:88) j =1 (cid:90) + ∞−∞ e iω ( t − (cid:126)n.(cid:126)r j ( t ) /c ) (cid:126)n × (cid:104) ( (cid:126)n − (cid:126)β j ) × ˙ (cid:126)β j (cid:105) (1 − (cid:126)β j .(cid:126)n ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (17)This formula expresses the coherent addition of the ra-diation field of each electron. It can be considerablysimplified by considering that all electrons follow simi-lar trajectories linked to each other by a spatiotemporaltranslation ( t j , (cid:126)R j ) of a reference trajectory (cid:126)r ( t ): (cid:126)r j ( t ) = (cid:126)R j + (cid:126)r ( t − t j ) ,d Idωd
Ω = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N e (cid:88) j =1 e iω ( t j − (cid:126)n. (cid:126)R j /c ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e π (cid:15) c × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) + ∞−∞ e iω ( t − (cid:126)n.(cid:126)r ( t ) /c ) (cid:126)n × (cid:104) ( (cid:126)n − (cid:126)β ) × ˙ (cid:126)β (cid:105) (1 − (cid:126)β.(cid:126)n ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (18)The radiated energy per unit frequency and unit solidangle from an electron bunch is equal to the radiatedenergy from a single electron following the trajectory (cid:126)r ( t )multiplied by the coherence factor c ( ω ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N e (cid:88) j =1 e iω ( t j − (cid:126)n. (cid:126)R j /c ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (19)The value of c ( ω ) depends on the electron distribution inthe ( t j , (cid:126)R j ) space. For a uniform distribution, c ( ω ) = 0,whereas for a random distribution, c ( ω ) = N e on aver-age. If the distribution is microbunched at the wave-length λ b = 2 πc/ω b , the summation is coherent for thefrequency ω b and its harmonics, c ( nω b ) = N e for n ∈ N ∗ .In large accelerators or in laser-plasma accelerators,electrons are randomly distributed inside the bunch atthe x-ray wavelength scale, and the radiation is incoher-ently summed, c ( ω ) = N e , (20) d Idωd Ω (cid:12)(cid:12) N e = N e d Idωd Ω (cid:12)(cid:12) N e =1 . (21)The spectrum shape and the radiation divergence θ r re-main unchanged for an electron bunch. The temporalprofile of the radiation is given by the convolution be-tween the electron bunch temporal profile and the ra-diation profile from a single electron. Since the typicalelectron bunch length is in the micron range, whereasthe radiation length from a single electron l r | Ne =1 = N λ is in the nanometer range for x rays, the duration of theradiation from an electron bunch is in most cases approx-imately equal to the bunch duration, τ r | Ne = τ b .In addition, the electron experiences the radiation fromother electrons in the case of a bunch, which can modifyits motion and its energy. This interaction of the bunchwith its own radiation can lead to a microbunching of theelectron distribution within the bunch at the fundamen-tal wavelength of the radiation and its harmonics. This isthe free-electron laser (FEL) process which produces co-herent radiation. Here the coherence factor c ( nω b ) is N e instead of N e , indicating that the FEL radiates orders ofmagnitude higher than conventional synchrotrons. How-ever, the FEL effect requires stringent conditions on theelectron beam quality and an important number N ofoscillations. At the present status of laser-plasma accel-erators, realizing a FEL represents a technological chal-lenge. In the following sections, the interaction betweenthe electron bunch and its radiation will not be taken intoaccount because in these schemes the conditions requiredfor the FEL are not fulfilled: the electron distribution re-mains random along the propagation and the radiationis incoherent. In Sec. VII, the underlying physics ofthe free-electron laser is presented in more detail and thepossible realization of such high-brightness coherent ra-diation using laser-plasma accelerators is discussed. G. Radiation reaction
The radiation reaction (RR) corresponds to the ef-fect of the electromagnetic field scattered by an electronon itself, the so-called self-interaction (Hartemann, 2002;Jackson, 2001; Landau and Lifshitz, 1994). RR effectscan modify the electron trajectory and its energy andit is therefore important to define the range of param-eters for which these effects come into play. The mainsteps, followed by Dirac to derive the relativistically co-variant form of the self-force associated with RR, areto solve for the self-quadripotential A sµ scattered by theelectron in terms of Green’s functions, and then to calcu-late the associated electromagnetic force on the electron, F sµ = − e ( ∂ µ A sν − ∂ ν A sµ ) u ν , where u ν = dx ν /dτ is theelectron quadrivelocity with τ the electron proper time.This leads to the Dirac-Lorentz equation of motion for apointlike electron, dp µ dτ = − eF µν u ν + τ (cid:20) d p µ dτ − p µ m c (cid:18) dp ν dτ dp ν dτ (cid:19)(cid:21) , (22) where the first term is the Lorentz force with F µν the ex-ternal electromagnetic field tensor, the last two termsthat define the RR self-force correspond, respectively,to the Schott term and the radiation damping term, p µ = mu µ is the electron quadrimomentum and τ =2 r e / (3 c ) = e / (6 π(cid:15) mc ) = 6 . × − s, with r e theclassical electron radius. The Schott term τ d τ p µ = − d τ G µ accounts for the change of energy momentum G µ of the external field, while the radiation dampingterm − τ p µ ( d τ p ν d τ p ν ) / ( m c ) = − d τ H µ accounts forthe change of energy-momentum H µ of the scattered elec-tromagnetic wave (Hartemann, 2002).Several difficulties appear in the point electron modeldescribed by the Dirac-Lorentz equation (22). First,it admits unphysical runaway solutions with exponen-tially increasing acceleration, which can be eliminatedby requiring the Dirac-Rohrlich asymptotic conditionlim τ →±∞ d τ p µ = 0. Second, physical solutions presentsacausal preacceleration, i.e. , that electron momentumchanges before an external force is suddenly applied, ona time scale τ . Equation (22) can be approximatedby evaluating the RR self-force with the solution of thezeroth-order equation d τ p µ = − eF µν u ν (Landau andLifshitz, 1994). This yields the Landau-Lifshitz equa-tion which neither admits runaway solutions nor presentsacausal preacceleration behavior.For the case of an electron undulating according to Eq.(2), it is important to define the range of parameters forwhich radiation reaction comes into play and has to beincluded in the description of the electron motion and itsradiation. For relativistic electrons, the dominant term inthe radiation reaction comes from the energy momentumtransferred to the scattered electromagnetic wave [whilefor rest electrons, the radiation reaction describes the di-rect exchange of energy momentum between the externalfield and the scattered wave (Hartemann, 2002)]. Therate of energy loss ν γ for the electron can be estimatedfrom mc dγ/dt = − P γ , with P γ the average power ra-diated by the electron given by Eq. (13). It leads to γ ( t ) = γ / (1 + ν γ t ) with ν γ = τ γ K (cid:18) πcλ u (cid:19) , (23)and γ is initial gamma factor of the electron (Esarey,2000; Huang and Ruth, 1998; Koga et al. , 2005; Michel et al. , 2006; Telnov, 1997). Therefore, the radiation re-action can be neglected when the interaction durationor equivalently the number of oscillations satisfy respec-tively, τ (cid:28) ν − γ and N (cid:28) N RR = λ u / (2 π cτ γ K ).With conventional undulators (see Sec. V), RR will al-ways be negligible; for ∼
10 GeV electrons, λ u ∼ K ∼
1, the limiting number of period N RR is on theorder of 1 . × . For current and short-term laser-basedbetatron experiments (see Sec. IV), RR is negligible butin the long term or for electron beam driven plasma ac-celerators [e.g., parameters of the Facility for AdvancedAccelerator Experimental Tests (FACET) (Hogan et al. ,2010)], ∼
10 GeV electrons, λ u ∼ K ∼ N RR ∼ . × . For Thomson backscattering(see Sec. VI), GeV electrons colliding with a laser pulse ofstrength parameter K ∼
10 and wavelength λ = 0 . µ m( λ u = λ/
2) leads to N RR ∼
60. The radiation reactioncan be neglected in Thomson backscattering for sub-GeVelectron beams and laser pulses of strength parameter onthe order of unity, for which N RR (cid:38) × .In the realm of quantum electrodynamics (QED), theradiation reaction corresponds to the recoil experiencedby an electron due to consecutive incoherent photonemissions (Di Piazza et al. , 2010). Quantum effects be-come important when the electron energy loss associ-ated with the emission of a photon is on the order ofthe electron energy. Signatures of quantum effects canbe observed before entering this quantum regime. In-deed, quantum fluctuations, which imply that differentelectrons emit a different number of photons (that carrydifferent energies) and hence lose a different amount ofenergy, can lead to an observable increase of the elec-tron beam energy spread (Esarey, 2000). Experimen-tally, a study of the quantum regime is accessible in theframework of Compton scattering (Sec. VI). QED ef-fects, such as nonlinear Compton scattering (Bula et al. ,1996) and the production of electron-positron pairs fromlight (Burke et al. , 1997), were observed in the SLACE-144 experiment, where 46.6 GeV electron beams col-lided with relativistic laser pulses with intensities of 10 W.cm − . H. Real electron bunch: longitudinal and transverseemittance
In the previous section, ideal bunches with electrons atthe same energy and same momentum have been consid-ered in order to simply discuss the effect of summationof the radiation from each electron. However, in realis-tic bunches, electrons have slightly different energies andmomenta. More precisely, inside the bunch, electronsthat are at the same location can have different energiesand momenta. This implies, for example, that the bunchcannot be focused and compressed on an infinitely smallpoint. This limitation is fundamental and inherent to thebunch, it does not depend on practical realization. Theparameter which accounts for that is called the emittanceand is related to the volume occupied by the electronsin the 6D phase space ( x, y, z, p x , p y , p z ) at a given time(Humphries, 1990). The 6D phase volume is constantin time if only smooth external forces are applied and ifcollisions are neglected (the phase volume conservation isa consequence of the collisionless Boltzmann equation).The emittance reflects the quality of the electron beambecause it quantitatively indicates if electrons have thesame coordinates, direction, and energy. For a relativistic electron beam traveling in the (cid:126)e z di-rection, an emittance is defined for each dimension: the z one is called longitudinal and two others are trans-verse ( x and y ). For a uniform distribution with sharpboundary, the normalized emittance (cid:15) aN is defined asthe area occupied by electrons in the ( a, p a /mc ) spacedivided by π ( a = x, y, z ). But because realistic elec-tron beams have diffuse boundaries, the normalized root-mean-square (rms) emittance is used and defined as (cid:15) aN = (cid:112) (cid:104) ∆ a (cid:105) (cid:104) ∆ p a (cid:105) − (cid:104) ∆ a ∆ p a (cid:105) /mc, (24)with a = x, y, z and where ∆ a = a −(cid:104) a (cid:105) , ∆ p a = p a −(cid:104) p a (cid:105) .For transverse dimensions, it is convenient to use the unnormalized emittance (cid:15) a , which is related to the areaoccupied by electrons in the ( a, a (cid:48) ) trace space ( a = x, y ),where a (cid:48) (cid:39) p a /p z is the transverse angle with respect tothe propagation axis z , (cid:15) a = (cid:112) (cid:104) ∆ a (cid:105) (cid:104) ∆ a (cid:48) (cid:105) − (cid:104) ∆ a ∆ a (cid:48) (cid:105) , a = x, y. (25)It is generally expressed in π. mm . mrad. For cylindricallysymmetric beams, the emittance (cid:15) r (defined in the sameway by putting a = r ) can be used. The normalizedemittance is related to the unnormalized one by (cid:15) N = γβ(cid:15) and has the advantage of being conserved duringacceleration (in systems which preserve the emittance).In a focal plane, where there is no correlation betweenposition and angle, the emittance is simply the product ofthe rms transverse size σ a by the rms angular dispersion σ a (cid:48) : (cid:15) a = σ a σ a (cid:48) , a = x, y, r. (26)For incoherent radiation of electrons oscillating in un-dulator or wiggler devices, the emittance and energyspread have the following effects. The electron beamis focused in the device, with a transverse size σ anda divergence θ satisfying (cid:15) = σθ . Because of the angu-lar spread, the radiation angular distributions from eachsingle electron are slightly shifted from one another, lead-ing to a redshifted broadening of harmonic bandwidths[an electron with direction θ with respect to the prop-agation axis contributes on axis with higher wavelength λ = λ θ =0 (1 + γ z θ ), see Eq. (5)]. The energy spreadeffect is straightforward: electrons with different ener-gies radiate at slightly different wavelengths, leading to abroadening of harmonic bandwidths. These effects re-sult in a modified bandwidth given by (∆ λ n /λ n ) =(1 / ( nN )) + (2∆ γ/γ ) + ( γ z (cid:15) /σ ) . Hence, the band-width of an harmonic at a given direction comes fromthree different effects: the finite number of periods, theenergy spread, and the angular spread (which dependson the emittance).The transverse emittance is essential for transport con-siderations and applications such as the free-electronlaser. Concerning the FEL application, required condi-tions on the transverse emittance and the energy spread0will be given in Sec. VII. A smaller transverse emittancepermits one to transport or focus the electron beam ona smaller focal spot size. III. ELECTRON ACCELERATION IN PLASMA
The possibility to accelerate electrons in laser-produced plasmas was originally proposed by Tajima andDawson (1979). They suggested to use the intense elec-tric field of a relativistic plasma wave, created in the wakeof an intense laser pulse, to accelerate electrons to rela-tivistic energies. The main advantage of plasmas relieson their ability to sustain an accelerating gradient muchlarger (on the order of 100 GeV/m) than a conventionalradio frequency accelerating module (on the order of 10MeV/m). This means that electrons could be acceleratedup to 1 GeV in millimeter- or centimeter-scale plasmas(Clayton et al. , 2010; Froula et al. , 2009; Hafz et al. , 2008;Kneip et al. , 2009; Leemans et al. , 2006) while a few tensof meters would be necessary to reach the same energyin conventional accelerators.This acceleration method has experienced a remark-able development over the past decades, mainly thanksto the advent of high-intensity lasers and to a better un-derstanding of the physical mechanisms driving the accel-eration. Different plasma accelerator schemes have beendeveloped over the years, leading to electron buncheswith ever-increasing quality. The most efficient to dateis the so-called bubble, blowout or cavitated wakefieldregime (Lu et al. , 2006a,b, 2007; Pukhov et al. , 2004a;Pukhov and Meyer-ter Vehn, 2002). In that regime, de-pending on the chosen parameters, electron bunches cannow be produced with tunable energy in the hundreds ofMeV range (Faure et al. , 2006), low divergence (mrad), arelatively high charge ( ∼
100 pC), and a bunch durationof less than 10 fs (Davoine et al. , 2008; Faure et al. , 2004;Geddes et al. , 2004; Glinec et al. , 2007; Lundh et al. ,2011; Mangles et al. , 2004, 2006; Thomas et al. , 2007;van Tilborg et al. , 2006; Tsung et al. , 2006, 2004). Be-cause the x-ray sources which will be reviewed are basedon laser-plasma accelerators, this section is dedicated toa short description of wakefield acceleration in the cavi-tated regime. In particular two important physical mech-anisms are introduced: the ponderomotive force and the For an overview of the historical development of the field, seeAmiranoff et al. (1998, 1995); Bingham et al. (2004); Clayton et al. (1993); Coverdale et al. (1995); Esarey et al. (2009, 1996b);Everett et al. (1994); Gahn et al. (1999); Gordon et al. (1998);Joshi (2007); Joshi et al. (1984); Kitagawa et al. (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 et al. (2001); Tajima and Dawson (1979); Ting et al. (1997);Umstadter et al. (1996a); and Wagner et al. (1997). plasma wave. After a brief description of the character-istics of the acceleration mechanism, recent experimentalprogress is presented. We refer the interested reader tothe recent article of Esarey et al. (2009) for a completereview of laser-plasma electron accelerators.
A. Ponderomotive force and plasma waves
The ponderomotive force is a force associated with theintensity gradients in the laser pulse, that pushes bothelectrons and ions out of the high-intensity regions. Ions,being much heavier than electrons, still remain for shortinteraction times whereas electrons are cast away. Thisleads, in an underdense plasma, to the formation of arelativistic plasma wave whose fields can accelerate elec-trons. Here a short description of the ponderomotiveforce and of the excitation of a plasma wave (Kruer, 1988)is given.Because of the mass of the plasma ions, they can beconsidered motionless for short interaction times. Con-sidering a fluid description for the plasma electrons, theequation of motion for a fluid element submitted to theelectromagnetic force reads ∂ ˆ (cid:126)p∂t = ∂(cid:126)a∂t + c(cid:126) ∇ ( φ − γ ) , (27)where ˆ (cid:126)p = (cid:126)p/mc is the normalized momentum of an elec-tron fluid element, γ is the relativistic factor of an elec-tron fluid element, and φ = eV /mc and (cid:126)a = e (cid:126)A/mc are, respectively, the normalized scalar potential and thenormalized vector potential of the electromagnetic fields. (cid:126)a describes the high-frequency laser pulse, c(cid:126) ∇ φ is theCoulomb force associated with the charge distribution,and − c(cid:126) ∇ γ is the relativistic ponderomotive force whichexpels electrons away from the laser pulse. In the absenceof Coulomb and ponderomotive forces, the equation sim-plifies to ˆ (cid:126)p = (cid:126)a , corresponding to the fast electron oscilla-tion in the laser pulse. Depending on the amplitude of thelaser pulse normalized vector potential a , plasma elec-trons oscillate with relativistic velocities | (cid:126)v | (cid:39) c ( a > c ( a (cid:28) γ distribution and to a ponderomotive forcethat pushes plasma electrons from the high γ region (cor-responding to high a ) to the low γ region (low a ). Thisslow drift motion of the plasma electrons leads to a chargedensity distribution ρ responsible for a Coulomb force c(cid:126) ∇ φ (by virtue of the Poisson equation (cid:52) φ = ρ/(cid:15) ).On the other hand, a small charge density perturba-tion in a plasma oscillates at a characteristic frequency, Equation (27) assumes that (cid:126) ∇ × (ˆ (cid:126)p − (cid:126)a ) is initially zero, whichis the case in practice because both ˆ (cid:126)p and (cid:126)a are zero before thepassage of the laser pulse in the plasma. ω p = (cid:112) n e e /m(cid:15) , where n e isthe electron density of the plasma. For example, if wetranslate an electron slice (continuously, without cross-ing between different electrons) from its initial position,the slice will oscillate around its initial position at theplasma frequency because of the restoring Coulomb forcefrom the plasma ions. These plasma oscillations, calledplasma waves, are excited by the ponderomotive force ofthe laser pulse, which creates the initial charge densityperturbation. The phase velocity of the plasma wave v φ ,excited in the wake of the laser pulse, is approximatelyequal to the laser group velocity.For a low-intensity laser pulse, with a normalized vec-tor potential amplitude a (cid:28)
1, the excited plasma waveis linear and has a sinusoidal shape at the plasma fre-quency, φ ( (cid:126)r, t ) = φ ( (cid:126)r ) sin( ω p t ). For a higher intensity, a >
1, the plasma wave becomes highly nonlinear andcan involve transverse currents and a quasistatic mag-netic field (Gorbunov et al. , 1996).Relativistic plasma waves can be produced in variousregimes depending on the laser and plasma parameters.For appropriately chosen parameters, electrons can betrapped at a proper phase of the plasma wave and ex-perience its field over a distance sufficiently long to beaccelerated up to relativistic energies.In practical units, the plasma wavelength λ p = 2 πc/ω p and the laser strength parameter a are given by λ p [ µ m] = 3 . × / (cid:112) n e [cm − ] , (28) a = 0 . (cid:113) I [10 W/cm ] λ L [ µ m] , (29)where I is the laser intensity and λ L is the laser wave-length. B. The cavitated wakefield or bubble regime
To date, the most efficient mechanism to accelerateelectrons in a plasma wave is called the bubble, blowoutor cavitated wakefield regime. In this regime the wakeconsists of an ion cavity having a spherical shape (Lu et al. , 2006a,b, 2007; Pukhov and Meyer-ter Vehn, 2002).This regime is reached when the waist w of the focusedlaser pulse becomes matched to the plasma ( k p w =2 √ a , with k p = ω p /c ) and if the pulse duration is of theorder of half a plasma wavelength ( cτ ∼ λ p / a > + + + + + + + + + + ++ + + + + + + + ++ + + + + + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + r b Z Laserpulse
Ion cavity X Electron bunch Y F z FIG. 4 Principle of the electron acceleration in the bubbleregime. The laser pulse expels all plasma electrons out ofthe focal spot, leaving in its wake an ion cavity of radius r b .The longitudinal force F z accelerates the self-injected elec-tron bunch in the first half of the bubble, while the bunchdecelerates in the second half of the bubble. propagate before they reach that point is called the de-phasing length L d and is much larger than the bubbleradius r b , as electrons travel almost at the same speedas the wave phase velocity. If the process is turned offat that time, by setting the acceleration length close tothe dephasing length, electrons exit the plasma with themaximum energy gain. The maximum electric field andthe radius of the cavity are, respectively (Lu et al. , 2007), E m = mω p c √ a /e, (30) r b = w = (2 /k p ) √ a . (31)For the typical experimental condition accessible withpresent lasers, a = 4 with a 30 fs laser, the maximumelectric field is E m ∼
600 GeV/m and the radius of thecavity is r b ∼ µ m for an electron density of n e = 1 × cm − . C. Experimental production of relativistic electron bunches
Several techniques have been explored to produceplasma waves appropriate to accelerate electrons. Atypical experiment for laser-plasma acceleration consistsin focusing a laser pulse into a gas jet, the interac-tion parameters being essentially the laser intensity, fo-cal spot size and duration, the propagation length, andthe plasma density. Depending on the choice of these pa-rameters the features of the produced electron bunch canbe very different. Prior to 2004, experiments used rela-tively high plasma densities ( > cm ) and the pro-duced electron bunches were characterized by broadbandspectra, extending up to about 100 MeV. These spectrawere either nearly Maxwellian in the direct laser accel-eration (DLA) regime (Gahn et al. , 1999; Kneip et al. ,2008; Mangles et al. , 2005; Pukhov, 2003; Pukhov et al. ,21999; Tsakiris et al. , 2000) or non-Maxwellian in the self-modulated laser wakefield accelerator (SM-LWFA) andforced laser wakefield (FLWF) regimes (Coverdale et al. ,1995; Malka et al. , 2002; Modena et al. , 1995; Najmudin et al. , 2003; Santala et al. , 2001; Ting et al. , 1997; Um-stadter et al. , 1996a; Wagner et al. , 1997). In 2004, ma-jor advances were made by setting the laser pulse dura-tion close to the plasma period, increasing the interactionlength and matching the dephasing to the propagationlength. Three groups reported simultaneously on theproduction of monoenergetic electrons in the 100 MeVrange (Faure et al. , 2004; Geddes et al. , 2004; Mangles et al. , 2004), collimated within a few milliradians andwith charge on the order of 100 pC. However, despitethese remarkable progresses, the high nonlinearity of themechanism resulted in electron bunches which were nei-ther stable nor tunable in energy; in addition electronenergies remained below the GeV level.To overcome these limitations, improved schemes haverecently been developed. An external and controlled elec-tron injection into the wakefield was proposed (Esarey et al. , 1997; Fubiani et al. , 2004; Umstadter et al. , 1996b)and recently demonstrated (Faure et al. , 2007, 2006;Malka et al. , 2009). It consists of colliding the main laserpulse generating the plasma wave with a second laserpulse, creating a beat wave whose ponderomotive forcecan preaccelerate and locally inject background electronsin the wakefield (Davoine et al. , 2008). The energy ofthe electron bunch can be tuned by varying the collisionposition and therefore the acceleration length. Exper-iments demonstrated that stable electron bunches canbe produced with energy continuously tunable from afew tens of MeV to above 200 MeV (Faure et al. , 2007,2006; Malka et al. , 2009) and an energy spread on theorder of 1% (Rechatin et al. , 2009b). The few femtosec-ond duration and the few kA current of these electronbunches were experimentally demonstrated by Lundh et al. (2011). A cold optical injection providing nar-row energy spread for GeV electrons has been proposed(Davoine et al. , 2009).Several other possibilities to control electron injectionhave been investigated: the use of a plasma density down-ramp (Brantov et al. , 2008; Bulanov et al. , 1998; Faure et al. , 2010; Geddes et al. , 2008; Hemker et al. , 2002; Suk et al. , 2001), ionization-induced injection (Clayton et al. ,2010; McGuffey et al. , 2010; Pak et al. , 2010; Pollock et al. , 2011) or magnetically controlled injection (Vieira et al. , 2011).Laser-plasma accelerators have also recently reached theGeV level, using either external laser guiding (Leemans et al. , 2006; Nakamura et al. , 2007) or higher laser power(Clayton et al. , 2010; Froula et al. , 2009; Hafz et al. , 2008;Kneip et al. , 2009). In the first case, the experiment re-lied on channeling a few tens of TW-class laser pulsesin a gas-filled capillary discharge waveguide in order toincrease the propagation distance, and so the accelera- tor length, to the centimeter scale (Leemans et al. , 2006;Nakamura et al. , 2007; Rowlands-Rees et al. , 2008). Inthe experiment of Leemans et al. (2006), electrons havebeen accelerated up to a GeV.In order to further improve the quality of electronsfrom laser-plasma accelerators, several routes are pro-posed from the use of a PW-class laser to multistagedacceleration schemes (Gordienko and Pukhov, 2005; Lif-schitz et al. , 2005; Lu et al. , 2007; Malka et al. , 2006;Martins et al. , 2010). These foreseen developments areof major importance for the production of a free-electronlaser based on a laser-plasma accelerator. IV. PLASMA ACCELERATOR AND PLASMAUNDULATOR: BETATRON RADIATION
In the bubble acceleration regime, the plasma cavitycan act as a wiggler in addition to being an accelera-tor, reproducing on a millimeter scale the principle ofa synchrotron to produce x rays (Kiselev et al. , 2004;Rousse et al. , 2004). In this section, we will show a laser-produced ion cavity drives the electron orbits in such away that a short pulse of collimated x-ray radiation isemitted. Figure 5 represents the principle of the mecha-nism. As discussed, the bubble regime is reached when anintense femtosecond laser pulse, propagating in an under-dense plasma, evacuates plasma electrons from the high-intensity regions and leaves an ion cavity in its wake. Inaddition to the longitudinal force, responsible for the ac-celeration discussed above, the spherical shape of the ioncavity results in a transverse electric field producing arestoring force directed toward the laser pulse propaga-tion axis. Therefore, electrons trapped and accelerated inthe cavity are also transversally wiggled. The conditionsfor an efficient production of accelerated charged particleradiation, discussed in Sec. II, are therefore met. A colli-mated beam of x-ray radiation is emitted by the electronbunch. This radiation, which can be directly comparedto a synchrotron emission in the wiggler regime, is calledbetatron radiation.The betatron radiation from laser-produced plasmaswas simultaneously proposed and demonstrated in 2004(Kiselev et al. , 2004; Rousse et al. , 2004). This repre-sented a major step forward in the field of plasma x-ray sources since it was the first method allowing one toproduce bright collimated x-ray (keV) beams from laser-plasma interactions. Since then, this radiation has beenmeasured and widely characterized in interaction regimesfrom multiterawatt (Albert et al. , 2008; Shah et al. , 2006;Ta Phuoc et al. , 2005a, 2008a, 2006, 2007) to petawattlasers (Kneip et al. , 2008). According to the theory, fem-tosecond pulses of x rays up to a few tens of keV couldbe produced within tens of milliradian beams. In thefollowing sections, the properties of the betatron mech-anism are reviewed. Following the approach of Sec. II,3 + + + + + + + + + + ++ + + + + + + + ++ + + + + + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + r b ϑ ∼ K/ γ X - r a y s Z Laserpulse
Ion cavity X ElectronBetatronorbit Y FIG. 5 Schematic of the betatron mechanism. When an elec-tron is injected in the ion cavity, it is submitted not onlyto the accelerating force, but also to a restoring transverseforce, resulting in its wiggling around the propagation axis.Because of this motion, the electron radiates x rays whosetypical divergence is θ = K/γ . the electron orbit is first calculated using an ideal modeland then the features of the emitted radiation are de-rived. Simulations based on this model and a particlein cell (PIC) simulation are presented. Finally, after asummary of the experimental results, we conclude withthe short term developments foreseen. A. Electron orbit in an ion cavity
An idealized model of a wakefield in the bubble regimebased on the phenomenological description developed byLu et al. (2006a,b, 2007) is assumed. The phase ve-locity of the wake is expressed as v φ = v g − v etch (cid:39) c (1 − ω p / ω L ), where v g = c (cid:113) − ω p /ω L is the lasergroup velocity in the plasma, v etch (cid:39) cω p /ω L is the etch-ing velocity due to local pump depletion (Decker andMori, 1994, 1995; Decker et al. , 1996; Lu et al. , 2007), ω p = (cid:112) n e e /m(cid:15) is the plasma frequency, n e the elec-tron density of the plasma, and ω L the central frequencyof the laser field. The ion cavity is assumed to be a sphereof radius r b = (2 /k p ) √ a ( k p = ω p /c ) and a cylindricalcoordinate system ( r, θ, z ), where r is the distance fromthe laser pulse propagation axis is used. The comovingvariable is defined as ζ = z − v φ t so that ( ζ = 0 , r = 0),( ζ = r b , r = 0), and ( ζ = − r b , r = 0), respectively, cor-respond to the center, the front, and the back of thecavity. The electromagnetic fields of the wake are anaxial electric field E z , a radial electric field E r , and anazimuthal magnetic field B θ . They are given by E z /E = k p ζ/ E r /E = k p r/
4, and B θ /E = − k p r/ (4 c ), where E = mω p c/e is the cold nonrelativistic wavebreakingfield (Kostyukov et al. , 2004; Lu et al. , 2006a,b; Pukhov et al. , 2004a; Xie et al. , 2007). The equation of motion of a test electron in the cavity is then: d(cid:126)pdt = − e ( (cid:126)E + (cid:126)v × (cid:126)B ) = (cid:126)F (cid:107) + (cid:126)F ⊥ (cid:39) − mω p ζ(cid:126)e z + r(cid:126)e r ) , (32)where (cid:126)p is the momentum of the electron. The last ex-pression assumes p ⊥ (cid:28) p z . The electron is initially in-jected at the back of the cavity with space-time coor-dinates ( t i = 0 , x i , y i , z i ) such that x i + y i + z i = r b and ζ i <
0, and with a energy-momentum quadrivector( γ i mc , (cid:126)p i ) such that v z > v φ .The term (cid:126)F (cid:107) is responsible for the electron accelerationin the longitudinal direction (cid:126)e z . As the electron becomesrelativistic, its velocity becomes greater than v φ ; the term − ζ = v φ t − z decreases and the accelerating force is re-duced. The length for the electron to reach the middle( ζ = 0) of the cavity and to become decelerated ( ζ > L d . The term (cid:126)F ⊥ isa linear restoring force that drives the transverse oscilla-tions of the electron across the cavity axis at the betatronfrequency ω β (cid:39) ω p / √ γ .To derive the analytical expressions of the electron or-bit, we start the integration of the equation of motionfrom an initial state where γ zi = 1 / (cid:112) − ( v zi /c ) (cid:29) γ φ in order to ensure that the slippage between the test elec-tron and the wakefield is constant in time. This approx-imation is realistic because an electron quickly attains γ z (cid:29) γ φ and the transitory period in which the approx-imation fails is small compared to the dephasing time L d /c . Hence dζ/dt = v z − v φ (cid:39) c − v φ (cid:39) c/ (2 γ φ ) if γ φ (cid:29)
1. In agreement with the assumption used in thelast expression of Eq. (32), we perform a perturbativetreatment in the variable p ⊥ /p z (cid:28)
1. The variables witha hat are normalized by the choice m = c = e = ω p = 1and (cid:126)β = (cid:126)v/c is the velocity normalized to the speed oflight c . Equation (32) projected on each axis reads d ( γβ x ) d ˆ t = − ˆ x , (33) d ( γβ y ) d ˆ t = − ˆ y , (34) d ( γβ z ) d ˆ t = − ˆ ζ i − ˆ t γ φ . (35)At zero order ( p ⊥ = 0), Eq. (35) can be directly inte-grated to γ (ˆ t ) (cid:39) ˆ p z (ˆ t ) = γ d (1 − τ ) with γ d = γ i + γ φ ˆ ζ i / τ = (ˆ t − ˆ t d ) / (cid:113) ˆ t d + 8 γ φ γ i and ˆ t d = − γ φ ˆ ζ i the dephasingtime (ˆ t d = k p L d ). This parabolic profile of γ ( τ ) can beinserted into Eqs. (33)-(34) to obtain the first-order so-lution for ˆ x ( τ ) and ˆ y ( τ ). Note that motion in the (cid:126)e x and (cid:126)e y directions are decoupled at this order of calculation.Each equation takes the form of a Legendre differential4equation, ddτ (cid:104) (1 − τ ) d ˆ xdτ (cid:105) + ν ( ν + 1)ˆ x = 0 , (36) ddτ (cid:104) (1 − τ ) d ˆ ydτ (cid:105) + ν ( ν + 1)ˆ y = 0 , (37)where ν ( ν + 1) = 4 γ φ . The solution space is generatedby the Legendre functions of the first and second kinds P ν ( τ ) and Q ν ( τ ), which have the following asymptoticexpressions for ν (cid:29) P ν ( τ ) = (cid:115) πν (1 − τ ) / cos[( ν + 1 /
2) cos − τ − π/ , (38) Q ν ( τ ) = (cid:114) π ν (1 − τ ) / cos[( ν + 1 /
2) cos − τ + π/ . (39)Hence, the solutions for ˆ x ( τ ) and ˆ y ( τ ) can be written asˆ x ( τ ) = A x (1 − τ ) / cos(2 γ φ cos − τ + ϕ x ) , (40)ˆ y ( τ ) = A y (1 − τ ) / cos(2 γ φ cos − τ + ϕ y ) , (41)where ( A x , ϕ x ) and ( A y , ϕ y ) are constants which have tobe determined by initial conditions ( x i , p xi ) and ( y i , p yi ).They can be put into the simpler formˆ x (ˆ t ) = ˆ x β (ˆ t ) cos (cid:32)(cid:90) ˆ t ˆ ω β (ˆ t (cid:48) ) d ˆ t (cid:48) + ϕ (cid:48) x (cid:33) , (42)ˆ y (ˆ t ) = ˆ y β (ˆ t ) cos (cid:32)(cid:90) ˆ t ˆ ω β (ˆ t (cid:48) ) d ˆ t (cid:48) + ϕ (cid:48) y (cid:33) , (43)ˆ ω β (ˆ t ) = 1 / (cid:113) γ (ˆ t ) , (44)ˆ x β (ˆ t ) = A (cid:48) x /γ (ˆ t ) / , (45)ˆ y β (ˆ t ) = A (cid:48) y /γ (ˆ t ) / . (46)The transverse motion consists of sinusoidal oscilla-tions in each direction with time-dependent amplitudeˆ x β (ˆ t ) , ˆ y β (ˆ t ), and frequency ˆ ω β (ˆ t ). They depend on ˆ t onlythrough γ . For an arbitrary γ (ˆ t ) profile, Eqs. (42)-(46)can be derived from the adiabatic approximation and theWKB method [valid if (1 /ω β ) dω β /dt (cid:28) et al. , 2004;Thomas, 2010). Thus, Eqs. (44)-(46) in terms of γ aremore general than the frame of the above calculation [ i.e. ,the case of a parabolic γ (ˆ t ) profile]. The betatron ampli-tudes x β , y β decrease as γ − / during acceleration, whilethe betatron frequency ω β decreases as γ − / . Accord-ing to Eqs. (40)-(41), the number of betatron oscillationsbetween ˆ t = 0 and the dephasing time ˆ t = ˆ t d is approx-imately γ φ /
2. Depending on the values of ϕ (cid:48) x and ϕ (cid:48) y ,the motion in each direction (cid:126)e x and (cid:126)e y can be in phase (motion confined in a plane) or not (helical motion). Thelatter case corresponds to an initial state with nonzeroangular momentum L z = xp y − yp x . Finally, the lon-gitudinal motion ˆ z (ˆ t ) can be obtained by integrating β z which is deduced from γ = 1 / (1 − β x − β y − β z ).This trajectory is noticeably different from the onestudied in Sec. II. First, the betatron amplitude andfrequency are time dependent. Nevertheless, in the wig-gler regime, the resulting radiated energy dI/dω canbe seen as an integral over time of the radiated power dP/dω ( ω β , x β , y β ) for each instantaneous amplitude andfrequency if the characteristic time scale of acceleration ismuch longer than the betatron period. Second, x ( t ) and y ( t ) are sinusoidal functions of t and not z . Even if thedetail of the electron dynamics is slightly different fromthe case of Sec. II, the betatron trajectory has the sameproperties. As in Sec. II, the trajectory is a figure-eightmotion in the electron local average rest frame. The radi-ation features derived in Sec. II from the parameters K , λ u , and γ remain therefore valid. Assuming dt (cid:39) dz/c inEqs. (42)-(43) and identifying the terms with the onesof Eq. (2), the local electron period λ u ( t ) and the localstrength parameter K ( t ) read in physical units λ u ( t ) = (cid:112) γ ( t ) λ p , (47) K ( t ) = r β ( t ) k p (cid:112) γ ( t ) / , (48)in which we assume a motion confined in a plane witha betatron amplitude r β ( t ) = (cid:112) x β ( t ) + y β ( t ) . We re-cover the results obtained by Esarey et al. (2002) andKostyukov et al. (2003) in the case of an ion channelwithout longitudinal acceleration, but in which the pa-rameters are time dependent through the variation of γ .In the articles of Esarey et al. (2002) and Kostyukov et al. (2003), the constant of motion γ z = (cid:112) p z ap-pears in Eqs. (47) and (48) instead of γ . The relative dif-ference between γ and γ z is of the order of p ⊥ /p z and canbe neglected at the order where the trajectory has beencalculated. In this section, the definition γ z = (cid:112) p z has not been used because it is different from the usualone γ z = 1 / (cid:112) − β z (which is small compared to γ forwigglers, as discussed in Sec. II.C), commonly used inthe synchrotron and FEL communities.In practical units, Eqs. (47) and (48) read λ u [ µ m] = 4 . × (cid:112) γ/n e [cm − ] , (49) K = 1 . × − (cid:112) γn e [cm − ] r β [ µ m] . (50)In the equation of motion (32), the electromagneticfields correspond to an idealized cavitated regime. It as-sumes three conditions: ( i ) the matching conditions interms of focal spot size, laser pulse duration, and den-sity are perfectly met; ( ii ) the cavity is free of electrons;and ( iii ) the trapped electrons are only submitted to the5wakefields (there is no interaction with the electromag-netic fields of the laser pulse). In addition, in this ap-proach, initial conditions have to be arbitrarily chosensince the injection mechanism is not taken into account.If the three above conditions are not met, the followingeffects occur.First, the interaction regime is slightly different whenthe plasma wavelength is smaller than the focal spot sizeand the pulse duration (slightly off the matching condi-tions). In this regime, called the forced laser wakefieldregime (Malka et al. , 2002), the shape of the wake is notspherical anymore, the laser pulse does not necessarily ex-pel all electrons from the focal spot. The wakefields aretherefore different from the case of a cavity free of elec-trons. Nevertheless, the bubble model remains a goodapproximation to roughly determine the electron trajec-tories and the betatron radiation features.Second, the cavity cannot be considered free of elec-trons when injection occurs. Trapped electrons can mod-ify the fields of the cavity by the beam loading mechanism(Rechatin et al. , 2009a; Tzoufras et al. , 2008, 2009): theymodify the motion of background electrons and hence theresulting wakefields. As a result the acceleration is re-duced and, for massive electron injection, the term (cid:126)F (cid:107) in(32) can be largely overestimated.Third, electrons that have been accelerated can catchup with the back of the laser pulse. The interaction ofan electron with the laser pulse can increase its oscilla-tion amplitude r β in the direction of the laser polariza-tion, enhancing the betatron radiation. The interactionis complex and strongly dependent on the phase velocityof the laser pulse. It corresponds to the mechanism ofdirect laser acceleration (DLA), in which some electronscan be in betatron resonance, if the induced transversemotion of the laser pulse fields is in the same directionas the betatron motion at all times (Gahn et al. , 1999;Kneip et al. , 2008; Mangles et al. , 2005; Pukhov, 2003;Pukhov et al. , 1999; Tsakiris et al. , 2000). In the forcedlaser wakefield regime, electrons can catch up with thelaser pulse more easily, resulting in a brighter betatronradiation.Three-dimensional PIC simulations can be used to ac-curately describe al these effects, and to compute theelectron orbits and the emitted radiation. Simple nu-merical simulation, integrating the equation of motion,can be used as well to provide the additional effects bymodeling the laser pulse and by extracting effective ac-celeration and transverse fields from a PIC simulation.N´emeth et al. (2008) followed this approach and showedthat the betatron motion can be driven by the laser pulsein the polarization direction if electrons interact with it.Such a model, without the laser pulse fields, has also beenconsidered by Wu et al. (2009) who, for a given config-uration with ultrahigh intensity ( a = 20), reduced theacceleration field by a factor of 0.35 and the transversefields by a factor of 0.9 in order to fit the electron tra- jectories with the ones extracted directly from the PICsimulation. This highlights how the wakefields are over-estimated when Eq. (32) is considered without any cor-rection. B. Radiation properties
The betatron motion derived in the above simplemodel leads to the emission of synchrotron radiation re-ferred to as betatron radiation. It can be calculatedwithin the general formalism of the radiation from a mov-ing charge. The ion cavity acts as an undulator or awiggler with a period λ u ( t ) and strength parameter K ( t )which depends on the electron initial conditions upon in-jection into the cavity. We first consider the radiationproperties for constant λ u , K , and γ and then considerthe radiation produced by the electron with acceleration.
1. Without acceleration
The spectrum of the emitted radiation depends on theamplitude of K . For a small amplitude of the betatronoscillation K (cid:28)
1, the radiation is emitted at the fun-damental photon energy (cid:126) ω with a narrow bandwidth inthe forward direction ( θ = 0). As K → K (cid:29) (cid:126) ω c . Thesequantities are given by (cid:126) ω = (2 γ hc/λ u ) / (1 + K /
2) for
K < , (cid:126) ω c = 32 Kγ hc/λ u for K (cid:29) , (51)which gives in practical units (cid:126) ω [eV] = 5 . × − γ . (cid:112) n e [cm − ] for K (cid:28) , (cid:126) ω c [eV] = 5 . × − γ n e [cm − ] r β [ µ m] for K (cid:29) . (52)According to Sec. II, the radiation is collimated withina cone of typical opening angle θ r = 1 /γ in the undulatorcase. For a wiggler, the radiation is collimated within atypical opening angle K/γ in the electron motion plane( (cid:126)e x , (cid:126)e z ) and 1 /γ in the orthogonal plane ( (cid:126)e y , (cid:126)e z ).The x-ray pulse duration equals the electron bunch du-ration. Depending on the parameters and the electron in-jection process, it can be extremely short, a few fs (Lundh et al. , 2011), corresponding to a bunch length equal to avery small fraction of the bubble radius, or using massiveself-injection at high density, the bunch length can be onthe order of the bubble radius and τ r ∼ r b /c .The number of emitted photons can be estimated usingthe expressions of Sec. II. In practical units, the numberof photons emitted per period and per electron (at the6mean photon energy (cid:104) (cid:126) ω (cid:105) = 0 . (cid:126) ω c for the wiggler case)is N γ = 1 . × − K for K < ,N γ = 3 . × − K for K (cid:29) . (53)From the above expressions, an estimation of the radia-tion properties for constant λ u , K , and γ can be obtainedfor a typical parameter regime. A 100 MeV electron( γ (cid:39) n e = 2 × cm − is considered. The spatialperiod of the motion is λ u (cid:39) µ m. For K (cid:28)
1, thiselectron will emit betatron radiation at a fundamentalenergy (cid:126) ω (cid:39)
650 eV. In the same parameter regime butwith K ∼
10, typical of our experimental conditions, thecritical energy of the radiation is (cid:126) ω c ∼ L acc , the electron accelerates and ra-diates but its main contribution in terms of energy to thebetatron radiation comes from the part of the trajectorywhere its energy is maximal. If the electron is aroundits maximal energy during ∼ (cid:126) ω ∼ . N γ ∼
1. Considering that thenumber of electrons trapped into the ion cavity is on theorder of 10 − , the number of x-ray photons expected isin the range 10 − as well. Finally, in a typical parameterregime where K ∼
10 and γ ∼ ×
2. With acceleration
For an electron accelerating, the situation is more com-plex. In what follows, we consider the wiggler limit andwe derive an approximation for the radiation spectrumtaking into account the acceleration. This will be usefulfor discussing the scalings of betatron properties whichwill be presented in Sec. IV.E. The acceleration has acharacteristic parabolic profile γ ( τ ) = γ d (1 − τ ), with τ = (ˆ t − ˆ t d ) / (cid:113) ˆ t d + 8 γ φ γ i and ˆ t d = − γ φ ˆ ζ i the dephasingtime (ˆ t d = k p L d ), as shown in Sec. IV.A. Thus, fromEqs. (13), (45)-(48), and (51), we deduce ω β ( τ ) = ω β,d (1 − τ ) − / , (54) K ( τ ) = K d (1 − τ ) / , (55) (cid:126) ω c ( τ ) = (cid:126) ω c,d (1 − τ ) / , (56) P γ ( τ ) = P γ,d (1 − τ ) / , (57)where ω β,d , K d , (cid:126) ω c,d , and P γ,d are the values of theparameters at the dephasing time τ = 0, and P γ ( τ )is the radiated power averaged over one oscillation pe-riod [whose expression is given in Eq. (13) for a pla-nar trajectory, and is increased by a factor of 2 for a helical trajectory]. We consider that the spectrum (in-tegrated over angles) radiated per unit time is given by dP/dω = ( P γ /ω c ) S ( ω/ω c ), i.e. , a synchrotron spectrum[see Eq. (8)] in which P γ is replaced by P γ , which is ex-act for a helical trajectory (for which the radius of cur-vature ρ does not depend on the phase of the oscillation)but only approximate for a sinusoidal trajectory (because ρ varies during an oscillation and ω c corresponds to theminimal value of ρ in the oscillation). The radiation spec-trum then reads dIdω ≈ (cid:90) t d dPdω ( t ) dt ≈ t d (cid:90) − dτ (cid:34) P γ ( τ ) ω c ( τ ) S ( ω/ω c ( τ )) (cid:35) = P γ,d t d ω c,d S (cid:48) ( ω/ω c,d ) , (58)where the function S (cid:48) is defined as S (cid:48) ( x ) = (cid:90) − dτ (1 − τ ) / S [(1 − τ ) − / x ] . (59)Equation (58) highlights the fact that the radiation prop-erties are encoded into the values of the parameters ω β,d , K d , γ d , (cid:126) ω c,d and t d . The shape of the spectrum isnot described by the usual universal function S of thesynchrotron spectrum anymore, but by the function S (cid:48) defined above. This latter function takes into accountthe photons emitted at low energies during the accelera-tion. Indeed, Eq. (53) shows that the number of emit-ted photons in one period has a weak dependence on γ (only through K ∝ γ / ), whereas the radiated energyis strongly dependent on γ because photons are emit-ted at much higher energies for higher γ . As a result,the angular distribution of the radiated energy is dom-inated by the electron oscillations around maximal val-ues of the parameters, i.e. , around the dephasing time.Hence, to obtain rough estimations of the radiation prop-erties, the formulas given above for constant parameterscan be used, in which the values at the dephasing timeare inserted. The information on the spectrum will bevalid at high photon energies, but will underestimate thenumber of low-energy photons.Finally, it is interesting to compare the scalings of dI/dω and N γ for both cases of constant parameters andacceleration, dIdω (cid:12)(cid:12) γ =constant ∝ N β KS ( ω/ω c ) , (60) dIdω (cid:12)(cid:12) acceleration ∝ ω β,d t d K d S (cid:48) ( ω/ω c,d ) , (61)which leads to, respectively, N γ (cid:12)(cid:12) γ =constant ∝ N β K, (62) N γ (cid:12)(cid:12) acceleration ∝ ω β,d t d K d , (63)7 Z ( μ m) K ∼ Y ( μ m) X ( μ m) Z ( μ m) FIG. 6 Electron trajectory on the left and its gamma factoras a function of the longitudinal coordinate z on the right.The set of initial conditions is x i = 2 µ m, y i = 0, z i = ζ i = − r b + 0 . µ m = − . µ m, p xi = 0, p yi = 0, p zi = 25 mc and the laser-plasma parameters are n e = 1 × cm − and a = 4. for the dependence of the total number of emitted pho-tons, where N β is the number of betatron oscillations inthe γ = const case. The value ω β,d t d is also characteristicof the number of betatron oscillations for the accelerationcase. Thus, we see that the scalings are identical for bothsituations (parametrized either by the constant parame-ters or by the parameters at the dephasing time). Theonly difference resides in the exact shape of the spectrum,given by the function S (cid:48) instead of S . C. Numerical results
In this section, the case of an electron experiencingthe longitudinal acceleration force is treated numerically.The complete equation of the electron motion (32) is firstintegrated and then the features of the emitted radia-tion are calculated using the general formula of movingcharge radiation (1). Finally, a particle in cell simulationof the betatron radiation taking into account all possi-ble effects, including the injection process, is presented.This latter simulation is based on the extraction of thetrapped electron trajectories from the PIC code and thecalculation of the corresponding radiation via Eq. (1) inpostprocessing.
1. Test-particle simulation
An interaction regime accessible with tens of TW-classlasers and parameters typical of current experiments isconsidered. The left panel of Fig. 6 represents the orbitof a test electron in the ion cavity obtained by integrat-ing Eq. (32). The electron density is n e = 1 × cm − , the propagation length is set at 1 . a = 4 (only used to calcu-late the bubble radius r b = 6 . µ m). The test electronenters at the back of the cavity with x i = 2 µ m, y i = 0, z i = ζ i = − r b + 0 . µ m = − . µ m and p xi = 0, p yi = 0, p zi = 25 mc . The electron drifts along the longitu-dinal direction and oscillates in the transverse direction(Ta Phuoc et al. , 2005a). The betatron amplitude r β θ Y ( m r a d ) θ X (mrad) R a d i a t e d e n e r g y ( - J s r - ) -20 -10 0 10 20 34 FIG. 7 X-ray angular distribution calculated from the trajec-tory of Fig. 6. The color scale represents the radiated energyper unit solid angle.
A AB B R a d i a t e d e n e r g y ( a r b . un i t s ) Y ( μ m ) Y ( μ m ) X ( μ m)X ( μ m) θ Y ( m r a d ) θ Y ( m r a d ) -60 -40 -20 0 20 40 60-60 -40 -20 0 20 40 606040200-20-40-606040200-20-40-60 θ X (mrad) θ X (mrad) R a d i a t e d e n e r g y ( a r b . un i t s ) FIG. 8 Transverse trajectory (on the left) and angular profileof the corresponding emitted radiation (on the right, the colorscale representing the radiated energy per unit solid angle) fortwo different cases. A: The trajectory is three dimensional andhelical (elliptical in the transverse plane). B: The trajectoryis planar. Both electrons are accelerated up to γ (cid:39)
380 andthe density is 2 × cm − . decreases as the electron gains energy ( r β ∝ γ − / ) andpresents a minimum at the maximum electron energy.The right panel of Fig. 6 shows the γ factor as a functionof the longitudinal coordinate z . The electron is rapidlyaccelerated and acquires an energy up to ∼
240 MeV. It isdecelerated after the dephasing length has been reached.As discussed in Sec. IV.A, the acceleration field may belargely overestimated and several effects not taken intoaccount here can affect the acceleration process. Thus,the predicted maximum electron energy can be largelyoverestimated in this test-particle simulation. The shapeof the trajectory is determined by the initial conditions( x i , y i , z i ) and ( p xi , p yi , p zi ) and can take various forms:planar as in the present simulation, circular, or helical(Ta Phuoc et al. , 2008a, 2006).The radiation has been calculated using the generalformula (1) in which the orbit calculated above is in-cluded. Figure 7 represents the spatial distribution ofthe radiated energy. The transverse profile of the emit-ted radiation has the same symmetry as the transverse8 Energy (keV) P h o t o n s / . % B W
0 4 8 12 162 × -4 × -4 × -4 × -5 FIG. 9 Betatron radiation spectrum, integrated over anglesand corresponding to the trajectory of Fig. 6, in number ofphotons per 0.1% bandwidth and per electron. orbit. In our particular case, the main divergence is in the( (cid:126)e x , (cid:126)e z ) plane and the radiation beam is confined withina typical angle 1 /γ in the perpendicular direction. Thetypical opening angles in each direction are θ Xr ∼ θ Y r ∼ et al. , 2008a, 2006). As an example,the spatial distributions of the radiation produced by anelectron undergoing two types of transverse orbits arerepresented in Fig. 8. For minimum radius of curvaturealong the helical orbit (case A), the transverse electronmomentum is along (cid:126)e y , so the radiation is emitted upand down and the radiated energy is maximal. For max-imum radius of curvature, the transverse momentum isalong (cid:126)e x , so the radiation is emitted left and right and theradiated energy is minimal. Therefore, the x-ray beamprofile is mapping out p x /p z and p y /p z . Also note thatthe increase of the radiated energy with the electron en-ergy appears in the radiation angular profile. Measure-ment of the spatial distribution in the x-ray range cantherefore provide information on the transverse electronmomentum during their oscillation in the wakefield cav-ity, provided that the energy and p z are known. It canalso provide some insight on the initial conditions of elec-tron injection that determines the type of trajectory.Figure 9 presents the spectrum of the radiation in-tegrated over the spatial distribution of Fig. 7. In-stead of the radiated energy primarily given by Eq. (1),Fig. 9 shows the number of emitted photons per 0.1%bandwidth and per electron. The spectrum extendswell above the keV range with a significant number ofx-ray photons per electron. An estimate of the totalnumber of photons can be obtained on the basis of thenumber of electrons that are currently accelerated in alaser-plasma accelerator. Assuming 10 electrons gives ∼ photons / .
1% bandwidth (BW) between 1 and 10keV. The total photon number is on the order of 10 .
2. Particle In Cell simulation
We now look at a 3D PIC simulation in a typical exper-imental parameters range. We are using the CALDER-Circ code (Lifschitz et al. , 2009), which, for the cost ofa few 2D simulations, can provide the fully 3D trajecto-ries of particles in the plasma. Indeed, this model usesFourier decomposition of the electromagnetic fields in theazimuthal direction. The axially symmetric (or cylin-drical) assumption consists in computing only the singlemode m = 0. In order to be able to account for planarfields, such as the linearly polarized laser, we assume theso-called quasicylindrical geometry and compute only oneadditional mode, m = 1. The linearly polarized laser’swakefield in a ultrahigh-intensity regime is well describedby those first two modes since fields never stray far fromthe axial symmetry. The macroparticles, each represent-ing an assembly of electrons, evolve in a full 3D spacein which the 3D fields have been reconstructed from thetwo known Fourier modes. Monitoring macroparticle tra-jectories gives us the same information as a test-particlecode but with realistic wakefield, energies, and injectionproperties in space and time. The injected particles inthe code are detected by their energy. As soon as a par-ticle reaches 45 MeV, it is considered as being injectedand we keep track of its trajectory for the rest of thesimulation time. These trajectories thus begin after theinjection time of each single particle and finish at simula-tion end, after the particles come out of the plasma. Ap-proximately 10 macroparticles are tracked. Their threeposition coordinates and their γ factor are recorded ev-ery ∆ t traj = 100 dt comput (cid:39) dt comput is thesimulation time step. Their contribution to the radia-tion is computed similarly to the test-particle code [ i.e. ,using Eq. (1)]. We note, however, that to reduce thecomputing time, it is possible to use the synchrotronradiation formulas instead of Eq. (1), since betatronradiation occurs in the wiggler regime. For each elec-tron, the angle-integrated spectrum can be calculated by dI/dω = (cid:82) dP/dω ( t ) dt , where dP/dω is the synchrotronspectrum given in Eq. (8) and depends on the radius ofcurvature which can be evaluated numerically along thearbitrary electron trajectory. The angle-dependent spec-trum d I/dωd
Ω can also be obtained in the wiggler limitby using the “saddle point” method (Kostyukov et al. ,2003). The saddle points are defined as the points in theelectron trajectories where the velocity vector (cid:126)v is paral-lel to a given observation direction (cid:126)n . The total radiationspectrum is then obtained as the sum of the synchrotron-like bursts of all saddle points.A 30 fs (FWHM) laser pulse is focused at the begin-ning of the density plateau of a 4.2 mm wide gas jet, toa peak normalized electric field of a = 1 .
2. It is linearlypolarized in the (cid:126)e x direction and the focal spot size is 18 µ m (FWHM). The gas jet density profile consists in adensity plateau of 3 mm at n e = 1 . × cm − and the9 Y ( μ m ) Z ( μ m) Y ( μ m ) Z ( μ m)
900 910 920 930 9401930
FIG. 10 Electronic density in the z − y plane averaged over5 µ m in the x direction. Densities are given in cm − . z is thedistance from the beginning of the first density ramp. Differ-ent steps of the simulation are shown. First the bubble forms,then injection starts, a whole bunch of accelerated electronsis created inside the bubble and finally the bunch comes outof the bubble and is scattered. The dynamics of the bunch issuch that the trailing electrons have a much higher oscillationamplitude than leading electrons. edges are described by ramps varying linearly from 0 to n e = 1 . × cm − over 600 µ m. The simulation de-scribes all the successive steps of the bubble regime (seeFig. 10): penetration of the laser pulse in the plasma,formation of the bubble, injection of electrons at the backof the bubble and acceleration, wiggling of the electronbunch in the bubble, laser energy depletion and scatter-ing of the bunch in the plasma when it comes out of thefocusing transverse electric field of the bubble.Figure 11 shows the electron beam spectrum beforescattering. Most of the electrons are around 100 MeVbut the tail extends up to 350 MeV. The total charge ofthe bunch is 640 pC, i.e. , approximately 4 × electrons.Figure 12 focuses on a single particle. It illustratesthe fact that a particle in the bunch, first acceleratedand then dephased, follows a sinusoidal orbit around thepropagation axis and radiates the most when it has botha high γ and a strong perpendicular acceleration (whenthe curvature radius is small).In a more global point of view, Fig. 13 shows typicaltrajectories of electrons during their acceleration insidethe bubble in the polarization plane. Each line repre-sents the trajectory of one macroparticle in the code.The color of a line changes as the macroparticle is ac-celerated to higher energy (represented here by its rela-tivistic γ factor). The 500 trajectories plotted in Fig. 13are computed from macroparticles injected around thesame instant and thus in the same conditions. As the Energy (MeV) -1 -2 -3 -4 -5 -6 -7 -8 d N e / d E ( a r b . un i t s )
0 50 100 150 200 250 300 350 400 450
FIG. 11 Normalized electron beam spectrum before scatter-ing at z (cid:39) µ m. -4 -3 Radiated power (J/s)
X Z
FIG. 12 Trajectory, energy and radiated power of a singleelectron before scattering. The tube shows the particle’s tra-jectory. Its radius is proportional to the particle’s energy andthe color measures the instantaneous radiated power of theparticle. The orbit of this particle is in the polarization plan. behavior of a particle strongly depends on its injectionconditions, it is not surprising to see that, in majority,the represented particles follow very close orbits duringthe acceleration phase. Moreover, the oscillation is mod-ulated and slightly amplified by the interaction with thelaser pulse. This interaction was neglected in the test-particle simulation.0
50 100 150 200 250 X ( μ m ) Z ( μ m)1440 1660 1880 2100 2320 2.50-2.5 FIG. 13 Projection in the z − x plane of trajectories and ener-gies of 500 particles from injection until scattering. The laseris polarized in the (cid:126)e x direction and modulates the particletrajectories. Note the different scales in x and z axes. -4 -4 -4 X ( μ m ) Z ( μ m)1440 1660 1880 2100 2320 2.50-2.5 Radiated power (J/s)
FIG. 14 Trajectories and radiated power of the same 500particles as Fig. 13. The laser is polarized in the (cid:126)e x direction. Figure 14 shows trajectories and radiated power of thesame particles. It can be directly compared to Fig. 13.As expected, most of the radiation is produced inside thebubble at high electron energies and far from the axis,where the transverse acceleration is the highest.As shown in Sec. II, the radiation is emitted in thedirection of the electron velocity. Figure 15 shows theangular distribution of the radiated energy. It is verywell centered on the propagation axis and has an angularwidth of 30 mrad (FWHM).The frequency distribution of the radiated energy in-tegrated over all angles also yields interesting informa-tion. Figure 16 represents the calculated x-ray spectrumemitted by all injected macroparticles. The spectral dis-tribution is given by the number of emitted photons per0.1% bandwidth. The total number of emitted photonsis N γ (cid:39) . × and we have ≈ photons / . ≈ × photons / . -4 -4 -4 -4 R a d i a t e d e n e r g y ( J s r - ) θ Y ( m r a d ) θ X (mrad) -50 0 50 FIG. 15 Angular profile of the emitted radiation. -1 Energy (keV) P h o t o n s / . % B W FIG. 16 Spectrum of the x-ray radiation emitted by all in-jected macroparticles.
D. Experimental results
The production of betatron radiation requires a rela-tivistic electron bunch and an ion cavity or channel. Thelaser-plasma interaction is not the only mechanism thatcan create this ion cavity. Another approach was inves-tigated first experimentally: an electron beam propagat-ing in a plasma can expel background electrons via therepulsing electric space-charge force and create an ioncavity if its own density is larger than the plasma den-sity. The cavity being created by the front edge of thebunch, its main body propagates in the self-producedcavity and experiences its fields. This scheme of beta-tron x-ray emission was demonstrated by Wang et al. Y ( mm ) X (mm)
10 20 30 40 403020100 25155-5
FIG. 17 Betatron x-ray beam (round spot), produced by the28.5 GeV SLAC electron beam propagating through a plasmaof density ∼ × cm − , and recorded by a fluorescentscreen imaged onto a visible CCD camera. From Wang et al. (2002). (2002). The experiment, carried out at the SLAC Na-tional Accelerator Laboratory, was based on the use ofthe 28.5 GeV SLAC electron beam, containing 1 . × electrons per bunch and focused near the entrance of apreformed lithium plasma of length 1.4 m. For a plasmadensity of 1 . × cm − , the strength parameter ofbetatron oscillations was K = 16 .
8, and the number ofemitted photons and the peak spectral brightness wereestimated to be respectively 6 × photons / . × photons / (s mrad mm . ∼ × cm − ) was found tobe collimated in the divergence angle of ∼ (1 − × − rad. The expected quadratic density dependence of thespontaneously emitted betatron x-ray radiation [ P γ ∝ γ K /λ u ∝ n e at constant γ and r β ; see Eqs. (13) and(49)-(50)] was also observed, with a detection sensible inthe spectral range 5 - 30 keV. These features were foundto be in good agreement with theoretical predictions.The first experiment based on a fully laser-plasma ap-proach discussed above was performed by Rousse et al. (2004) using a 50 TW / 30 fs laser focused onto a 3 mmhelium gas jet at an electron density of ∼ cm − .A typical betatron experimental setup is shown in Fig.18. An intense femtosecond laser pulse is focused onto asupersonic helium gas jet. Electrons accelerated duringthe interaction are deflected toward a phosphor screenusing a permanent magnet with a field approaching 1T. Because the magnet bends the electron trajectories inonly one direction, the image on the screen reveals theelectron spectrum and the beam divergence. The x-rayradiation is observed using an x-ray CCD placed on axis. CC D
50 TW / 30 fs laser Betatron X-ray beamDeviated electrons
He gas jet Magnet x - r a y CC D Filter: Be, Al, ..Phosphor screen L e a d w a ll FIG. 18 Experimental setup for the generation and observa-tion of betatron radiation. A 50 TW / 30 fs class laser isfocused onto the front edge of a gas jet. Electron spectra areobtained by deflecting electrons with a magnet and using aLANEX phosphor screen imaged onto a CCD. Betatron ra-diation is recorded on axis using a charged coupled devicesensitive up to 10 keV (CCD X). A filter (Be, Al, etc.) isplaced before the CCD to block infrared light.
20 mrad
E > 3 keV CCD image
10 20 30 40 50 60 70 1 1.5 2 2.5 3 ! X ( m r ad ) n e (cm -3 )ExperimentModel Electron density (10 cm -3 ) θ X ( m r a d )
1 1.5 2 2.5 3 70605040302010
FIG. 19 Left: A raw image of the angular distribution of thebetatron radiation beyond 3 keV recorded by the CCD in asingle shot. Right, divergence (full width at half maximum)of radiation as a function of density. The dotted line showsthe dependence of θ on n e for constant r β and γ according tothe equations of Secs. IV.A and Sec. IV.B. With the camera sensitive in a broad spectral range, allradiation from the laser and the plasma below the keVrange is blocked using filters (beryllium and aluminum).The left part of Fig. 19 shows the x-ray beam profileobserved on the camera (at energy > µ mberyllium filter was placed in front of the camera). Theright part of the figure represents the evolution of thebeam divergence (full width at half maximum) as a func-tion of the electron density. The radiation is collimatedon axis with a divergence in the range [10 : 100] mrad.The divergence increases with the electron density dueto the reductions of both the betatron period and the2 Electron density /10 (cm -3 ) R a d i a t e d e n e r g y ( a . u ) Electron density (10 cm -3 ) R a d i a t e d e n e r g y ( a r b . un i t s ) FIG. 20 X-ray signal (proportional to radiated energy beyond1 keV per unit solid angle) as a function of plasma electrondensity. Each point corresponds to an average value over tenshots. The dotted line corresponds to the results obtainedusing 3D PIC simulations. From Rousse et al. (2004). electron energy as observed in the experiment. Theseexperimental observations are in good agreement withthe numerical predictions presented above (Albert et al. ,2008; Ta Phuoc et al. , 2005a, 2008a, 2006) and with PICsimulations (Kiselev et al. , 2004; Pukhov et al. , 2004b;Rousse et al. , 2004; Ta Phuoc et al. , 2005a).The betatron radiation has a specific dependence withplasma density (for given laser parameters) (Rousse et al. , 2004; Ta Phuoc et al. , 2005a). Figure 20 showsthe maximum of the radiated energy per unit solid anglebeyond 1 keV (in arbitrary units), as a function of theelectron density, obtained in the experiment and in PICsimulations. The observed x-ray emission is peaked at n e ∼ . × cm − . Below 5 × cm − , the x-ray sig-nal vanishes simply because no electron is trapped (thelaser intensity is too low for such densities in order to en-sure electron trapping). This is confirmed in the experi-ment for which no electron was detected by the spectrom-eter. Just above this threshold, electrons are trapped inthe bubble regime and monoenergetic electron bunchesas well as betatron radiation of relatively low intensitycan be obtained. When the density is higher, the plasmawavelength is smaller and the focal spot size and pulseduration are not exactly matched. It first corresponds tothe FLWF regime (Malka et al. , 2002; Najmudin et al. ,2003), and then for even higher density to SM-LWFA(Coverdale et al. , 1995; Modena et al. , 1995; Najmudin et al. , 2003; Santala et al. , 2001; Ting et al. , 1997; Um-stadter et al. , 1996a; Wagner et al. , 1997). This resultsin acceleration of electron bunches of poor quality com-pared to the bubble regime (broad spectrum, high diver-gence, and large shot-to-shot fluctuations). The coun-terpart of this poor quality electron beam is that the E (MeV) −
30 0 30 A ng l e ( m r ad ) n e =7.5 10 cm − −
30 0 30 n e =1.25 10 cm − −
30 0 30 n e =1.5 10 cm − xxx Energy (MeV) A n g l e ( m r a d )
25 50 100 200
FIG. 21 Raw electron spectra for three different densities andfixed laser parameters. Horizontal axis, electron energy; ver-tical axis, exit angle; and color scale, number of counts. Thislatter gives an indication of the beam charge. No electron isobserved below the trapping threshold ( n e = 7 . × cm − ),a quasimonoenergetic electron beam is produced just above( n e = 1 . × cm − ), and a broadband spectrum isrecorded for a higher density ( n e = 1 . × cm − ). Thethreshold density is not the same as in Fig. 20 because thelaser parameters are different. number of electrons and the amplitude of motion r β arehigher and the betatron period is smaller, such that thebetatron radiation is brighter even if the mean electronenergy is smaller. An additional effect which can occuris the interaction of the electron bunch with the laserpulse. From n e ∼ . × to 2 . × cm − , thex-ray signal drops down and a plateau is reached. ThePIC numerical simulations (Rousse et al. , 2004; Ta Phuoc et al. , 2005a) shown in Fig. 20 reproduce this experi-mental behavior: a sharp increase of the x-ray intensityfollowed by a smoother decrease of the signal. Threetypical electron spectra which can be obtained simulta-neously with the measurement of betatron radiation arerepresented in Fig. 21. The first shows a broadband spec-trum obtained at relatively high density, characteristic ofthe forced laser wakefield regime, and the second corre-sponds to a monoenergetic electron bunch (signature ofthe bubble regime) obtained for a density just above thetrapping threshold. Below this threshold, no electron isobserved (third spectrum). For some broadband spectra,a correlation of the electron output angle with the elec-tron energy can be observed. It is attributed to off-axisinjection of the electron beam due to asymmetric laserpulse intensity profile or tilted energy front. For theseshots, direct measurement of the betatron motion can beobtained from the oscillation observed in the spectrum(Glinec et al. , 2008).The spectrum of the betatron radiation was first mea-sured using a set of filters (Rousse et al. , 2004; Ta Phuoc et al. , 2005a), for an electron density n e = 1 × cm − .The top panel of Fig. 22 presents the number of x-ray photons obtained experimentally, integrated over thebeam divergence and over the spectral bandwidths deter-3 Photon energy (keV) Ph o t o n s / . % B W / s r / s h o t Photon energy (eV)
Experiment Synchrotron fit (Ec) Fit (Ec-1 keV) Fit (Ec+1 keV)
Energy (keV)
8 10
16 18 Energy (keV)
1 2
5 6
7 8
9 10 P h o t o n s / ( s r . % B W ) P h o t o n nu m b e r Ph o t o n s / . % B W / s r / s h o t Photon energy (eV)
Experiment Synchrotron fit (Ec) Fit (Ec-1 keV) Fit (Ec+1 keV)
FIG. 22 Top: Total number of photons obtained experimen-tally within the spectral bandwidths determined by filters: 25 µ m Be (1 < E <
10 keV), 25 µ m Be + 40 µ m Al (4 < E < µ m Be + 25 µ m Cu (6 < E <
10 keV). FromRousse et al. (2004). Bottom: Averaged spectrum (in redline) of the radiation obtained by photon counting, with a fit(in green line) by the function S ( ω/ω c ) with (cid:126) ω c = 5 . ± et al. (2011b). mined by (1) 25 µ m Be (1 < E <
10 keV), (2) 25 µ m Be+ 40 µ m Al (4 < E <
10 keV), and (3) 25 µ m Be + 25 µ m Cu (6 < E <
10 keV). The bottom part of the figurerepresents a measurement of the spectrum performed byphoton counting, at higher resolution (350 eV), in therange 8 to 21 keV. In this experiment, betatron radia-tion was produced with a 80 TW laser system (2.5 J and30 fs) at the Advanced Laser Light Source (Fourmaux et al. , 2011b), for an electron density n e = 5 . × cm − . The experimental spectrum (averaged over 10shots) was well described by a synchrotron function at (cid:126) ω c = 5 . ± et al. , 2011b). Depending on the experimentand available laser energy in the focal spot, the totalnumber of photons range from 10 (Rousse et al. , 2004)to 10 with 2 . × photons / (sr 0 . et al. , 2011a). The latter result also showed Energy (keV) d I / d ω d S ( a r b . un i t s ) FIG. 23 Synchrotron spectrum fitted from transmission val-ues through six different filters. The fit uses the synchrotronfunction ( ω/ω c ) K / ( ω/ω c ) instead of S ( ω/ω c ), and the fit-ted critical energy is 29 ±
13 keV. From Kneip et al. (2010). a critical energy above 10 keV (Fourmaux et al. , 2011a).Betatron radiation has also been observed in a recentexperiment performed on the Michigan Hercules lasersystem at the University of Michigan (Kneip et al. , 2010).The results demonstrated high photon energies and smallx-ray beam divergence. The betatron radiation had abroadband spectrum, characterized using a set of filters.For an electron density n e = 1 × cm − , the trans-mission values through the six filters were fitted with thesynchrotron function ( ω/ω c ) K / ( ω/ω c ) [which differsfrom S ( ω/ω c )] with a critical energy (cid:126) ω c ∼ ±
13 keV(see Fig. 23). The divergence angle was found to be ∼
10 mrad (FWHM), and the electron spectrum, simul-taneously recorded, indicated electron acceleration up to400 MeV. The peak spectral brightness was estimatedto be ∼ photons / (s mrad mm . et al. , 2007), an ultrafast phasetransition (nonthermal melting of InSb) was used as aBragg switch to sample the x-ray pulse duration. Thedynamics of this specific transition is well known. TheInSb is initially crystalline and diffracts the x-ray radia-tion according to the Bragg law. However, when irradi-ated by a femtosecond laser pulse (at a fluence in the 100mJ/cm range), the surface of the InSb is melted and dis-orders appears in a few tens of femtoseconds. As a conse-4quence, x-ray radiation can no longer be diffracted. Theduration of this phase transition has been determinedin several experiments based on the use of femtosecondsynchrotron radiation (Lindenberg et al. , 2005) and fem-tosecond K α radiation (Rousse et al. , 2001a). Here, as-suming the dynamics of the phase transition is known,the duration of the betatron x-ray pulse was estimatedto be less than 1 ps with a best fit below 100 fs (Ta Phuoc et al. , 2007). Recently, the electron bunch duration wasmeasured (Lundh et al. , 2011) using spectral characteri-zation of coherent transition radiation (CTR), in a laser-plasma accelerator working with external colliding pulseinjection (Faure et al. , 2006). The CTR result demon-strates a root-mean-square duration of 1.5 fs for the elec-tron bunch. Betatron radiation was shown to be stronglycorrelated with the properties of these electron bunches(Corde et al. , 2011a), showing that the radiation is effec-tively produced by these ultrashort electron bunches andthat it inherits its duration of a few femtoseconds.The source size has been determined with two differ-ent methods. The first uses the Fresnel edge diffractionof the x-ray beam and is based on the spatial coherenceof the radiation (Shah et al. , 2006). Because electronsemit x rays incoherently (see Sec. II.F) in the betatronmechanism, the source size determines the degree of spa-tial coherence which results in interference fringes in theFresnel diffraction experiment. This latter experimentcorresponds to the measurement of the intensity profileof the shadow made by a knife edge (here the edge of acleaved crystal) placed in the x-ray beam onto a detectorat a distance sufficient to ensure an appropriate resolu-tion. Using this technique, it was demonstrated that thesource size was less than 8 µ m (FWHM). A knife-edgetechnique was also used in betatron experiments withthe Michigan Hercules laser system at the University ofMichigan (Kneip et al. , 2010), and with the AdvancedLaser Light Source laser system (Fourmaux et al. , 2011a),showing an x-ray source size of ∼ − µ m (FWHM). Thesecond method relies on the information contained in thespatial distribution of the radiation. Indeed, because thebetatron radiation is emitted in the direction of the elec-tron velocity, its profile represents a direct signature ofthe electron orbits in the cavity. In addition, the diver-gence and the structure of the distribution depends onthe amplitude of the electrons orbits. This method, dis-cussed in the previous section, is presented in referencesTa Phuoc et al. (2008a, 2006). It is shown that the spa-tial distribution of the radiation implies a limited choiceof electron orbits. Therefore, by determining the electronorbits, the source size was deduced and the result was inthe range of ∼ − µ m (FWHM).Betatron x-ray radiation has interesting properties forapplication experiments: high peak spectral brightness,ultrashort duration, very small source size, as well asfemtosecond time-scale synchronization in pump-probeexperiments. The potential of betatron radiation was R e l a t i v e i n t e n s i t y Position (pixels)
Position (pixel) R e l a t i v e i n t e n s i t y
80 1001.21.00.80.60.4
A BC
FIG. 24 Bee imaged with the betatron x-ray beam with anedge line out indicated by the rectangular area: (a) one lasershot; (b) 13 laser shots. (c) Line out of the images taken at theindicated rectangular area. From Fourmaux et al. (2011a). demonstrated with the example of single shot phase con-trast imaging of biological samples (Fourmaux et al. ,2011a; Kneip et al. , 2011), where the experiment takesadvantage of the very small source size and high spatialcoherence of the x-ray beam. For an x-ray source size of ∼ − µ m, the coherence length at 1 m from the source is ∼ µ m, which allows to perform phase contrast imagingwith a compact setup. Figure 24 shows a phase contrastof a bee in single shot [Fig. 24(a)] as well as with accu-mulation over 13 shots [Fig. 24(b)] and the contrast isfound to be 68 % in single shot for the indicated line out(Fourmaux et al. , 2011a). Betatron x-ray radiation is alsoa powerful tool to study laser-plasma accelerator physicsand for noninvasive measurement of these properties. Anovel method has shown the possibility to measure thelongitudinal profile of the x-ray emission region, givinginsight into the history of the laser-plasma interaction(Corde et al. , 2011b). Betatron radiation from opticallyinjected, tunable, and monoenergetic electron buncheshave demonstrated the strong correlation between elec-tron and x-ray properties (Corde et al. , 2011a), showingthat betatron radiation can be used to perform noninva-sive measurements of some of the electron beam param-eters, such as the normalized transverse emittance of theelectron beam (Kneip et al. , 2012).Betatron radiation was also observed in the petawattregime using a laser delivering 300 J within 600 fs pulses(Kneip et al. , 2008). In this parameter regime, the laserpulse is much longer than the plasma period and thephysical mechanisms are different. For a long laser pulse5( τ L (cid:29) /ω p ), electrons can be accelerated and wiggledin the SM-LWFA and/or DLA regimes, depending on thelaser strength parameter a . Electrons experience boththe transverse focusing force of the channel and the elec-tromagnetic field of the laser. As a consequence, thetransverse amplitude of the electron orbits can be signif-icantly increased due to betatron resonance. The resultsshowed the possibility to produce betatron radiation witha spectrum fitted by the synchrotron function S ( ω/ω c )with a critical energy up to (cid:126) ω c = 14 . et al. (2008), their convention for (cid:126) ω c leads toa value of 29 keV] and with a divergence of ∼ × photons / (mrad . K were deduced tobe ∼ µ m and ∼
130 respectively, in agreement withPIC simulations.
E. Scalings and perspectives
The betatron source offers several routes of develop-ment in terms of number of photons, spectral range anddivergence. According to Sec. IV.B, these quantitiesscale in the wiggler limit as (Rousse et al. , 2007) N γ ∝ N e N β K ∝ N e N β √ γn e r β , (64) (cid:126) ω c ∝ γ . √ n e K ∝ γ n e r β , (65) θ r ∝ K/γ ∝ √ n e r β / √ γ, (66)where N β is the number of betatron periods and N e isthe number of electrons. As discussed in Sec. IV.B, theseformulas are valid for time-constant parameters, but canbe used with the values of the parameters at the de-phasing time for the realistic case of an electron beingaccelerated in the ion cavity. In addition to these formu-las, the scalings of the acceleration process are required.Considering the phenomenological model of the bubbleregime described by Lu et al. (2006a,b, 2007), higher elec-tron energy gain requires lower plasma density. Its scal-ing is given by γ (cid:39) ∆ E /mc = (2 ω L / ω p ) a ∝ a /n e .The matched spot size condition reads k p w = 2 √ a and the bubble radius should equal the laser pulse waist r b = w ∝ a / n − / e . The number of trapped electrons,obtained from an energy balance between the electronbeam and the bubble electromagnetic fields, scales as theionic charge of the bubble: N e ∝ r b n e ∝ a / n − / e . Thedephasing length scales as L d ∝ a / n − / e and the beta-tron period as λ u ∝ a / n − e , such that the number of be-tatron oscillations obeys N β ∝ n − / e . Different electroninjection mechanisms can lead to different scalings for r β ,but betatron radiation is optimized when r β is maximum, and this latter is limited by r b . We therefore assume thatthe transverse size of the electron bunch roughly scales asthe bubble radius for optimized conditions for betatronradiation: r β ∝ r b ∝ a / n − / e .The scalings of Eqs. (64), (65) and (66) can then bereformulated as functions of a and n e as N γ ∝ a / n − / e , (67) (cid:126) ω c ∝ a / n − / e , (68) θ r ∝ n / e , (69)while the needed laser energy E laser ∝ a w τ obeys E laser ∝ a / n − / e . (70)This expression uses the optimal pulse duration cτ =2 w / L d matches the pump depletion length L pd .These scalings take into account neither the possible in-teraction between the electron beam and the laser pulsenor beam loading effects, and describe the betatron per-spectives in the purely bubble regime. However, theyhighlight the route toward harder x rays and higher radi-ated energies: increasing the laser pulse strength param-eter and/or decreasing the density. Equation (70) indi-cates that it is more efficient in terms of laser energy todecrease the plasma density and not to increase the laserstrength parameter. The efficiency η X of the conversionof laser energy into x rays, defined as the ratio of the radi-ated energy by the laser energy, scales as a / n − / e . Thebetatron mechanism will become more and more efficientwhen going to higher and higher photon energies.Petawatt class lasers and capillaries both allow one todecrease the plasma density. With petawatt lasers pro-ducing sub-100 fs pulses, the higher laser energy can befocused into a larger focal spot matched to the lowerplasma density according to k p w = 2 √ a . For self-injection to occur, a ∼ − a (cid:38) ( n c /n e ) / , where n c = m(cid:15) ω L /e is the critical density (this condition ex-presses that energy loss at the front of the laser pulse dueto pump depletion is higher than the loss due to diffrac-tion) (Lu et al. , 2007). Capillaries allow one to guide thelaser pulse over distances much larger than the Rayleighlength such that no condition apply on a . This tech-nique has already been used to accelerate electrons upto 1 GeV in a few centimeters capillary (Leemans et al. ,2006). Combined with external injection, a strength pa-rameter as small as a = 2 can be used, which permitsone again to focus the laser energy into a larger focal spotand to decrease the plasma density. A drawback of thesenatural scalings is the increase of the x-ray source sizeand pulse duration as r β , τ r ∝ a / n − / e . As a conse-quence, the peak spectral brightness (number of photonsdivided by the pulse duration, by the solid angle, andby the source size in 0.1% spectral bandwidth) scales as6 a /n e . Nevertheless, an external injection could providesmall duration and therefore better time resolution andbrightness for application experiments.Using 15 J of laser energy focused into a gas jet of den-sity n e = 1 . × cm − with a waist w = 21 µ m andcompressed to a duration τ = 48 fs (300TW on target),the laser strength parameter is a = 4 .
4, ensuring self-guiding and self-injection, and trapped electrons reachthe maximum energy of 2.4 GeV after 23 mm of plasma(Rechatin, 2009). Assuming a reasonable value for thebetatron amplitude r β = 3 µ m, we expect K ∼
30 andx-ray photons emitted up to (cid:126) ω c ∼
400 keV, collimatedwithin a typical opening angle of 6 mrad, and containing N γ ∼ n e = 5 . × cm − , w = 21 µ m, τ = 47 fs (60TW ontarget) corresponding to a = 2 and assisted by exter-nal guiding (capillaries) and external injection [collidingpulse injection (Faure et al. , 2006) or density gradientinjection (Geddes et al. , 2008)] provides the same out-put electron energy of 2.4 GeV after 52 mm of plasma.With the same value of betatron amplitude r β = 3 µ m, xrays are emitted with the following properties: K ∼ (cid:126) ω c ∼
200 keV, θ r ∼ N γ ∼ N e is 2 times smaller, the number of betatronperiods N β is 1.4 times higher (see their respective scal-ings above), and the total number of emitted photonsis 2 times smaller, but within a better divergence ( θ r is1.4 times smaller) and for a needed laser power 5 timessmaller. For such configurations, the number of betatronphotons could reasonably reach the level of 10 photons.Another route toward brighter betatron radiation andhigher photon energies makes use of density-tailored plas-mas (Layer et al. , 2007) as recently suggested (Ta Phuoc et al. , 2008b). This method relies on the control ofthe electron orbits by varying the density, and thereforethe forces acting on the electrons along the propagation.For appropriately chosen density modulations, numeri-cal simulations show that the amplitude of the betatronoscillations r β can be significantly increased. The basicidea is to freeze the wakefield during a short durationin order to let the electron beam defocus and acquirelarger betatron amplitude. An additional effect is that,for large betatron amplitude, the mean longitudinal ve-locity is lowered and electrons dephase slower and canreach higher energies. In an optimistic configuration de-scribed by Ta Phuoc et al. (2008b), the critical energyof the radiation can be increased by a factor of ≈
9, thenumber of photons by a factor of ≈
5, but with an in-crease of the divergence by a factor ≈ ZXY X - r a y s ϑ ∼ K/ γ Relativistic Electronfrom LWFAPermanent magnet undulator / wiggler λ u Plasma acceleratorgas
FIG. 25 Schematic of the undulator source. A laser-plasmaaccelerated electron beam is injected in a conventional undu-lator. Electrons oscillate transversally and emit x rays due tothis motion.
V. PLASMA ACCELERATOR AND CONVENTIONALUNDULATOR: SYNCHROTRON RADIATION
Synchrotron radiation can be produced from laser-plasma accelerated electrons propagating and oscillatingin a conventional undulator or wiggler. A conventionalundulator or wiggler is a periodical structure of magnetsgenerating a periodic static magnetic field (Clarke, 2004).Figure 25 represents the principle of the scheme. Elec-trons are first accelerated in a plasma accelerator (gas jet,capillary or steady-state-flow gas cell) and then trans-ported into an undulator or wiggler. The size of sucha device is on the meter scale, with a typical magneticperiod in the centimeter range. Moreover, the values of λ u , technologically limited, imply that GeV electrons arenecessary to produce radiation in the x-ray range. Theserepresent the disadvantages of this scheme, which how-ever is of crucial interest because it represents a possibleroute toward the production of a compact free-electronlaser based on a laser-plasma accelerator (Gr¨uner et al. ,2007; Malka et al. , 2008; Nakajima, 2008) (see Sec. VII).In the following sections, the electron orbits in undulatorsand the main properties of synchrotron radiation pro-duced from laser-plasma electron bunches are describedin a presently realizable case. After a summary of the ex-perimental results, we conclude with the short term de-velopments foreseen. For detailed properties of undulatorradiation, such as the angular distribution of individualundulator harmonics, the different undulator types (pla-nar or helical), the polarization properties and the spatialor temporal coherence properties, see Chao and Tigner(2009); Clarke (2004); and Wiedemann (2007a). A. Electron motion
We consider a laser wakefield accelerator producing amonoenergetic electron bunch of energy E = γ i mc withvelocity directed in the (cid:126)e z axis. The simplest model todescribe the electron dynamics in the undulator is to ap-proximate the static magnetic field near the undulator7axis by (cid:126)B = B sin( k u z ) (cid:126)e y , (71)where k u = 2 π/λ u with λ u the magnetic period of theundulator. The equation of motion for a test electron inthis idealized model is given by d(cid:126)pdt = − e(cid:126)v × (cid:126)B. (72)The normalized Hamiltonian describing the electron dy-namics is given byˆ H (ˆ (cid:126)r, ˆ (cid:126)P, ˆ t ) = γ = (cid:113) (cid:126)P + (cid:126)a ) , (73)where ˆ (cid:126)P is the normalized canonical electron momentumand (cid:126)a = − a cos( k u z ) (cid:126)e x is the normalized vector poten-tial verifying (cid:126)B = ( mc/e ) ∇× (cid:126)a . ˆ H does not depend on ˆ x ,ˆ y , and ˆ t . The transverse canonical momentum ˆ (cid:126)P ⊥ andthe normalized energy γ are therefore conserved. Inte-grating these constants of motion with the initial condi-tions ˆ (cid:126)P ⊥ = 0 and γ = γ i leads to, in physical units, γ = cte = γ i , (74) x ( z ) (cid:39) − Kγk u sin( k u z ) , (75) K = a = eB k u mc . (76)The general description developed in Sec. II is recoveredand the practical expression of K in terms of experimen-tal parameters and fundamental constants has been ob-tained. In particular, K does not depend on γ and theradiation regime (undulator or wiggler) is the same forall electron energies. The strength parameter K is deter-mined by the amplitude of the magnetic field B and theperiod λ u and is, in practical units, given by K = 0 . λ u [cm] B [T] . (77) B. Radiation properties
The radiation properties are the same as in Sec. II.The spectrum of the emitted radiation depends on K .For a small amplitude of oscillation K (cid:28)
1, radiationis emitted at the fundamental photon energy (cid:126) ω with anarrow bandwidth in the forward direction ( θ = 0). As K → K (cid:29) (cid:126) ω c . These quantities are given by (cid:126) ω = (2 γ hc/λ u ) / (1 + K /
2) for
K < , (cid:126) ω c = 32 Kγ hc/λ u for K (cid:29) . (78) In practical units, this gives (cid:126) ω [eV] = 2 . × − γ /λ u [cm] for K (cid:28) , (cid:126) ω c [eV] = 1 . × − γ B [T] for K (cid:29) . (79)The radiation is collimated within a cone of typicalopening angle θ r = 1 /γ in the undulator case. For a wig-gler, the radiation is collimated within a typical openingangle K/γ in the electron motion plane ( (cid:126)e x , (cid:126)e z ) and 1 /γ in the orthogonal plane ( (cid:126)e y , (cid:126)e z ).The radiation temporal profile is simply given by theconvolution between the electron bunch temporal pro-file and the radiation profile from a single electron. For xrays, the radiation length from a single electron l r | Ne =1 = N λ is in the nanometer range, much smaller than the typ-ical electron bunch length which is in the micron range,and therefore τ r | Ne (cid:39) τ b . However, for UV and XUV ra-diation, the slippage between the radiation and the elec-tron bunch is not necessarily negligible, and the radiationduration can be stretched by the single electron radia-tion duration. The electron bunch duration depends onthe acceleration mechanism and on its transport from itssource to the undulator. Presently, the shortest electronbunches are obtained using a laser-driven plasma-basedaccelerator with an external optical injection mechanism(Davoine et al. , 2008; Faure et al. , 2007, 2006; Malka et al. , 2009; Rechatin et al. , 2009b). In this scheme, elec-tron bunches with rms (root-mean-square) duration assmall as 1.5 fs can be produced (Lundh et al. , 2011).However, the necessary transport of the electron bunchin the undulator can deteriorate the duration (and thetransverse emittance), because different electrons in thebunch travel different distances. For example, consider-ing usual transport devices (quadrupoles), placed at ∼
1m from the laser-plasma accelerator, an initially 1- µ m-long bunch has its length increased to ∼ µ m duringits transport to an undulator placed several meters afterthe laser-plasma accelerator. An ultracompact transportsystem, placed as closed as possible to the plasma source,is required to avoid such a deterioration.The number of emitted photons follows the expressionsgiven in Sec. II. In practical units, the number of photonsemitted per period and per electron (at the mean photonenergy (cid:104) (cid:126) ω (cid:105) = 0 . (cid:126) ω c for the wiggler limit) is given by N γ = 1 . × − K for K < ,N γ = 3 . × − K for K (cid:29) . (80)From these expressions the features of the produced ra-diation can be obtained for a realizable case: an electronbunch at an energy of 1 GeV containing 100 pC and anundulator of 100 periods with λ u = 1 cm and K = 0 . E X ∼ θ = 0), iscollimated within a typical opening angle of ∼ µ rad,and contains ∼ × photons.8 -4-3-2-1 0 1 2 3 4 0 20000 40000 60000 80000 100000 X ( μ m ) Z (cm) FIG. 26 Electron trajectories. The orbit is plotted as a redsolid line for the undulator case ( K = 0 .
2) and green dashedline for the wiggler case ( K = 2). At the entrance, the electronhas γ = 1000 and (cid:126)p = (cid:112) γ − (cid:126)e z in both situations. Theundulator or wiggler is ten periods long. C. Numerical results
In this section, the electron motion and the emittedradiation are calculated numerically. Both the cases ofan undulator and a wiggler are considered, with strengthparameter of K = 0 . λ u = 1 cm and the test electronpropagates in the (cid:126)e z direction with γ = 1000 ( (cid:39) N = 10oscillation periods (the number of periods of an undula-tor or wiggler is usually on the order of 100). The motion,with γ constant, consists of a transverse oscillation at aperiod λ u combined with a longitudinal drift. The am-plitude of the transverse motion is given by λ u K/ πγ (here (cid:39) . × K [ µ m]). For K = 0 . .
32 and 3 . µ m, respectively.The spatial distributions of the radiation produced bythe electron undergoing these trajectories are presentedon the left of Fig. 27. For K = 0 . /γ ∼ K = 2 (bottom), the radiation has a diver-gence of typical opening angle K/γ ∼ /γ ∼ . × .
3, 0 . × .
8, and 2 × ), whereas for the wiggler case it is integrated overthe overall spatial distribution. For K = 0 . (cid:126) ω ∼ λ = λ/N (where N = 10 is the number ofoscillation periods), and it is stretched after integrationover the emission angles due to the angular dependenceof the radiated wavelength λ (cid:39) ( λ u / γ )(1 + K + γ θ )[see Sec. II, Eq. (5)]. For K = 2 the spectrum becomesbroadband with a critical energy of around 300 eV. Anestimate of the total photon number can be obtained on θ Y ( m r a d ) θ X (mrad) θ Y ( m r a d ) R a d i a t e d e n e r g y ( × - J s r - ) P h o t o n s / . % B W P h o t o n s / . % B W Energy (keV) × -4 × -4 K = 2K = 0.2 K = 2K = 0.2 R a d i a t e d e n e r g y ( × - J s r - )
0 0.4 0.8 1.20 0.2 0.4 0.66 × -6 × -6 × -6 FIG. 27 On the left is represented the angular distributionsof the radiation (radiated energy per unit solid angle) andthe corresponding spectra are shown on the right (in num-ber of photons per 0.1% bandwidth, per electron, and pershot). The undulator case, K = 0 .
2, is on the top whereasthe wiggler case, K = 2, is on the bottom. In the first sit-uation, three different solid angles of integration have beenconsidered to highlight the stretching of the spectrum. In thewiggler situation, the integration is performed over the totalangular distribution. the basis of the number of electrons that are currently ac-celerated in a laser-plasma accelerator. Assuming 1 × electrons gives ∼ photons at 200 eV within a spectralbandwidth of 10% in the undulator case (selection of the0 . × . solid angle). In the wiggler case thereare ∼ × photons / . − . D. Experimental results
Synchrotron facilities have been developed worldwideand have been providing x-ray radiation to users for sev-eral decades. These installations are very robust and thex-ray beams have high quality. They are based on con-ventional accelerator technology.The first demonstration of the production of syn-chrotron radiation from laser-accelerated electrons wasperformed by Schlenvoigt et al. (2008a,b). In this ex-periment, laser-produced electron bunches between 55and 75 MeV were injected, without transport, into a1-m-long undulator having a period λ u = 2 cm anda strength parameter K = 0 .
6. They obtained syn-chrotron radiation in the visible and infrared part of thespectrum (the wavelength was in the range 700 − . × photons / (s mrad mm . . × . The radiation wavelengthwas observed to scale with the electron bunch energy asexpected.More recently, synchrotron radiation from laser-plasma9accelerated electrons was produced up to ∼
130 eV(Fuchs et al. , 2009). This experiment represents an im-portant advance compared to the first demonstration.First, a stable electron beam was produced by the in-teraction of a laser at ∼ × W.cm − with a 15-mm-long hydrogen-filled steady-state-flow gas cell (Osterhoff et al. , 2008). The electron bunch was in the 150 - 220MeV energy range and contained 30 pC. Second, a pairof miniature permanent-magnet quadrupole lenses wasimplemented in order to make the electron beam colli-mated at a selected energy and to transport it througha 30-cm-long undulator having a period λ u = 5 mm anda strength parameter K = 0 .
55 (Eichner et al. , 2007).Thanks to the high energy of the electron and the shortperiod of the undulator, the radiation was produced downto the XUV range with a tunable wavelength. Notethat this tunability resides in the possibility to select,with the magnetic lenses, a specific electron energy inthe 150 - 220 MeV range (electrons with different en-ergies are not collimated and will not produce a col-limated radiation beam). The radiation was analyzedusing a spectrometer based on a transmission grating.The fundamental wavelength ( λ = 18 nm) as well asthe second harmonic were measured. In addition, theparabolic dependence of the radiation wavelength on theangular direction λ = ( λ u / γ )(1 + K / γ θ ) wasobserved with good agreement. This soft-x-ray undu-lator source delivers ∼ photons per shot integratedover the fundamental within a detection cone of ± ∼ . × photons / (s mrad mm . E. Perspectives
The experiments performed open perspectives for theproduction of a tunable synchrotron-type x-ray source.However, further progress is necessary to produce tun-able and reproducible monoenergetic electron bunches inthe GeV range. The foreseen possibilities to achieve thesefeatures rely on the use of higher power lasers (petawatt)and/or capillaries. This gain in laser power compared toactual systems would allow one to decrease the plasmadensity, increase the acceleration length, and as a conse-quence increase the electron energy. The use of an exter-nal injection will be required as well to easily control theelectron energy (optical injection (Faure et al. , 2006) orplasma density gradient injection (Geddes et al. , 2008),as presented in Sec. III). Finally, a compact electrontransport system from the plasma source to the undula-tor has to be implemented in order to avoid deteriorationof the electron beam parameters and to reduce the radi-ation source size.In the short term, this approach will allow university-scale laboratories to access x-ray sources with angstrom wavelengths and sub-10-fs pulse durations for four-dimensional imaging with atomic resolution. Besides thisfuture development, the main and more challenging per-spective relies on the possibility to produce a free-electronlaser based on a laser-plasma accelerator combined withan undulator. This radiation source has a brightness or-ders of magnitude higher than a synchrotron, thanks tothe coherent addition of radiation from each electron.Electron bunches produced with laser-plasma accelera-tors are interesting candidates for a FEL source sincesuch bunches are naturally femtosecond and have highcurrent. In Sec. VII, we discuss the principle of the free-electron laser and its possible realization using electronsfrom laser-plasma accelerators which could have revolu-tionary impacts in many fields of science, technology, andmedicine.
VI. ELECTROMAGNETIC WAVE UNDULATOR:NONLINEAR THOMSON SCATTERING ANDTHOMSON BACKSCATTERING
In this section, the production of femtosecond x-rayand gamma-ray beams from electrons oscillating in anelectromagnetic wave is reviewed. The involved radiativemechanism is referred to as Compton scattering. In theframework of quantum electrodynamics, it correspondsto the absorption of one (linear) or several (nonlinear)photons by an electron and to the emission of a singlephoton. When, in the rest frame, the electron experiencesa negligible recoil (which happens when (cid:126) ω (cid:28) mc in therest frame), the mechanism is referred to as Thomsonscattering, the low-energy limit of Compton scattering.In the following, we consider Thomson scattering anduse a classical description, assuming that quantum andradiation reaction effects are negligible (see Sec. II.G).At the laser-plasma interaction, two schemes of fem-tosecond x-ray or gamma-ray sources based on Thomsonscattering have been proposed and demonstrated. Basedon the same principle, the two methods differ by the ini-tial energy of the electron and the intensity of the laserpulse scattering off the electrons.In the first scheme, electrons are initially at rest andlaser wakefield acceleration is not invoked. Here, Thom-son scattering occurs in a highly nonlinear regime. In-deed, for a (cid:29)
1, electrons have a highly non linearmotion and the emitted radiation consists of high-orderharmonics. The harmonic spectrum can extend up tothe x-ray range. This scheme offers simplicity and thepossibility to produce a large x-ray photon flux becausethe number of electrons participating in the emission ison the order of the number of electrons in the focal vol-ume. However, reaching the keV energy range requiresvery high laser strength parameters, typically a > γ with respect to thelaboratory frame, the frequency of the incident electro-magnetic wave (the laser pulse) becomes ω (cid:48) i = 2¯ γω i . Thelight is scattered by the electrons at the same frequency ω (cid:48) r = ω (cid:48) i (for a low intensity electromagnetic wave corre-sponding to the linear Thomson scattering and the undu-lator regime) or at harmonics ω (cid:48) r = nω (cid:48) i (for high intensitycorresponding to nonlinear Thomson scattering and thewiggler regime). When observed in the laboratory frame,this reflected wave is Doppler shifted and its frequencybecomes ω r = 4¯ γ ω i (linear Thomson backscattering) orits harmonics (nonlinear Thomson backscattering). Thisscheme offers the possibility to produce x rays in the keVrange even with modest energy electrons (tens of MeV)or gamma rays if 100 MeV range electrons are used.In the following sections the two schemes are discussed.For both, electron orbits, radiation properties, numerical,and experimental results are presented. A. Nonlinear Thomson scattering
In this section, the production of femtosecond x-rayradiation from electrons initially at rest is discussed. Inthis scheme, Thomson scattering occurs in a stronglynonlinear regime and electrons are directly acceleratedand wiggled within an intense laser field. This schemeoffers the possibility to produce a large photon flux be-cause the number of electrons participating in emissioncan be much larger than the number of electrons trappedand accelerated in a laser-plasma accelerator. It is ofthe order of the number of electrons in the focal vol-ume. However, producing x-ray radiation requires a laserstrength parameter a much greater than unity. Indeed,in that regime, the term − e(cid:126)v × (cid:126)B of the Lorentz force,negligible at low intensity, becomes comparable to the − e (cid:126)E component and the motion is a nonlinear functionof the driving field in addition to becoming relativistic.The electron motion is no longer harmonic and the radi-ation emitted consists of high-order harmonics forming abroadband spectrum that extends up to the x-ray range.The spectrum shifts to higher energies as a increases.This radiative mechanism is commonly called nonlinearor relativistic Thomson scattering. The principle of thissource is displayed in Fig. 28. It simply consists of fo-cusing an intense laser on a target (here an underdenseplasma is considered).In the following sections, the properties of nonlinearThomson scattering are reviewed. The electron orbit is For an overview of the subject see Bardsley et al. (1989); ϑ ∼ K/ γ X - r a y s Z Laserpulse X ElectronOrbit fs pulse Y FIG. 28 Schematic of the nonlinear Thomson scatteringsource. A free electron is submitted to a relativistic laserpulse ( a (cid:29) derived in the case of a circularly polarized laser pulse,because it leads to a situation where the concept of wig-gling, developed in Sec. II, can be applied. For a linearlypolarized laser pulse, analytical results can be easily ob-tained for the electron orbit, but not for the radiationfeatures. In that latter case, the electron movement doesnot correspond to a transverse wiggling around a rel-ativistic drift motion along the propagation axis. Theelectron performs violent accelerations from γ = 1 to γ = 1 + a / γ = 1 for each half pe-riod of the movement. The emitted radiation comes fromthe longitudinal acceleration, and not from the transverseone anymore. Hence, in the Secs. VI.A.1 and VI.A.2 an-alytical expressions are derived for the circular polariza-tion but the linear polarization will be discussed in thenumerical Sec. VI.A.3. After a brief summary of the ex-perimental results obtained, we finally conclude with theshort-term developments foreseen.
1. Electron orbit in an intense laser pulse
We consider a test electron initially at rest ( γ i = 1)submitted to an intense laser pulse modeled by a circu-larly polarized plane electromagnetic wave propagating Brown and Kibble (1964); Castillo-Herrera and Johnston (1993);Esarey et al. (1993a,c); Gibbon (1997); He et al. (2002); Lau et al. (2003); Salamin and Faisal (1996); Sarachik and Schappert(1970); Shen et al. (1997); Tian et al. (2006); Ueshima et al. (1999); and Vachaspati (1962). (cid:126)e z axis, with a wave vector (cid:126)k i = 2 π/λ L (cid:126)e z anda frequency ω i = 2 πc/λ L , and with a normalized vectorpotential given by (cid:126)a = a (cid:20) √ ω i t − k i z ) (cid:126)e x + 1 √ ω i t − k i z ) (cid:126)e y (cid:21) . (81)The equation of motion of the test electron is d(cid:126)pdt = − e ( (cid:126)E + (cid:126)v × (cid:126)B ) . (82)All quantities with a hat are normalized by the followingchoice of units: m = c = e = ω i = 1. The Hamiltoniandescribing the test electron dynamics in the electromag-netic wave is written asˆ H (ˆ (cid:126)r, ˆ (cid:126)P, ˆ t ) = γ = (cid:113) (cid:126)p = (cid:113) (cid:126)P + (cid:126)a ) . (83)This system of an electron in a electromagnetic planewave is integrable and action-angle variables can be foundto solve the electron motion. Here, for simplicity, weanalyze the symmetries of the system to directly yield theconserved quantities according to Noether theorem. TheHamiltonian ˆ H depends on the canonical momentum ˆ (cid:126)P and on the potential vector (cid:126)a ( ϕ ) through the variable ϕ =ˆ t − ˆ z . Hence ˆ H is independent of ˆ x and ˆ y , which impliesthat the transverse canonical momentum is a constant ofmotion. The first constant of motion readsˆ (cid:126)P ⊥ = ˆ (cid:126)p ⊥ − (cid:126)a = (cid:126) . (84)In addition, ˆ H depends on ˆ t and ˆ z only through ϕ =ˆ t − ˆ z . Thus ∂ ˆ H /∂ ˆ t = − ∂ ˆ H /∂ ˆ z , which leads to the secondconstant of motion C, γ − ˆ p z = C . (85)For an electron initially at rest, C = 1, and the testelectron orbit is obtained by integrating the constants ofmotion, ˆ x ( ϕ ) = a √ ϕ ) , (86)ˆ y ( ϕ ) = − a √ ϕ ) , (87)ˆ z ( ϕ ) = a ϕ, (88) γ = 1 + a . (89)The electron motion in the laser field consists of a driftin the longitudinal direction combined with a transverseoscillation. The electron orbit is mainly longitudinal andrelativistic for a > a <
1. For a <
1, the test electronmotion is a simple nonrelativistic dipole motion which isnot relevant for the production of x rays since radiation is emitted at the same wavelength as the incident field.For a >
1, the longitudinal motion becomes relativisticand nonlinear effects and Doppler shift occur. In thefollowing, the interesting nonlinear case a >
1, whichresults in the generation of smaller wavelength radiationthan the incident laser pulse, is studied.The period of the motion is given by λ u = a λ L . (90)The electron orbit is similar to the standard orbits dis-cussed in Sec. II since the normalized energy γ is con-stant. However, the trajectory takes place in three di-mensions and is helical. The qualitative regime and theangular distribution can be derived from the parameters K X = γψ X and K Y = γψ Y which are given by K X = K Y = 4 √ a (1 + a / (cid:39) a / √ , (91)where ψ X and ψ Y are the maximum deflection angles ofthe orbits in the (cid:126)e x and (cid:126)e y directions.
2. Radiation properties
For a >
1, the relativistic electron motion describedabove leads to the emission of nonlinear Thomson scat-tering radiation whose features can be calculated withinthe frame of the general formalism of the radiation froma moving charge. As seen in Sec. II, the spectrum of theproduced radiation depends on the parameter K . Ac-cording to the above expressions of K X and K Y , the mo-tion is driven in the wiggler regime since the nonlinearcase a > (cid:126) ω c = 32 γ (cid:126) cρ , (92)where ρ = ( λ L / π )( a / √ √ a / (cid:39) ( λ L / π ) ×√ a /
16 is the instantaneous radius of curvature of theelectron orbit obtained from the trajectory. Basically,the critical energy grows as a for a (cid:29)
1. In practicalunits, it reads for a (cid:29) (cid:126) ω c [eV] = 0 . a λ L [ µ m] . (93)Being an emission from a relativistically moving elec-tron, nonlinear Thomson scattering radiation is emittedin the direction of the electron velocity and has the samesymmetry as the electron orbit. For circular polariza-tion, the radiation is emitted symmetrically around thelaser propagation axis at an angle θ (cid:39) √ /a , within atypical angular width ∆ θ = 1 /γ (cid:39) /a .The number of photons can be estimated by integrat-ing, over one oscillation period, the expression of the ra-diated power P ( t ) = ( e / π(cid:15) c ) γ [( d ˆ (cid:126)p/dt ) − ( dγ/dt ) ]2 X ( μ m ) Z ( μ m)
80 16040 120010-1 a = 5 a = 10 γ FIG. 29 Top: The electron orbit in an intense linearly po-larized laser pulse with a = 5 (red solid line) and a = 10(green dashed line). Bottom: The γ factor as a function of thelongitudinal position for each case. The electron is initiallyat rest and is submitted only to the linearly polarized laserpulse whose temporal profile is Gaussian with 20 fs duration(FWHM). in which the trajectory calculated above is inserted. Inpractical units, the number of photons emitted at themean energy (cid:104) (cid:126) ω (cid:105) = 0 . (cid:126) ω c per one electron undergoingone oscillation period is, for a (cid:29) N γ = 4 . × − a . (94)From the above expressions an estimation of the ra-diation properties can be obtained for a typical parame-ter regime. The laser pulse is focused along a Rayleighlength z r = 500 µ m with a = 10, and has a dura-tion of 20 fs FWHM and a wavelength λ L = 800 nm.The period of the electron motion is λ u ∼ µ m andthe electron executes about ∼ (cid:126) ω c ∼
400 eV. The emission is emitted at an angle θ ∼
280 mrad with respect to the laser propagation axis.For a plasma density of 1 × cm − , and if we con-sider that the number of electrons participating to theemission is the number of electrons in the focal volume ∼ × × µ m (which is equal to 10 electrons),the number of x-ray photons per shot is ∼ × .
3. Numerical results
The features of the radiation have been described sofar for a free electron submitted to a circularly polarizedplane wave for which the concept of wiggling applies andanalytical solutions are straightforward. In this section,the case of a linearly polarized laser field is studied us-ing numerical simulations. Electron orbits and radiationfeatures are presented.We consider a laser pulse with a Gaussian temporal a = 10a = 5 Y ( mm ) Y ( mm ) X (mm) R a d i a t e d e n e r g y ( a r b . un i t s ) R a d i a t e d e n e r g y ( a r b . un i t s ) FIG. 30 Spatial distribution of the nonlinear Thomson scat-tering, corresponding to trajectories of Fig. 29. The radiatedenergy per unit surface in the plane situated at 1 m fromthe source and perpendicular to the propagation axis is rep-resented. On the top is the case a = 5 and on the bottomthe case a = 10. profile, whose normalized vector potential reads (cid:126)a = a exp[ − (2 ln 2)( z − ct ) / ( c τ )] cos( ω i t − k i z ) (cid:126)e x . (95)The laser pulse duration (FWHM) is τ = 20 fs and thepolarization is linear along the (cid:126)e x direction. The testelectron is initially at rest. Figure 29 displays the elec-tron orbits for a = 5 and 10 and the evolution of therelativistic factor of the electron γ along its motion. Thetrajectory consists in successive straight lines with vio-lent longitudinal acceleration or deceleration, the velocityvanishing at each corner. The electron acquires energy inthe tens of MeV range while oscillating in the laser pulse,and is again at rest once the laser has passed. There isno net energy gain.The radiation has been calculated using the generalformula (1) in which the orbits calculated above are in-serted. In Fig. 30, the spatial distribution of the radiatedenergy is represented. Because the polarization is linear,the radiation consists in two lobes corresponding to thetwo directions pointed by the straight lines of the orbit.These two lobes form, with respect to the laser propa-gation axis, an angle θ X ∼
450 mrad for a = 5 and θ X ∼
230 mrad for a = 10. As expected, the radiationis more collimated for stronger a . Figure 31 displays thespectra of the radiation integrated over the spatial dis-tribution (the photon number per 0.1% bandwidth perelectron and per shot). An estimate of the total photonnumber can be obtained assuming that all electrons inthe focal volume participate to the emission. Assuminga focal volume of dimension 10 × × µ m and a den-sity of 10 cm − gives 10 electrons which radiate ∼ photon/0.1%BW at 50 eV for a = 5 and ∼ . × photon/0.1%BW at 400 eV for a = 10. Comparing3 P h o t o n s / . % B W Energy (keV) × -4 × -4 × -4 × -4 × -4 a = 5 a = 10 Energy (keV) d I / d ω d Ω ( a r b . un i t s ) a = 10 FIG. 31 Nonlinear Thomson scattering spectrum, integratedover angles and corresponding to the trajectories of Fig. 29.The number of photons per 0.1% bandwidth for one electron isdisplayed for the two cases: a = 5 in red solid line and a =10 in green dashed line. The spectrum near the maximumangle of emission is represented in the inset for the case a =10. these numerical results with the analytical expressionsobtained for the circular polarization, it appears that thespectral properties depend on the observation angle (seethe inset of Fig. 31). The shape of the spectrum at themaximum angle of emission is not synchrotronlike, be-cause it does not correspond to a wiggling motion. Theradiation for linear polarization is due to the longitudinalacceleration from γ = 1 to γ = 1 + a / γ = 1 occurring for each straight line section of themotion. The time profile of the radiated field in a singlestraight line section consists of a double peak structure:the first corresponds to the moment when the acceler-ation is maximal, and the second to the moment whenthe deceleration is maximal. This complex behavior isat the origin of the spectral modulation observed at themaximum angle of emission in the inset of Fig. 31 (Lee et al. , 2003).It is important to note that the most simple case ofnonlinear Thomson scattering has been considered. Fora more accurate description of the mechanism, severaleffects must be taken into account. In the tightly fo-cused case, the exact laser field can strongly modify theelectron orbit. In particular, the electron is expelledfrom the high-intensity region. This results in a radialdrift and limits the number of electron oscillations in thehigh-intensity regions. Therefore, the radiated energy issmaller than in the ideal case and the divergence of thex-ray beam is higher and broadened. The presence ofthe ions from the plasma, or more generally, the collec-tive fields of the plasma, can as well have a significantimpact on the electron orbit. They can reduce the longi-tudinal drift or accelerate the electrons. In the first case,the longitudinal drift being reduced, the divergence is in-creased. Oppositely, if the electron gains energy from theplasma, the radiation becomes more collimated. Finally,the laser pulse propagates in the plasma with a group
50 TW / 30 fs laser Deviated electronsHe gas jet Magnet x - r a y CC D x - r a y CC D x - r a y CC D GratingMirrorFilter a = 5 FIG. 32 Experimental setup for the nonlinear Thomson scat-tering experiment. Radiation is produced from the interactionof a laser pulse with a ∼ velocity v g smaller than c , which can strongly modifythe electron orbit if the electron longitudinal velocity v z becomes comparable or greater than v g .
4. Experimental results
While nonlinear Thomson scattering was anticipatedin the 1960s (Brown and Kibble, 1964; Sarachik andSchappert, 1970; Vachaspati, 1962), the first experimen-tal demonstration of nonlinear Thomson scattering wasperformed in 1998 only (Chen et al. , 1998). Using a 4TW laser pulse with a duration of 400 fs, λ L = 1 µ m,and a ∼
2, focused into a helium gas jet with a density ofa few 10 cm − , nonlinear Thomson scattering radiationwas observed for the first time. The measured radiationfollowed the specific features of nonlinear Thomson scat-tering and other radiative processes were ruled out. Inparticular, the spatial distributions measured were con-sistent with the nonlinear Thomson scattering propertiesobtained from the picture of the single free electron sub-mitted to a plane wave. In this experiment, radiationwas detected at wavelengths up to the third harmonic ofthe laser (few eV range). The efficiency was estimatedto be of the order of a few 10 − photons per electronper pulse. In latter experiments (Banerjee et al. , 2002,2003), the nonlinear Thomson scattering radiation wasobserved up to the 30th harmonic with a spatial distri-bution collimated in the direction of the laser propaga-tion axis (and maximum on axis, in contradiction withthe single free-electron model). In 2003, thanks to theadvent of high-intensity lasers, nonlinear Thomson scat-tering was demonstrated in the XUV range (Ta Phuoc et al. , 2005b, 2003a,b). This experiment was based onthe interaction of a 50 TW laser (with a ∼
5, 30 fs, 8004 L a s e r a x i s R a d i a t e d En e r g y ( a r b . un i t s ) Observation angle (deg)
Energy (keV)
Energy (keV) N u m b e r o f ph o t o n s ( a r b . un i t s ) R a d i a t e d En e r g y ( a r b . un i t s ) FIG. 33 On the left is represented the angular dependence ofthe radiated energy recorded experimentally by rotating thewhole detection system. The spectrum of the radiation onaxis is given in logarithmic scales on the right of the figureand the inset shows the total number of photons integratedover several spectral bands determined by the choice of filters(the horizontal extension of the lines indicates the spectralband). nm) and a helium gas (density in the range 10 − cm − ). The experimental setup is presented in Fig. 32.The laser was focused within a 6 µ m focal spot (FWHM)onto the front edge of a supersonic helium gas jet (3 mmin diameter). Radiation was collected using a grazingincidence spherical mirror and spectrally analyzed usinga transmission grating. The whole system was mountedon a rotation stage centered on the gas jet in order tomeasure the spatial distribution of the radiation. Non-linear Thomson scattering radiation was observed in thefew tens of eV range up to 1 keV using a set of filters (Al,Zr, Ni, and Be). As at previous experiments, radiativemechanisms others than nonlinear Thomson scatteringwere ruled out by the variations of the XUV signal asa function of experimental parameters. The signal wasfound to increase linearly with the plasma density, the ra-diation was anisotropic and the energy range fitted whatis expected for nonlinear Thomson scattering.However, the results differ from the expected emissionwith the model of free electrons in a plane electromag-netic wave. The radiation spatial distribution, presentedin Fig. 33 (left), was collimated within a large angle(within a cone of half angle ∼
30 deg) and was maxi-mum on axis instead of consisting in two lobes centeredat +23 deg and −
23 deg. This result can be explainedconsidering the multitude of electron orbits, the focusingof the laser and the effect of plasma fields. This was sup-ported by the fact that electrons at relativistic energieswere observed during the experiment. In addition, laserfilamentation was observed (Faure et al. , 2002; Thomas et al. , 2009, 2007) indicating a complex propagation ofthe laser.The spectrum, measured on axis in a limited spectralrange (30 - 120 eV), is presented in Fig. 33 (right). Themeasurement was performed using a transmission grat-ing. In the inset is presented the spectrum measured withfilters (Al, Zr, Ni, and Be) over a larger bandwidth andthe number of photons. The radiation detected was in the spectral range expected for nonlinear Thomson scat-tering. A total of 10 photons per shot were measured(integrated over the spectrum and the spatial distribu-tion).
5. Perspectives
Nonlinear Thomson scattering from the laser-plasmainteraction remains a complex research topic and onlya few experiments have been performed. Importantnumerical and experimental work has to be done to un-derstand the mechanism. This research will be motivatedby the fact that this source of radiation can becomea powerful x-ray source. Indeed, the development ofpetawatt-class lasers will open novel perspectives fornonlinear Thomson scattering. Assuming that a = 20can be reached, the produced radiation is expected tohave the following features. The source is extended fromthe XUV to the x-ray part of the spectrum with photonenergies up to ∼
10 keV. In addition, the number ofphotons will be significantly increased: ∼ ∼
150 mrad.Such a source has the main advantage to be very simpleto realize. Further developments would rely on thecontrol of the preacceleration of the electron inside theplasma in order reduce the divergence of the x-ray beam,enhancing the brightness. Flattop or annular transverselaser profile should be also interesting to maintainelectrons in the high-intensity regions and increasethe number of x-ray photons. Finally, several groupsreported on the possibility to produce attosecond pulsesvia nonlinear Thomson scattering (Lan et al. , 2006; Lee et al. , 2003). In fact, the nonlinear Thomson scatteringfrom a single electron is naturally a train of attosecondpulses (Lee et al. , 2003). However, when summing theradiation from all electrons, the property disappearsunless electrons radiate coherently (instead of incoherentradiation, as presented throughout this section). If thislatter condition is satisfied, it becomes possible to gener-ate single intense attosecond pulse (Lan et al. , 2006). Tofulfill the coherence condition, the use of ultrathin solidtargets has been suggested (Lee et al. , 2005). A schemenamed “lasetron” for generating coherent zeptosecondpulses has also been proposed by Kaplan and Shkolnikov(2002), and is based on the wiggling of electrons in asubwavelength-size solid particle or thin wire submittedto two counterpropagating circularly polarized petawattlaser pulses.5 Z ϑ ∼ K/ γ XY Relativistic Electronfrom LWFAPlasma acceleratorgas X - r a y s Counter-propagating laser pulse
FIG. 34 Schematic of the all-optically driven Thomsonbackscattering source. A laser-plasma accelerated electronbeam is injected in a counterpropagating laser pulse. Elec-trons oscillate transversally and emit x rays or γ rays due tothis motion. B. Thomson backscattering
X-ray or γ -ray radiation can be produced by scatteringan electromagnetic wave off a counterpropagating rela-tivistic electron bunch. This scheme was first proposed in1963 (Arutyunian and Tumanian, 1963; Milburn, 1963)and reinvestigated in the 1990s thanks to the advancein laser technology and high-brightness electron accelera-tors, suggesting the possibility to realize a source of high-brightness pulsed x rays or γ rays (Esarey et al. , 1996a,1993a,b; Gao, 2004; Hartemann, 2002; Hartemann et al. ,2001; Hartemann and Kerman, 1996; Hartemann et al. ,1996; Kim et al. , 1994; Krafft et al. , 2005; Li et al. , 2002;Ride et al. , 1995; Sprangle et al. , 1992; Yang et al. , 1999).More recently, when the research on electron acceler-ation from laser-plasma interaction became mature, itwas proposed to use these laser-plasma accelerated elec-tron bunches to develop an all-optically driven scheme(Catravas et al. , 2001; Hafz et al. , 2003; Hartemann et al. ,2007; Leemans et al. , 2005; Tomassini et al. , 2005). Theprinciple of this scheme is represented in Fig. 34. Twolaser pulses are required: the first drives the plasma ac-celerator, as discussed in Sec. III, and the second scat-ters off the accelerated electrons. In order to derive thetypical properties and analytical features of the Thomsonbackscattered radiation, the electron dynamics in a coun-terpropagating laser pulse in the simplest model of planeelectromagnetic wave is presented. Then, numerical andexperimental results are discussed. Foreseen perspectivesof this method are described in the conclusion.
1. Electron orbit in a counterpropagating laser pulse
We consider a laser-plasma accelerator producing amonoenergetic electron bunch of energy E = γ i mc witha velocity directed along (cid:126)e z (toward the positive val-ues). The counterpropagating laser pulse is modeled bya linearly polarized plane electromagnetic wave of wavevector (cid:126)k i = − π/λ L (cid:126)e z and frequency ω i = 2 πc/λ L , and with a normalized vector potential given by (cid:126)a = a cos( ω i t + k i z ) (cid:126)e x . The equation of motion of the testelectron and the Hamiltonian ˆ H describing the test elec-tron dynamics in the electromagnetic wave are given byEqs. (82) and (83). We use the same normalized units,with the choice m = c = e = ω i = 1. As in Sec. VI.A.1,the system is integrable and action-angle variables can befound to solve the electron motion. Here we analyze thesymmetries of the system to directly yield the conservedquantities according to Noether theorem. The Hamilto-nian ˆ H depends on the canonical momentum ˆ (cid:126)P and onthe potential vector (cid:126)a ( ϕ ) through the variable ϕ = ˆ t + ˆ z .Hence ˆ H is independent of ˆ x and ˆ y , which implies thatthe transverse canonical momentum is a constant of mo-tion. In our problem, it is assumed that the electron hasa velocity directed along (cid:126)e z exclusively before enteringthe laser pulse. Therefore the first constant of motionreads ˆ (cid:126)P ⊥ = ˆ (cid:126)p ⊥ − (cid:126)a = (cid:126) . (96)In addition, ˆ H depends on ˆ t and ˆ z only through ϕ =ˆ t + ˆ z . Thus ∂ ˆ H /∂ ˆ t = ∂ ˆ H /∂ ˆ z , which leads to the secondconstant of motion C, γ + ˆ p z = C . (97)The constant C is obtained using the initial conditions γ = γ i , ˆ p z = (cid:112) γ i −
1. The integration of the constantsof motion gives the electron orbit:ˆ x ( ϕ ) = a C sin( ϕ ) , (98)ˆ y ( ϕ ) = 0 , (99)ˆ z ( ϕ ) = (cid:40) − a / (cid:41) ϕ − a sin(2 ϕ ) , (100) γ ( ϕ ) = C2 + 1 + a cos ( ϕ )2C , (101)where C = γ i + (cid:112) γ i − γ i − / (2 γ i ) + o (1 /γ i ).The trajectory obtained cannot exactly take the form x ( z ) = K/ ( γk u ) sin( k u z ) corresponding to the standardsinusoidal trajectory studied in Sec. II. Here γ is notconstant along the orbit and Eq. (97) implies that γ and p z vary oppositely. The oscillation motion z takes, how-ever, a form similar to the sinusoidal trajectory, leadingto the same characteristic figure-eight motion in the aver-age rest frame of the electron. The description of Sec. IIcan be recovered with the approximation ϕ (cid:39) z (cid:39) t .The counterpropagating laser pulse can be seen as anundulator with a spatial period given by λ u = λ L / , (102)and a strength parameter K given by K = a = 0 . (cid:113) I [10 W/cm ] λ L [ µ m] . (103)6
2. Radiation properties
The qualitative features of the radiation remain thesame as in Sec. II and can be parametrized by K , λ u ,and γ (cid:39) γ i (for a (cid:28) γ i ). The spectrum of the emittedradiation depends on the amplitude of the parameter K .For K (cid:28)
1, the electromagnetic wave acts as an undula-tor. In the average rest frame of the electron, Thomsonscattering occurs in the linear regime. In the laboratoryframe, the radiation is emitted at the Doppler shifted fun-damental frequency corresponding to a photon energy (cid:126) ω in the forward direction ( θ = 0). As K →
1, harmon-ics of the fundamental start to appear in the spectrum.For K (cid:29)
1, the electromagnetic wave acts as a wig-gler, and the spectrum contains many harmonics closelyspaced and extends up to a critical energy (cid:126) ω c (a nonlin-ear Thomson scattering occurs in the average rest frameof the electron). These energies are given by (cid:126) ω = (4 γ hc/λ L ) / (1 + K /
2) for
K < , (cid:126) ω c = 3 Kγ hc/λ L for K (cid:29) . (104)This gives in practical units (cid:126) ω [eV] = 4 . γ /λ L [ µ m] for K (cid:28) , (cid:126) ω c [eV] = 3 . γ (cid:113) I [10 W/cm ] for K (cid:29) . (105)The radiation is collimated within a cone of typicalopening angle θ r = 1 /γ in the undulator case. For a wig-gler, the radiation is collimated within a typical openingangle K/γ in the electron motion plane ( (cid:126)e x , (cid:126)e z ) and 1 /γ in the orthogonal plane ( (cid:126)e y , (cid:126)e z ).As described in Sec. II, the duration of the emittedradiation is approximately equal to the duration of theelectron bunch. The duration of the electron bunch beingfemtoseconds at the position of interaction (Lundh et al. ,2011), the duration of the x-ray pulse is femtoseconds aswell.The number of emitted photons follows the expressionsof Sec. II. In practical units, the number of photonsemitted per period and per electron (at the mean photonenergy E = 0 . (cid:126) ω c for the wiggler limit) is given by N γ = 1 . × − K for K < ,N γ = 3 . × − K for K (cid:29) . (106)From these expressions the features of the emitted ra-diation can be estimated. For a 100 MeV electron bunch( γ (cid:39) a = 0 . λ L = 800 nm,the electron oscillation period is λ u = 400 nm and theemitted radiation has an energy (cid:126) ω ∼
200 keV. The ra-diation is collimated within a cone of typical solid angleof 5 mrad × × photons are emitted peroscillation period. Considering a laser pulse containingaround ten optical cycles, the number of emitted photonsis ∼ . -0.0008-0.0004 0 0.0004 0.0008 0 2 4 6 8 10 12 X ( μ m ) Z ( μ m) -4 -4 -4 -8x10 -4 FIG. 35 Electron trajectories. The orbit is plotted as thered solid line for the undulator case ( a = 0 .
2) and as thegreen dashed line for the wiggler case ( a = 2). Before theinteraction, the electron has γ = 200 and (cid:126)p = (cid:112) γ − (cid:126)e z in both situations. The counterpropagating laser pulse has aGaussian temporal profile with FWHM duration of 20 fs anda central wavelength of 800 nm.
3. Numerical results
The electron equation of motion can be numericallyintegrated. The case of an electron initially propagat-ing in the (cid:126)e z direction (toward the positive values) with γ = 200 is investigated. The counterpropagating laserpulse is modeled by a linearly polarized plane wave with aGaussian temporal envelope with τ L = 20 fs at full widthat half maximum (FWHM), at the wavelength 0.8 µ m.Two laser strength parameters are considered: a = 0 . a = 2 for the wiggler case.Figure 35 shows the orbits of a test electron travelingin the counterpropagating laser pulse for each situation.The transverse motion consists of an oscillation at a pe-riod λ u = 0 . µ m and its amplitude follows the envelopeof the laser pulse and increases as a increases. The max-imum amplitudes are 6 . × − and 6 . × − µ m for a = 0 . a = 0 . /γ ∼ a = 2 (bottom), the radiation has a divergence of typ-ical opening angle a /γ ∼
10 mrad in the plane of theelectron motion and 1 /γ ∼ a = 0 . . × .
5, 4 ×
4, and10 ×
10 mrad ), and over the overall angular distributionfor the wiggler case a = 2. For a = 0 .
2, the spectrumis nearly monochromatic at the energy (cid:126) ω ∼
200 keV inthe forward direction, with a width of the emission linegiven by ∆ λ = λ/N (where N ∼ − λ (cid:39) ( λ u / γ )(1 + K / γ θ ) [seeSec. II, Eq. (5)]. For a = 2, the spectrum becomesbroadband with a critical energy of about 300 keV. Anestimate of the total photon number can be obtained onthe basis of the number of electrons that are currently ac-7 θ X (mrad) -10 0 100-10100-10 210 P h o t o n s / . % B W × -6 × -6 Energy (keV)
10 x 10 mrad × -4 × -5 × -5 R a d i a t e d e n e r g y ( - J s r - ) a = 0.2a = 2 a = 0.2a = 2 θ Y ( m r a d ) θ Y ( m r a d ) -10 0 10 R a d i a t e d e n e r g y ( - J s r - ) P h o t o n s / . % B W
0 400 800 12000 200
400 600
FIG. 36 On the left are represented the angular distributionsof the radiation (radiated energy per unit solid angle) andthe corresponding spectra are shown on the right (in num-ber of photons per 0.1% bandwidth, per electron, and pershot). The undulator case, a = 0 .
2, is on the top whereasthe wiggler case, a = 2, is on the bottom. In the first sit-uation, three different solid angles of integration have beenconsidered to highlight the broadening of the spectrum. Inthe wiggler situation, the integration is performed over thetotal angular distribution. celerated in a laser-plasma accelerator. Assuming 1 × electrons gives ∼ × photons within a spectral band-width of ∼ . × . solid angle). In the wiggler case, ∼ × photons / . − are expected.
4. Experimental results
Many experiments of Thomson backscattering havebeen performed using electrons from conventional ac-celerators. The first experiment to actually use laser-backscattered photons as a beam in a physics measure-ment was conducted at SLAC in 1969 (Ballam et al. ,1969). Many other experiments followed this approachto produce x rays and γ rays (Albert et al. , 2010; Ander-son et al. , 2004; Babzien et al. , 2006; Bula et al. , 1996;Burke et al. , 1997; Glotin et al. , 1996; Kotaki et al. , 2000;Leemans et al. , 1997; Litvinenko et al. , 1997; Pogorel-sky et al. , 2000; Sakai et al. , 2003; Sandorfi et al. , 1983;Schoenlein et al. , 1996; Ting et al. , 1995; Yorozu et al. ,2003, 2001), and nowadays several γ -ray facilities arebased on this method.A significant milestone has been achieved with thefirst demonstration of femtosecond x-ray generation byThomson backscattering (using electrons from conven-tional accelerators) in 1996, by Schoenlein and coworkers(Leemans et al. , 1997; Schoenlein et al. , 1996). They re-ported on the production of femtosecond x-ray pulses bycrossing, at a right angle, a 50 MeV electron beam with a Relativistic e- bunch Laser pulse (fs) S o li d f o il Back-reflected laser pulse High energy x-ray beamWakefield cavity z Plasma mirrorGas
FIG. 37 Principle of the all-optical Thomson source based onthe combination of a laser-plasma accelerator and a plasmamirror.
100 fs duration, 0.8 µ m wavelength, terawatt laser. Thelaser pulse contained 60 mJ and was focused on a 90- µ m-diameter electron bunch (1.3 nC in a 20 ps FWHMbunch). In this experiment, ∼ × photons were pro-duced with a maximum energy of 30 keV. However, dueto the geometry and the mismatch of the electron bunchand laser pulse durations, the number of electrons par-ticipating in the production of radiation was limited tothe portion of the bunch selected by the laser pulse. Thex-ray pulse duration was theoretically predicted to be ∼
300 fs due to the extension of the overlapping regionand the radiation was found to be collimated within ∼ et al. , 2006; Ta Phuoc et al. , 2012). The first, performed in 2006, was based onthe use of two laser pulses in a counterpropagating geom-etry (Schwoerer et al. , 2006). The first pulse (85 fs, 800nm, and a = 3) was used to produce an electron bunchwith a Maxwellian spectrum whose temperature equals 6MeV and with a total charge in the picocoulomb range.The second laser pulse (85 fs, 800 nm, and a = 0 .
8) wascounterpropagating and focused on the electron bunch.In this experiment, about 3 × x-ray photons in the 400eV - 2 keV range were observed within a 80 µ sr detectioncone at 60 mrad with respect to the laser axis. However,this type of experiment remains challenging because of8 b P h o t o n s / . % B W ( ) Energy (keV)
50 100 150 25020031 300 3505
0 10 20 -10 -20 -20 -10 0 10 20 θ x (mrad)
0 180 60 120 θ y ( m r a d ) FIG. 38 Spectrum of the Thomson backscattering radiationmeasured using a set of copper and aluminum filters. The in-set is the x-ray beam angular profile measured using a phos-phor screen. the difficulty to overlap in time and space the electronbunch and the counterpropagating laser pulse.Recently, a much more simple and efficient method,based on the use of one laser pulse only, has been demon-strated (Ta Phuoc et al. , 2012). This method has, for thefirst time, allowed the generation of all-optically drivenhigh-energy femtosecond x-ray beams. The scheme re-lies on the marriage of a laser-plasma accelerator anda plasma mirror. The principle is illustrated in Fig.37. An intense femtosecond laser pulse, focused into amillimeter-scale gas jet, drives a wakefield cavity in whichelectrons are trapped and accelerated. Then, the laserstrikes a foil placed at nearly normal incidence with re-spect to the laser and electron beam axis. Ionized by therising edge of the laser pulse, the foil turns into a plasmamirror which efficiently reflects the laser pulse (the reflec-tivity is up to 70%). Naturally overlapped in time andspace with the backreflected laser pulse, the relativisticelectrons oscillate in the laser field and emit a bright fem-tosecond x-ray beam by Thomson backscattering.The experiment was performed at the Laboratoired’Optique Appliqu´ee. A laser pulse (1 J, 35 fs, 810 nm,and a (cid:39) .
5) was focused onto the front edge of a super-sonic helium gas jet (2.1 mm plateau and 600 µ m gradi-ents on both sides) to generate the electron bunch. Theelectrons were accelerated in the forced laser wakefieldregime (Malka et al. , 2002). The electron spectrum wasbroadband with a maximum energy of 100 MeV and a to-tal charge of about 120 pC. A foil (either glass 1 mm orCH 300 µ m) was placed in the region of the exit gradientof the gas jet. At the interaction of the backreflected laserpulse with the electron bunch, a bright high-energy fem-tosecond x-ray beam is generated via Thomson backscat-tering. The x-ray spectrum, obtained by measuring the x-ray signal transmitted by filters of different thicknessesand materials, is presented in Fig. 38. With the electronspectrum broadband, the produced radiation spectrumwas broadband as well. The spectrum extends, as ex-pected, up to a few hundreds of keV and the total num-ber of photons is about 10 . The x-ray spectrum wasreproduced numerically considering the measured elec-tron spectrum and charge, and a 15 fs (FWHM) dura-tion and a = 1 . µ m (FWHM). The peak brightness is estimated to beof the order of 10 photons / (s mm mrad . γ -ray sources from conventional accelerators (Albert et al. , 2010; Schoenlein et al. , 1996). This high bright-ness is obtained thanks to the micron source size and thefemtosecond duration reached in this all-optical scheme.
5. Perspectives
Thomson backscattering is a remarkable source for theproduction of energetic radiation, since it allows one toconvert electrons into x rays or γ rays in an efficientmanner. Among the foreseen development, a promisingroute would be to take advantage of the latest progressmade in laser-plasma acceleration. Monoenergetic elec-tron bunches with tunable energy have recently been pro-duced with an optical injection scheme (Faure et al. ,2006). The all-optically driven Thomson backscatter-ing source, based on these electron bunches, would allowone to produce tunable and nearly monochromatic radi-ation. This would represent significant progress since todate none of the existing laser-based femtosecond x-raysources can produce monochromatic x-ray beams. Thom-son backscattering offers as well the possibility to gener-ate γ -ray radiation. For example, considering an electronenergy of 1 GeV and a counterpropagating laser pulse ofwavelength λ L = 800 nm, γ rays with an energy as highas 24 MeV can be produced. In this photon energy range,100 keV and beyond, radiography applications, of inter-est for industrial or medical applications, will be possi-ble. Taking advantage of the very small source size, high-resolution images can be obtained. Such high-energy ra-diation sources would also have many applications in nu-clear physics. An additional foreseen source developmentrelies on the production of a high repetition rate x-raysource, delivering bright photon beams in the tens of keVrange. In that case, electron bunches in the few tens ofMeV will be produced using compact 100 Hz-kHz lasersystems. Generating such a compact femtosecond x rayssource at high repetition rate, accessible to a university-scale laboratory, will certainly have unprecedented im-9pact for applications in time-resolved x-ray science, high-resolution radiography, and phase contrast imaging.Finally, a fully optical free-electron laser scheme basedon Thomson backscattering has also been proposed(Petrillo et al. , 2008) and will be discussed in Sec. VII.This scheme remains a feasibility study since it requiresan electron bunch quality which has not been achievedyet. However, such a source, if it becomes realizable,would revolutionize work in the community of x-ray ra-diation. VII. COHERENT RADIATION: TOWARD A COMPACTX-RAY FREE-ELECTRON LASER
In this section, the topic of free-electron laser (FEL)is introduced, especially in the XUV to x-ray range, andwe discuss the perspectives to get FEL light from laser-accelerated electron bunches. The free-electron laser con-cept was first proposed by Madey (1971) who, havingexperience with insertion devices (undulators and wig-glers) for light sources, realized that a laserlike amplifiercould be constructed by combining a high-quality elec-tron beam with an undulator or wiggler and an inputradiation beam. A few years later, the principle wasdemonstrated experimentally by Madey and co-workers(Deacon et al. , 1977).The electron beam inside the undulator acts as an ac-tive medium and provides the amplification of the radia-tion beam. The undulator is the coupling component,which permits energy exchange between the radiationand electrons. In practice, injecting an electron beamin an undulator will naturally lead to incoherent syn-chrotron emission, as discussed in the previous sections.This incoherent emission provides the input radiationbeam and can be amplified by the FEL process whencertain conditions are fulfilled (Bonifacio et al. , 1994,1984; Kondratenko and Saldin, 1980; Murphy and Pel-legrini, 1985; Saldin et al. , 1998). The mechanism can beexplained as follows. A single electron experiences theradiation from other electrons in the bunch. This inter-action between the bunch and its own radiation can leadto a microbunching of the electron distribution withinthe bunch at the fundamental wavelength of the radiationand its harmonics. This microbunching in return leads tocoherent emission from the overall electron bunch, whosepower is orders of magnitude higher than the usual inco-herent synchrotron radiation. However, this free-electronlaser mechanism requires restrictive conditions on theelectron beam quality and an important number N ofoscillations to take place. This situation, where the inco-herent synchrotron or spontaneous emission is amplifiedin a single pass, is commonly named the self-amplifiedspontaneous emission (SASE) configuration, and bringsa fluctuating radiation output (shot to shot) with poortemporal coherence. In the opposite way, injecting in the undulator an input coherent radiation beam, i.e. , an ex-ternal seeding beam, gives a stable radiation output withgood temporal coherence. Nevertheless, the latter caserequires a coherent source at a given radiation wavelengthwith a power higher than the synchrotron one (Lambert et al. , 2009), and such coherent sources are not yet avail-able in the x-ray domain. Hence, the SASE configurationseems to be the simplest route toward the production ofhard x rays.Many projects with a compact free-electron laser in thex-ray range of the spectrum (X-FEL) and operating FELfacilities are based on the SASE approach. The first facil-ity which has provided SASE radiation in the soft x-raydomain is the Free-Electron Laser at Hamburg (FLASH)which is currently working at wavelengths down to λ = 6nm (Tiedtke et al. , 2009). The Linac Coherent LightSource (Arthur et al. , 2002; Emma et al. , 2010; Mc-Neil, 2009) and the Spring-8 Angstrom Compact Free-Electron Laser (Ishikawa et al. , 2012; Pile, 2011) havedemonstrated FEL saturation at angstrom wavelengths,while the European X-ray Free-Electron Laser Altarelli et al. (2007) is under preparation. On the other hand,seeded free-electron lasers are considered to improve thebeam quality [for example, seeded FLASH (sFLASH) andFERMI@Elettra (Bocchetta et al. , 2007)], but they gen-erally aim to less energetic photons (soft x rays).For our purposes, the electron beam is generated us-ing a laser-plasma accelerator and the undulator can be aconventional meter-scale undulator (see Sec. V), an elec-tromagnetic wave undulator (see Sec. VI), or the plasmaundulator of the blowout regime (see Sec. IV). This latterdiffers significantly from the others because the strengthparameter of the undulator depends on the amplitude ofthe betatron oscillation and on the electron energy. Theamplification process has been studied for electrons in aion channel, and has been named ion channel laser (ICL)for this case. In addition, the spectral ranges of interestare from XUV to x rays.In the following, we first present a short description ofthe underlying physics of the FEL process and summa-rize the conditions needed to be applied on the electronbunch for setting the FEL amplification. The two differ-ent modes of operation are discussed: the amplificationof a seeding radiation beam and the amplification of theshot noise, i.e. , of the spontaneous synchrotron radiation(SASE regime). Finally, the required conditions for theFEL amplification are used to investigate its feasibilityfrom laser-accelerated electron bunches for three specificcases of undulator: the conventional one, the electromag-netic wave one, and the ICL one. We focus especially onthe conventional undulator, since it represents the mostpromising route for near-future developments toward acompact x-ray free-electron laser.0 A. The FEL amplifier
1. Principle of the free-electron laser process
To understand the physical mechanism of FEL ampli-fication, it is instructive to begin by the interaction ofa single electron with the radiation beam and the undu-lator (Brau, 1990a; Huang and Kim, 2007; Saldin et al. ,2000). In the following, a conventional undulator is con-sidered, i.e. , a periodic structure of magnets generatinga periodic static magnetic field (as shown in Sec. V), inorder to present the FEL amplification process and therequired conditions on the electron beam parameters.The undulator forces the electron to follow a sinusoidaltrajectory whose period corresponds to the undulator pe-riod λ u and whose amplitude is equal to Kλ u / (2 πγ ). Theterm K = eλ u B / (2 πmc ) is the undulator strength pa-rameter, B is the peak magnetic field generated by theundulator, and γ = E /mc is the relativistic factor of theelectron with E the electron energy. The electron prop-agates in the z direction and the magnetic field of theundulator is oriented in the y direction, such that themotion of the electron is confined in the Oxz plane. Interms of velocity, the motion of the electron inside theundulator is written as β x ( z ) = Kγ cos( k u z ) , (107)where k u = 2 π/λ u , β x = (cid:126)β.(cid:126)e x , and (cid:126)β = (cid:126)v/c is the elec-tron velocity normalized to the speed of light c . We thenconsider the effect of the radiation beam. This latteris modeled by a linearly polarized plane electromagneticwave whose electric field is (cid:126)E = E sin( kz − ωt ) (cid:126)e x (one-dimensional analysis), where k = 2 π/λ is the norm of thewave vector, ω = 2 πc/λ is the frequency, λ is the radi-ation wavelength, and E is the electric field amplitude.Assuming that the radiation field is small compared tothe undulator field ( K (cid:29) a = eE /mcω ), the expres-sion of the electron velocity (107) can be used in orderto obtain the rate of electron energy change using thekinetic energy theorem, d E dz (cid:39) d E cdt = − e(cid:126)β. (cid:126)E = − eKE γ cos( k u z ) sin( kz − ωt )= − eKE γ [sin ψ + sin( kz − ωt − k u z )] , (108)where ψ = kz − ωt + k u z is the ponderomotive phase. Anenergy exchange can occur between the electron and theradiation if the sine terms are nonzero after averagingover z . The phase ψ can remain constant if the increaseof the undulator phase k u z compensates the decrease ofthe radiation phase kz − ωt , so that the first sine term ismanifestly nonzero after averaging. This latter conditionis written, for the phase ¯ ψ = kz − ω ¯ t + k u z defined with the averaged arrival time ¯ t = (cid:82) dz/ ¯ v z , as d ¯ ψdz = 0 = k u + k − ω ¯ v z . (109)The above resonance condition states that when the elec-tron advances one undulator period, the radiation fieldhas slipped by one radiation wavelength λ . It can be putin the following form: λ = λ u γ (1 + K / . (110)We emphasize here that this equation is nothing else butthe fundamental wavelength of the electromagnetic fieldradiated by the electron moving in the undulator [see Eq.(5) in Sec. II]. This reinforces the physical intuition thatenergy exchange can occur between the electron and theradiation when the condition (110) is fulfilled. A morecareful analysis of the sine terms of Eq. (108) [see Brau(1990b) and Wiedemann (2007b)] shows that energy ex-change can also occur if λ n = ( λ u / nγ )(1 + K / i.e. ,for harmonics of the electromagnetic field radiated by theelectron. This exchange at harmonics of the fundamentalwavelength is possible because of the z oscillatory motionof the electron in the undulator. In addition, the analysisshows that Bessel functions appear in the multiplicativefactor, so that near the resonance at the fundamentalwavelength Eq. (108) becomes d E dz = − eA u E √ γ sin ψ, (111)where A u = K [ J ( x ) − J ( x )] / √ x = K / [4(1 + K / J and J are Bessel functions (for K (cid:28) A u (cid:39) K/ √ K (cid:29) A u (cid:39) K/ ψ is now implicitly assumed. In the following, we will notconsider harmonics and concentrate on the resonance atthe fundamental wavelength.The ponderomotive phase ψ plays a major role in theFEL dynamics and it is relevant to locate the electronusing the variable ψ instead of z , and to use z as the newtime variable. If E is the energy at resonance satisfying(110) and ∆ E = E − E is the energy of the electronrelative to the resonance energy, then for ∆ E (cid:28) E Eqs.(109) and (111) become dψdz = kγ z E ∆ E , (112) d ∆ E dz = − eA u E √ γ sin ψ, (113)where γ z = 1 / (cid:113) − β z . By differentiating Eq. (112),the pendulum equation is obtained, d ψdz + ekA u E √ γγ z E sin ψ = 0 . (114)1Hence, the electric field of the radiation E (all other pa-rameters being constant) can be interpreted as the am-plitude of the effective or ponderomotive potential of thependulum. There are two classes of trajectory for thependulum equation: trapped and untrapped. The curvein phase space ( ψ , E ) at the limit of both regimes is theseparatrix. The higher the electric field is, the higher thephase space volume of trapped conditions (volume insidethe separatrix) and the induced acceleration or decelera-tion for the electron.This analysis of the interaction of a single electron withthe radiation beam and the undulator can be used to de-scribe qualitatively the collective effect of FEL amplifi-cation occurring with a bunch of electrons. The initialcondition is the electron distribution in the phase space( ψ , E ). As discussed in Sec. II.F, the electron bunchcoming from a laser-plasma accelerator has a randomdistribution. For simplicity, we consider the case of auniform distribution in phase ψ (such that there is nosynchrotron radiation) and electrons at exact resonance E = E .Imposing that the electric field of the radiation E isconstant (the so-called small signal gain regime), thereare as many electrons which lose energy as electronswhich gain energy. As a result, there is no energy gainfor the radiation. In this regime, increasing the radia-tion energy requires E > E and decreasing the radiationenergy requires E < E .However, the FEL amplification takes place in thehigh-gain regime and the process significantly differs.When the electrons of the initially uniform beam startto move inside the ponderomotive potential, the electrondistribution is modulated with a period 2 π in ψ , whichcorresponds at a fixed time t to a modulation with period λ in z . This small modulation implies that the electronbeam radiates partially coherently, such that the electricfield of the radiation E increases and the mean energyof the electron beam decreases [in Eq. (113), the en-ergy variation averaged over all electrons is nonzero be-cause (cid:104) sin ψ (cid:105) (cid:54) = 0]. As a consequence, on the one hand,the amplitude of the ponderomotive potential increasesand, on the other hand, electrons lose more energy onaverage than predicted by the single electron model. Be-cause the ponderomotive amplitude increases, the volumeof trapped conditions also increases and electrons whichinitially were on untrapped trajectories can be finallytrapped in the ponderomotive potential well. Therefore,after a stage of modulation of the electron distribution,electrons tend to be trapped in the wells of the pon-deromotive potential. The amplification process can besummarized as the following: • The electromagnetic radiation induces a modula-tion of the electron beam. At the beginning, themodulation is sinusoidal and the regime is linear. E l e c t r i c fi e l d a m p li t u d e E ( a r b . un i t s ) Exponential growthNon-linear growthSaturation Oscillations
Position inside the undulator z (arb. units) -1 -2 -3 -4 -5 FIG. 39 Schematic of the FEL amplification process. A typ-ical profile for the electric field amplitude E along the un-dulator axis in linear scale (red solid line, left y axis) andlogarithmic scale (green dashed line, right y axis). The higher the electric field amplitude E is, thefaster the amplitude of the sinusoidal modulationgrows. • The electric field amplitude growth is proportionalto the beam modulation. Hence, it results in anexponential amplification, E ∝ exp( z/ l g ), where l g is the gain length in power. • The modulation of the electron density becomes onthe order of the mean electron density, i.e. , becomesnonlinear. This happens when electrons becometrapped in the wells of the ponderomotive poten-tial. The electron beam is microbunched at the ra-diation wavelength λ and radiates coherently. Thelargest fraction of output radiation is generatedduring this last stage. This leads to an importantloss of energy for the electron beam in such a waythat it becomes off-resonance. This corresponds tothe saturation: the radiation energy cannot be in-creased further in normal operation. • In addition, letting the process go on, electrons off-resonance fall in accelerating phases and extractenergy from the electromagnetic radiation. Whenthey recover sufficient energy to satisfy the reso-nance condition, they can amplify the radiationagain. Hence, the radiation amplitude starts to os-cillate after the saturation point.This FEL amplification process is illustrated in Fig. 39,where the radiation amplitude is plotted as a functionof z . This emphasizes the different stages of the process:2linear mode of amplification, nonlinear mode, saturation,and then oscillations.The efficiency of the amplification process, defined asthe ratio of the output total radiation energy to thetotal energy of the electron bunch, is limited at satu-ration by the fact that electrons lose energy and be-come off-resonance. As presented next, in the ideal1D case the efficiency at saturation equals the Pierceparameter ρ which is small compared to unity in allcases. Recalling that the resonance condition reads λ = ( λ u / γ )(1 + K / γ ) can be compensated by varying undulatorparameters (for example, the undulator gap and so K ),which allows the process to remain in resonance after thesaturation point and to continue to radiate at the wave-length λ in a fully coherent manner. This approach iscalled tapering and it permits the FEL efficiency η to beincreased well above the untapered one η (cid:39) ρ and evento approach efficiency of unity.
2. Required conditions on the electron beam parameters
In our simple illustration of the amplification process,any practical effects such as energy spread in the elec-tron beam have not been considered and tolerances ofthe FEL process to deviations from the ideal case havenot been discussed. Here the conditions requested fromthe electron beam parameters in order to allow amplifi-cation of an electromagnetic field are summarized. ThePierce parameter ρ (also named efficiency parameter andFEL universal parameter) is defined as ρ = 1 γ (cid:18) A u λ u λ p (cid:19) / = 12 γ (cid:18) II A (cid:19) / (cid:18) A u λ u πσ (cid:19) / , (115)where λ p = 2 πc/ (cid:112) n e e /m(cid:15) is the plasma wavelength(without relativistic correction), I is the electron beamcurrent, I A = 4 π(cid:15) mc /e (cid:39) . σ is the rms radius of the transverse distribution of theelectron beam, and A u = K [ J ( x ) − J ( x )] / √ ρ = 1 . × − A / u γ λ / u [cm] n / e [cm − ] (116)= 2 . × A / u γ λ / u [cm] I / [kA] σ / [ µ m] . (117) a. Steady-state 1D FEL theory In the steady-state 1D FEL theory, the Pierce parame-ter provides the following conditions applied on the elec-tron beam and the output values of the main parameters: • the rms relative energy spread of the electron beamhas to be smaller than ρ ,∆ EE (cid:46) ρ, (118) • the relative detuning of the mean energy E m of theelectron beam from the resonance ( E is the en-ergy at resonance corresponding to a given radia-tion wavelength λ ) has to be smaller than ρ , E m − E E (cid:46) ρ, (119) • the power gain length l g [defined by P ∝ exp( z/l g ),where P is the radiation power] is given by l g = λ u π √ ρ , (120) • the efficiency of energy conversion from the electronbeam to the radiation η = I radiation / ( N e γmc ) isfinally given at saturation by η ≈ ρ. (121)The output radiation does not depend on the energy ofthe input radiation if the length of the undulator is cho-sen in such a way that saturation is attained at its exit.This is because, whatever the input radiation is, the pro-cess ends when electrons have lost sufficient energy to beoff-resonance, and the amount of energy that they lose tobecome off-resonance is roughly given by ρ E . The totalradiation energy I radiation scales as N / e since the Pierceparameter ρ depends on the electron current as I / .The undulator length providing the saturation dependson the energy of the input radiation, and there is no sim-ple expression to estimate it. Nevertheless, its typicalvalue is on the order of 20 gain lengths, l sat ∼ l g .According to the above formulas, the Pierce parameter ρ has to be as high as possible. Indeed, this leads toless stringent conditions on the energy spread and detun-ing, smaller gain length, and so smaller undulator lengthand higher efficiency at saturation. The parameter ρ in-creases with the current or density as I / or n / e , withthe strength parameter as A / u and decreases with γ as1 /γ for constant undulator period λ u . However, for con-stant radiation wavelength λ , ρ increases with γ as γ / (the dependence on current or density is the same, whilethe dependence on K becomes more complex).Our next step will be to determine the limits of valid-ity of the steady-state 1D FEL theory. Working insidethe limits and the conditions of the 1D theory constitutesthe best configuration for a high FEL performance. Forout-of-limit configurations, one can use the Ming Xie for-mulas (Xie, 2000) (given below) which coarsely account3for 3D effects, or simulations with codes such as Gen-esis (Reiche, 1999) for example, which give precise andquantitative answers. b. Three-dimensional effects Three-dimensional effects and the associated limits ofthe 1D theory arise when considering the transverse di-mensions. First, do the electron beam and the radiationcorrectly overlap? Indeed, in three dimensions diffrac-tion occurs and radiation can escape the electron beamand become lost for the amplification process. To avoidthis, the Rayleigh length of the radiation has to be higherthan the gain length (to ensure gain guiding). Second,the electron beam having a finite emittance, it has an an-gular spread: electrons have slightly different directions,resulting in different values for γ z = 1 / (cid:113) − β z and inan effective energy spread. Imposing that this effectiveenergy spread is smaller than ρ provides a condition onthe unnormalized rms transverse beam emittance (cid:15) (seeSec. II.H). The above conditions or limits of the 1D the-ory can be written as z r ∼ π σ λ (cid:38) l g , (122) γ z (cid:15) σ (cid:46) ρ −→ (cid:15) (cid:46) λσ π √ l g . (123)The Ming Xie formulas allow these effects on the FELprocess to be estimated, by giving the modified gainlength and the modified efficiency at saturation (Xie,2000), l g −→ l Ming Xie g = l g (1 + Λ) , (124)Λ = 0 . . d + 0 . . (cid:15) + 3Λ γ , (125) η −→ η Ming Xie ∼ η (cid:32) l g l Ming Xie g (cid:33) , (126)where Λ d , Λ (cid:15) , and Λ γ are parameters taking into account,respectively, diffraction, emittance, and energy spread ef-fects, and are given byΛ d = l g λ πσ , (127)Λ (cid:15) = 4 π (cid:15) l g λσ , (128)Λ γ = 1 √ ρ ∆ EE . (129)These formulas show how 3D effects degrade the FELperformance, increasing the gain length and decreasingthe efficiency. However, they do not take into accounttime-dependent effects and space-charge effects. c. Time-dependent effects Time-dependent effects describe how the short dura-tion of electron bunches and variation of electron beamparameters inside the bunch influence the FEL amplifi-cation process. The previous formulas assume a steady-state electron beam, i.e. , infinitely long with constantdensity, for which slippage between radiation and elec-trons can be neglected. Realistic bunches have finite du-ration and the electron density varies along the bunch.The cooperation length is defined as the distance slippedby the radiation with respect to electrons in one gainlength, l c = l g ( c − v z ) /c = λ u π √ ρ γ z (130)= λ π √ ρ . (131)Electrons interact with each other via the radiationfield. However, this radiation field slowly slips on thebunch. The cooperation length indicates the size insidethe bunch where interaction between electrons can occurduring a gain length. As a direct consequence of thisnotion, one could consider that the bunch is made of in-dependent sub-bunches of length l c .The steady-state model is valid if the electron beam pa-rameters are approximately constant in a cooperationlength inside the bunch, and if the electron bunch lengthis much longer that the cooperation length. This lattercondition reads cτ (cid:29) λ π √ ρ , (132)where τ is the duration of the electron bunch.For bunch durations smaller than the above limit, theFEL process is highly degraded because radiation es-capes the bunch in less than a gain length (edge effects).This regime is called weak superradiance (Bonifacio andCasagrande, 1985; Bonifacio et al. , 1990, 1988, 1989). Su-perradiance, as defined by Dicke (1954), is a spontaneousemission from a coherently prepared system, whose en-ergy scales as the square of the number of electrons. FELsuperradiance is slightly different, because electrons be-gin to bunch when interacting with the spontaneous emis-sion and evolve toward a coherently prepared system.The radiation energy of the weak superradiance regimethen scales as N e instead of the N / e steady-state depen-dence, but is much lower than the steady-state radiationenergy. In contrast, for long electron bunches, strongsuperradiance can occur if the weak superradiance radia-tion emitted by the trailing edge of the bunch propagatesand slips through the overall bunch and is subsequentlyamplified. In that case, the peak power is much greaterthan the steady-state saturation value.The notion of cooperation length also indicates thatthe condition on the energy spread does not have to beapplied on the overall electron bunch. It is sufficient that4each sub-bunch of length l c has an energy spread smallerthan ρ . The energy spread calculated within a coopera-tion length is called the slice energy spread. The condi-tion (118) is relaxed and replaced by (cid:16) ∆ EE (cid:17) slice (cid:46) ρ. (133)Requiring transverse emittance can also be replaced withthe same conditions applying slice transverse emittance,defined in the same way as slice energy spread. d. Space-charge effects Another effect which influences the FEL performanceis space charge. When the FEL amplification processtends to microbunch the electron beam at the radiationwavelength, longitudinal space-charge forces, on the con-trary, tend to debunch the beam. The inverse plasmafrequency ω − p (time scale of plasma oscillations) can in-dicate over which distance debunching occurs, while thegain length characterizes the distance over which bunch-ing forces are significant. In the electron rest frame, thelongitudinal distance is γ times higher than in the labora-tory frame, so the density is divided by γ and the plasmafrequency by γ / . Hence plasma oscillations take placeon a time scale γ / /ω p in the rest frame, and because oftime dilatation, they take place on a time scale γ / /ω p in the laboratory frame, corresponding to a covered dis-tance of γ / c/ω p . Longitudinal space-charge forces arethen negligible when γ / c/ω p (cid:38) l g , (134)where ω p = (cid:112) n e e /m(cid:15) is the plasma frequency withoutany relativistic correction.Transverse space-charge forces can also influence the FELperformance. In the envelope equation for the elec-tron beam, the focusing field, emittance term, and thespace-charge term appeared. To neglect transverse space-charge forces, the corresponding term in the envelopeequation has to be small compared to the other terms, i.e. , the focusing one and the emittance one (Humphries,1990). e. Undulator errors and wakefields The practical realization of a free-electron laser con-sists in many undulator sections and transport devices.The magnetic field of undulators has small errors andsections can be slightly misaligned, resulting in a degra-dation of the FEL performance. In addition, a high-current electron beam propagating between the two re-sistive walls of small-gap undulators induces a wakefieldwhich can modify the beam properties along the propa-gation, altering the FEL process. See Huang and Kim(2007) for a description of these effects. f. Quantum FEL
The FEL process described above is based on a clas-sical treatment, so it is relevant to ensure that quantumphenomena are unimportant by determining the limitfor which the dynamics is effectively described by clas-sical mechanics. The FEL dynamics becomes quantumif the momentum recoils ∆ (cid:126)p = − (cid:126) (cid:126)k of an electron emit-ting a photon of wave vector (cid:126)k is greater than the FELbandwidth ρ | (cid:126)p | = ρ E /c , i.e , if the electron becomes off-resonance after emitting a photon. Hence, in a quantumFEL, each electron radiates only one photon. In all real-izable cases, (cid:126) k (cid:28) ρp and quantum effects are completelynegligible so that the classical FEL theory correctly de-scribes the process.To conclude, we emphasize that 1D formulas are use-ful for simple estimations, and Ming Xie formulas (Xie,2000) for corrections due to energy spread, diffraction,and emittance. For quantitative study of a given config-uration and for time-dependent effects and space-chargeforce effects, self-consistent, 3D, and time-dependentcodes such as Genesis can be used (Reiche, 1999).
3. Seeding or self-amplified spontaneous emissionconfigurations
In the above description of the FEL amplification pro-cess we did not discuss where the input radiation fieldcame from. The FEL radiation can be started either byan initial seeding field or by a random modulation of theelectron beam (SASE regime).The above description of the FEL amplification processdirectly applies to the seeding case. This case provideshigh-quality output radiation, but requires that a rela-tively intense coherent radiation field at the given wave-length λ can already be generated, which is not the casefor high-energy photons as x rays. In principle, high-order harmonics (HHG) generated in gas allow one toseed FEL up to the extreme ultraviolet part of the spec-trum at the present state of the art. A recent experi-ment has demonstrated the FEL amplification of a seed-ing beam at 160 nm by 3 orders of magnitude (Lambert et al. , 2008). These schemes, and, in particular, HHGon solid targets (Dromey et al. , 2006; Tarasevitch et al. ,2000; Teubner and Gibbon, 2009), could be extended tox-ray photons in the near future and could provide a seedfor X-FEL. However, the seeding power has to be higherthan the power of the incoherent synchrotron or sponta-neous emission (Lambert et al. , 2009), and this latter isproportional to the desired photon energy. This meansthat it will be more and more difficult to have a suit-able seeding beam for smaller radiation wavelength andhigher photon energy because the required power will behigher.The case of an initially randomly modulated electronbeam (SASE configuration) is different and the theoret-5ical description is more complex. No radiation input en-ters with the electron beam inside the undulator. As dis-cussed in Sec. II.F, electrons inside bunches delivered byconventional or laser-plasma accelerators are randomlydistributed in space. Whereas if the electron distribu-tion in space is uniform no radiation is emitted, for arandom one, incoherent radiation is emitted (synchrotronradiation, whose power is proportional to the number ofelectrons) and this incoherent radiation provides a seedfor the FEL amplification process. Moreover, the inco-herent radiation is emitted at the wavelength given byEq. (110), and therefore the resonance condition is auto-matically satisfied. The main advantage of this methodis straightforward: it allows one to extend FEL radia-tion to high-energy photons because no external coher-ent radiation is needed. A major experiment demon-strating self-amplified spontaneous emission gain, expo-nential growth, and saturation was achieved by Milton et al. (2001) and highlighted the possibility to developan operational X-FEL. Nowadays, most of the X-FELprojects ( λ < l c /c whilethe spikes of the spectrum have typical relative widthof λ u /l b , where l b is the bunch length (the FEL processsmooths the parameters of the electron beam and theradiation over the scale of the cooperation length). B. Free-electron laser from a laser-plasma accelerator
From the different required conditions on the electronbeam parameters (see Sec. VII.A.2), the feasibility ofa FEL from a laser-plasma accelerator can be investi-gated for each specific undulator or wiggler type. Theabove results which are strictly valid for the conventionalundulator give only good estimations for the other ones[for example, see Bacci et al. (2008) for the comparisonbetween electromagnetic wave and conventional undula-tors].
1. With a conventional undulator
Conventional undulators have typically centimeter pe-riod and strength parameter of unity. Hence, accordingto the resonance condition (110), using electron bunchesat the 100 MeV level, the FEL radiation wavelength isin the UV to XUV range, whereas at the 1 GeV level itreaches the soft x-ray and hard x-ray ranges. In the fol- lowing, we consider two different scenarios, X-FEL 1 and2, which are speculative and can be seen as future per-spectives in the short term and long term, respectively, oflaser-plasma accelerators. All input parameters of thesescenarios and the corresponding output FEL parametersare given in Table I. Table I contains the results obtainedby 3D time-dependent Genesis simulations in the SASEconfiguration.The choice of the electron beam parameters is moti-vated by extension of the 100 MeV range state-of-the-art laser-plasma accelerator performance to higher elec-tron energies. The electron beam peak current is main-tained constant when going to higher electron energiesbecause the requirement of small energy spread imposesan optimized beam loading which flattens the accelerat-ing electric field along the bunch (Rechatin et al. , 2009a).The optimized peak current scales as the normalized vec-tor potential a of the laser pulse and is independent ofthe electron density of the plasma n e . Because the elec-tron accelerator efficiency scales as 1 /a , a , and there-fore the electron beam peak current, should not be in-creased when designing a higher energy laser-plasma ac-celerator. The absolute energy spread and the normal-ized emittance can remain constant in the accelerationprocess if beam loading is optimized and if quasista-tionary forces are applied to the electron beam. There-fore, in principle, the low absolute energy spread (a fewMeV) (Rechatin et al. , 2009b) and normalized emittance (cid:15) N = 1 π. mm . mrad (Brunetti et al. , 2010) achievable forelectrons at the 100 MeV range could also be obtainedat 1 and 5 GeV.Note that the bunch rms transverse size σ is exper-imentally chosen by the transport configuration. Ofcourse, this choice determines the rms angular spread θ = (cid:15)/σ . In order to have the Pierce parameter as highas possible, σ has to be small, but, on the other hand,emittance, diffraction, and space-charge effects increasewhile σ decreases. In addition, practical considerationson the transport realization along the undulator put alimit on the smallest transverse size achievable.Since the topic of short wavelength free-electron laserusing conventional undulators has been extensively stud-ied by means of conventional accelerators, we discuss herethe differences which arise when using electron bunchesproduced by laser-plasma accelerators. The first differ-ence is straightforward: because of the short duration ofthe electron bunch, currents can be higher which result inhigher ρ , smaller gain lengths, smaller undulator lengths,higher efficiency at saturation, less stringent conditionson the energy spread and detuning, but also in higherspace-charge effects.Second, the short duration of the electron bunch can be-come comparable or smaller than the cooperation lengthdivided by c , leading to important edge effects which de-grade the FEL performance.6 TABLE I Parameters of different scenarios for realizing free-electron lasers from laser-driven plasma-based accelerators in theXUV and x-ray domains with the corresponding FEL quantities obtained from Genesis simulations in SASE configuration. Forall scenarios, the undulator period λ u is equal to 1 cm and the strength parameter K is equal to 1.X-FEL 1 X-FEL 2Electron beam input parametersElectron energy 1 GeV 5 GeVPeak current 10 kA 10 kABunch rms duration 4 fs 4 fsBunch rms transverse size 30 µ m 30 µ mrms energy spread 0.2% 0.04%Normalized rms transverse emittance 1 π. mm . mrad 1 π. mm . mradFEL output parametersRadiation wavelength λ ρ µ J 25 µ Jrms pulse duration 2 fs 2 fs
Third, the electron bunch transport from the plasmasource to the undulator is specific: even if the emittanceis comparable to that of conventional accelerators, at theexit of the plasma, the transverse size is very small (afew microns) when the divergence is higher (a few mil-liradians). The transverse size grows fast: 10 cm afterthe plasma it reaches a few hundreds of microns, and 1m after it reaches a few millimeters. For such size, chro-matic aberrations of quadrupoles (which are proportionalto the electron beam energy spread) are large, leading toan enormous growth of emittance, and because electronscover different distances, the longitudinal bunch lengthalso grows. It is therefore necessary to design an ultra-compact transport system which acts near the plasmasource and does not let the beam transverse size grow.Space-charge forces, higher in the case of laser-plasmaaccelerators, also strongly affect the beam transport, in-creasing emittance, longitudinal bunch length, and totalenergy spread [they induce a linear energy chirp: thehead of the bunch moves faster than the tail (Gr¨uner et al. , 2007, 2009)].We comment now on the results obtained for the sce-narios considered in Table I. They are illustrated inFig. 40, which shows the longitudinal profile of radiationpower as a function of the position inside the undula-tor for each scenario. In both configurations, X-FEL 1and 2, the small values of the Pierce parameter imposestringent conditions on the electron beam energy spread.As can be seen in Table I, the energy spread is close to ρ in both cases, and using a higher electron beam en-ergy spread strongly reduces the FEL performance. InX-FEL 1, the output FEL radiation consists of a fewSASE spikes, meaning that the bunch length is of theorder of a few cooperation lengths and that edge effectsdo not degrade the FEL performance. Moreover, diffrac-tion, emittance, and longitudinal space-charge effects are FIG. 40 Longitudinal profile of radiation power as a functionof the position inside the undulator obtained from Genesissimulations. The corresponding electron beam and undulatorparameters are indicated in Table I. Top: X-FEL 1. Bottom:X-FEL 2. The position relative to the electron beam centeris defined as s = ct e − ct , where t e and t are, respectively, thearrival time of the electron beam center and of the radiation,at a given position z in the undulator. µ J pulse energy, is produced at a wavelength of 1.96 nmwith an undulator length of 12.5 m.In X-FEL 2, the radiation wavelength reaches theangstrom range, a spectral region of great interest fordiffraction experiments, more particularly on biologicalsamples. In these conditions, the bunch length equalsmany cooperation lengths and the notion of slice energyspread is helpful: it is sufficient that only the slice en-ergy spread is below 0.04%, not the total energy spread.Here, because of the higher γ and the smaller ρ , the re-quirements on the relative slice energy spread and on theundulator length are more stringent. The output FEL ra-diation has a 15 GW peak power and 25 µ J pulse energyat a wavelength of 0.0783 nm, for an undulator length of100 m. In this situation, diffraction, longitudinal space-charge and edge effects are found to be negligible whileboth emittance and energy spread have an impact on theFEL performance, which may explain the slightly lowerperformance than in the previous scenario.In summary, laser-plasma accelerators have promisingperspectives in the realization of compact free-electronlasers, by combining them with conventional undulators.The scenarios X-FEL 1 and 2 highlight the possibilityto produce strong FEL radiation from a laser-plasma ac-celerator. The most challenging condition is the rela-tive energy spread of the electron beam, which has to bedecreased to attain the x-ray part of the spectrum. Inaddition, high-quality electron beams at the GeV levelare required [1 GeV electron beams have already beengenerated (Leemans et al. , 2006) but not with the samequality, i.e. , energy spread and stability, as at the 100MeV level]. The notion of slice energy spread and sliceemittance can facilitate reaching the FEL requirementsfor the highest-energy case.
2. With an electromagnetic wave undulator
In Sec. VI, the incoherent radiation of an electronbeam submitted to a counterpropagating electromagneticwave was presented. This wave acts as an undulator orwiggler whose strength parameter K equals the dimen-sionless vector potential amplitude a of the laser pulseand whose period equals λ L /
2, where λ L is the wave-length of the counterpropagating laser pulse. Currently,only two laser systems can deliver such a pulse with a onthe order of unity: Ti:Sa and CO , with wavelengths of0.8 and 10 µ m, respectively. From the resonance condi-tion (110), electron beams at the 10 MeV level radiate inthe keV (nanometer) range for CO lasers and in the 10keV (angstrom) range for Ti:Sa lasers. At the 100 MeVlevel, the radiation is in the 100 keV range and 1 MeV range for, respectively, CO and Ti:Sa lasers, and at theGeV level the radiation is in the 10 and 100 MeV rangefor, respectively, CO and Ti:Sa lasers. The use of anelectromagnetic wave undulator in the FEL process wasproposed in the 1980s (Danly et al. , 1987; Gallardo et al. ,1988; Gea-Banacloche et al. , 1987; Mima et al. , 1988) andreinvestigated recently (Bacci et al. , 2006, 2008; Maroli et al. , 2007; Petrillo et al. , 2008). It was also suggestedto couple an electron beam coming from a laser-plasmaaccelerator and a relativistic laser pulse, in the conditionswhere FEL collective amplification takes place, and thissystem has been called an all-optical free-electron laser(AOFEL) (Petrillo et al. , 2008).According to Pierce parameter scalings (115), for agiven radiation wavelength λ , using a smaller undulatorperiod λ u and smaller γ is unfavorable: the Pierce pa-rameter ρ decreases and the requirements on the electronbeam parameters are more stringent for the electromag-netic undulator than for the conventional undulator (re-member that for constant λ , ρ scales as γ / ). The mostrealistic scenarios for the electromagnetic undulator usehigh values of λ , i.e. CO laser and electron beams atthe 10MeV level.Petrillo et al. (2008) simulated the FEL collective am-plification taking place in the Thomson backscattering ofa relativistic laser pulse by an electron beam, this lattergenerated by a laser-plasma accelerator. First, they usea 2.5D particle in cell simulation (3D in the fields and2D in the coordinates) to obtain electron beam parame-ters. They considered the case of an injection realized bya down ramp in the plasma density profile and showedthat, after selecting a particular slice of the electron beamof longitudinal size 0.5 µ m, it is possible to obtain a sliceenergy spread of 0.4%, a slice transverse emittance of0 . π. mm . mrad, and a current of 20 kA at electron ener-gies of about 30 MeV. Then, inserting the exact shape ofthe slice in the FEL code Genesis 1.3 [using the equiva-lence between conventional and electromagnetic undula-tors developed by Bacci et al. (2008)], they obtained forthe SASE case the radiation output as a function of theundulator length: for a CO laser pulse with normalizedvector potential a = 0 .
8, coherent radiation at wave-length λ = 1 .
35 nm with subfemtosecond duration and200 MW peak power is generated for a saturation lengthof a few millimeters. These results open perspectives foran x-ray AOFEL using millimeter-scale electromagneticwave undulators. However, their simulation does not takeinto account the short propagation from the plasma exitto the electromagnetic wave undulator, which could de-grade the beam parameters at such low energies and highcurrents.To conclude, the AOFEL has the main advantage ofbeing compact and not needing a transport system. Be-cause of the short undulator period, high photon energiescan be achieved with small electron energies. Although8it allows one to theoretically generate hard x rays and γ rays, the FEL amplification for such wavelength de-mands extremely stringent conditions on energy spreadand emittance (the Pierce parameter is smaller for higherphoton energies). Hence, it is more reasonable to use elec-tron energies at the 10 MeV level and CO laser pulsesin order to succeed in operating a free-electron laser. Ex-perimentally, good quality electron beams (with percentenergy spread) have been demonstrated at the 100 MeVlevel (Rechatin et al. , 2009a), but not yet at the 10 MeVlevel. It is generally thought that, because accelerationconserves absolute energy spread, smaller relative energyspread is easier to obtain at higher electron energies.Hence, achieving very small relative energy spread at thelow-energy 10 MeV level seems to be difficult. From theperspective of an AOFEL, it is therefore necessary to ex-perimentally investigate the production of high-qualityelectron beams at rather small electron energies, whencurrently the tendency is toward higher and higher elec-tron energies.
3. With a plasma undulator
As shown in Sec. IV, the ion cavity, generated be-hind the laser pulse when the bubble or blowout regimeis attained, acts as an undulator or wiggler whose pe-riod is λ u = √ γλ p and whose strength parameter is K = r β k p (cid:112) γ/
2. Assuming that we can prepare exper-imental conditions in which the electron beam keeps aconstant energy inside the ion cavity ( γ is constant and λ u is too), then FEL amplification can in principle takeplace. For the case of a ion channel, the process has beenstudied (Whittum, 1992; Whittum et al. , 1990) and iscalled an ion channel laser (ICL). Additional conditionshave to be fulfilled: all electrons have to oscillate in thesame plane (for the radiation to be polarized identicallyfor all electrons) and with the same strength parameter(to radiate at the same wavelength) (Esarey et al. , 2002;Kostyukov et al. , 2003).The main difference between ICL and FEL is that, inICL, the strength parameter K depends on both the en-ergy and the transverse amplitude of motion r β , whenFEL it is a constant defined uniquely by the undulatorparameters. In ICL, different values of K lead to an ef-fective energy spread in γ z = γ/ (cid:112) K /
2. Imposingthat this energy spread is smaller than the Pierce param-eter ρ leads to the condition ∆ r β /r β (cid:46) ρ (2 + K ) /K forthe admissible relative spread in the amplitude of motion r β . Hence, the realization of ICL needs a configuration inwhich there is no acceleration to satisfy the condition ofconstant γ , where there is an off-axis injection with am-plitude r β and corresponding relative amplitude spreadroughly below ρ . To our knowledge, a method to im-plement such a configuration is currently not known. Inaddition, conditions analogous to those presented in Sec. VII.A.2 regarding the natural energy spread, emittance,and undulator length, also have to be fulfilled. VIII. CONCLUSION
In this paper, mechanisms that can produce short-pulse x-ray or gamma-ray radiation using laser-accelerated electrons have been described. We have seenthat all these sources are based on the radiation from arelativistically moving charge, and that even if the de-tails of the electron orbits are different for each source,the main features can be obtained using only five param-eters: the relativistic factor γ of electrons, the number ofelectrons N e , the strength parameter K , the period λ u ,and the number of periods N of the undulator or wiggler.In addition, the incoherent radiation provided by thesesources can eventually be amplified to coherent radia-tion via the FEL mechanism for sufficiently high-qualityelectron beams; the use of conventional undulators rep-resents the most promising route toward such goal. Herea summary of the features of these sources is given.Table II gives typical parameters and features as wellas scaling laws for the incoherent sources based on theuse of a plasma undulator (betatron radiation), a con-ventional undulator or an electromagnetic wave undula-tor (Thomson backscattering). For the produced sources,interaction parameters accessible with currently availabletens of TW-class lasers have been considered. The pho-ton spectra obtained with each mechanism show that anenergy range from eV to MeV can be covered. The maindifference between the different sources reviewed residesin their photon energy. Depending on the spectral rangerequired for a desired application, Table II helps to selectthe most suitable type of source.In the XUV domain, the conventional undulator cou-pled with a laser-plasma accelerator has been discussed.The emitted radiation can be monochromatic if the elec-tron bunch is monoenergetic. Up to ∼ photons col-limated within a few milliradians could be produced ina single shot using 10 electrons (160 pC) in the hun-dreds of MeV energy range and an undulator of strengthparameter K = 1 with 100 periods. The radiationwavelength can be tuned by varying the electron energy.Because the acceleration and the wiggling are dissoci-ated, a stable and/or tunable regime of acceleration canbe used, resulting in a stable and/or tunable radiationsource. From the perspective of GeV electron beams,this scheme will allow one to reach the keV range. How-ever, this source is more complex to realize because theelectron bunch must be transported into the undulator.The transport can degrade the electron bunch duration(and so the x-ray pulse duration) and the transverse emit-tance, leading to a broader bandwidth in the radiationspectrum and a smaller brightness. Finally, this scheme9 TABLE II The first subtable gives typical features for the different sources achievable with a 50 TW-class laser. The valuesrepresent the orders of magnitude of the parameters. The second subtable provides the scalings of the radiation features withthe relevant parameters K , λ u , and γ derived in Sec. II for sinusoidal trajectories. The last subtable presents the scalingsof the relevant parameters with the practical parameters for each schemes derived from simple ideal models which have beendescribed in each corresponding section. The practical parameters of each source are indicated in brackets. We refer to eachsection for the definition of the quantities used in this table.Typical features γ λ u K N N e (cid:126) ω/ (cid:126) ω c θ r N γ Betatron 200 150 µ m 10 3 10 ∼ Conventional undulator 400 1 cm 1 100 10
25 eV 2.5 mrad ∼ Thomson backscattering 400 0.4 µ m 1 10 10
650 keV 2.5 mrad ∼ Scalings of the radiation featuresFundamental radiation energy (cid:126) ω for undulators ( K <
1) (2 γ hc/λ u ) / (1 + K / (cid:126) ω c for wigglers ( K (cid:29) Kγ hc/λ u Typical opening angle of the radiation θ r for undulators ( K <
1) 1 /γ Typical opening angle of the radiation θ r for wigglers ( K > K/γ
Number of emitted photons per electron and per period N γ ,for undulators ( K <
1) 1 . × − K for wigglers ( K (cid:29)
1) 3 . × − K Scalings of the relevant parameters
K λ u Betatron ( n e , r β , and γ ) (cid:112) γ/ k p r β √ γλ p Conventional undulator ( B , λ u , and γ ) eB λ u / (2 πmc ) λ u Thomson backscattering ( a , λ L , and γ ) a λ L / deserves to be developed as it is on the path toward afree-electron laser based on laser-plasma accelerators.In the soft x-ray range, we discussed nonlinear Thom-son scattering. This radiation is produced by electronsoscillating in an intense laser field. While this source hasa spectrum extending in the few hundreds of eV rangeusing tens of TW-class lasers, it may become an inter-esting mechanism to generate keV x rays with PW-classlasers.In the range of a few keV, the betatron mechanism hasbeen demonstrated to be an efficient and simple methodto produce short x-ray pulses. This source can produceup to 10 photons per shot, collimated within less than50 mrad (FWHM). On the one hand, controlling the elec-tron orbits within the cavity in order to ensure a betterenergy transfer from the electron to the radiation couldallow one to extend the spectral range to the tens of keVrange, while still working with 50 TW-class lasers. Onthe other hand, petawatt-class lasers and/or capillarieswould be suitable to create larger ion cavities with lowerplasma density and to accelerate electrons to the GeVrange. This should allow one to produce x-ray beamswhose divergence is in the milliradian range and photonenergies in the tens and hundreds of keV range, with ahigher number of photons.Above 10 keV and up to a few MeV, the mechanismused is Thomson backscattering. Here high-energy ra-diation is produced thanks to the short period of theelectron motion. This scheme is similar to the case of alaser-plasma accelerator coupled with a conventional un- dulator, except that it benefits from the short period ofthe electron motion resulting in emitted photons of highenergies and it does not need transport. The photon en-ergy can be chosen by tuning the electron energy. Thisscheme has been demonstrated recently and it is the mostpromising for generating tunable incoherent bright hardx-ray and γ -ray radiation.The different schemes reviewed here produce incoher-ent radiation whose energy is proportional to the numberof electrons contributing to the emission. These sourcescan be compared, in some ways, to conventional syn-chrotron radiation. The main advantage provided bythe laser-plasma approach is, of course, the reduced size(from hundreds of meters or kilometers to the laboratoryroom) and the reduced cost (from multihundred milliondollars or multibillion dollars to multimillion dollars).Moreover, the pulse duration of laser-based sources isfemtosecond when it is commonly picosecond for conven-tional synchrotrons even if, with the latest technology,it can be reduced down to femtosecond (and this hasa certain cost). Another advantage is the natural syn-chronization of these sources with exciting laser pulsesneeded in pump-probe experiments, since both the pumpand the probe come from the same laser system. Again,external femtosecond synchronization in synchrotron fa-cilities is feasible with recent methods, but it remainsdifficult. The femtosecond x-ray sources based on therelativistic laser-plasma interaction presented in this re-view are attractive mainly because of their low cost, and0this is why perspectives cannot be extended indefinitelyto the use of higher intensity, since the cost will be-come comparable with conventional accelerators and syn-chrotron facilities. However, compared to synchrotrons,laser- and plasma-based sources present significant shot-to-shot fluctuations in terms of photon energy, numberof photons, and divergence. They can therefore not covera range of applications as large as synchrotrons. Thesesources remain at the research stage with limited appli-cations for the moment. Important challenging develop-ments are required to produced stable and robust x-raybeams appropriate for applications. In addition, the rep-etition rate is problematic: while synchrotron facilitieswork at MHz repetition rate allowing one to consider-ably increase the signal-to-noise ratio, laser-based sourceswork at 10 Hz on paper, but in practice they work in thesingle-shot regime in most 50 TW-class laser facilities toprevent rapid damage on optics such as in compressors.This means that theses sources are efficient for a single-shot experiment or experiments requiring not too muchdata accumulation.In Fig. 41, the x-ray sources are compared to otherexisting x-ray sources in terms of their peak brightness.This includes x-ray sources based on the laser-plasmainteraction such as high harmonic generation from rela-tivistic laser and overdense plasma interaction (Dromey et al. , 2007, 2006; Tarasevitch et al. , 2000; Teubner andGibbon, 2009; Thaury et al. , 2007), and K α radiationfrom laser-produced plasmas (Kieffer et al. , 1993; Mur-nane et al. , 1991; Rousse et al. , 1994), as well as x-raysources based on conventional accelerators: the undu-lator radiation source from the European SynchrotronRadiation Facility (ESRF), the femto-slicing-based un-dulator radiation source from the femto-laser station atthe Paul Scherrer Institut (PSI), and the Thomson γ -raysource from the Lawrence Livermore National Labora-tory (LLNL) (Albert et al. , 2010). Figure 41 shows thatlaser-plasma-based x-ray sources have competitive peakbrightness, in particular, the betatron radiation and theall-optical Thomson source, because of their femtosecondduration and their micron source size.As free-electron lasers are the natural evolution of syn-chrotrons, FEL based on laser-plasma interaction couldbe the next step in the development of laser- and plasma-based x-ray sources. In the free-electron laser process,the emitted radiation is amplified and becomes coher-ent with the number of photons increased by orders ofmagnitude. The most promising scenario uses a con-ventional undulator coupled to a laser-plasma acceler-ator. Because stringent conditions apply to the electronbeam parameters such as energy spread and transverseemittance, challenging developments are required in thelaser-plasma accelerator domain, including ultracompacttransport system. Thanks to the high brightness of FELradiation, many experiments could be performed in singleshot or with small data accumulation such that the low repetition rate of laser facilities becomes a more benign ~ ! (keV) P e a k b r i g h t n e ss [ ph o t o n s / ( s mm m r a d . % B W ) ] FIG. 41 Peak brightness for different types of x-ray sources:high harmonic generation from relativistic laser and overdenseplasma interaction (triangle and yellow line) [experiment con-ducted with the Vulcan Petawatt laser system at Ruther-ford Appleton Laboratory (RAL) by Dromey et al. (2006)], K α radiation from laser-produced plasmas (black star) [ex-periment conducted at the Laboratoire d’Optique Appliqu´ee(LOA) by Rousse et al. (1994)], nonlinear Thomson scat-tering (NTS) from relativistic laser and underdense plasmainteraction (diamond and cyan line) [experiment conductedat LOA by Ta Phuoc et al. (2003b)], conventional undula-tor radiation from laser-plasma accelerators (green square)[experiment conducted with the ATLAS laser system at theMax-Planck-Institut f¨ur Quantenoptik (MPQ) by Fuchs et al. (2009)], betatron radiation from laser-plasma accelerators(pentagon and blue line) [experiment conducted with the Her-cules laser system at the University of Michigan (U-Mich) byKneip et al. (2010)], Thomson backscattering (TB) radiationfrom laser-plasma accelerators (circle and red line) [experi-ment conducted at LOA by Ta Phuoc et al. (2012)], undu-lator radiation from conventional accelerators (green dottedline) [data from the European Synchrotron Radiation Facility(ESRF); see the ESRF website], femto-slicing-based undula-tor radiation from conventional accelerators (magenta dash-dotted line) [data from the femto-laser station at the PaulScherrer Institut (PSI); see the PSI website and Beaud et al. (2007)] and Thomson backscattering (TB) from conventionalaccelerators (black dashed line) [experiment conducted at theLawrence Livermore National Laboratory (LLNL) by Albert et al. (2010)]. The repetition rate and the pulse duration usedin the peak brightness estimation are given in the legend, andpermit one to convert peak brightness into average brightness. K , λ u , and γ . Finally, thepossibility to implement a free-electron laser using theseconfigurations has been discussed and the electron beamquality required for such realization has been presented. ACKNOWLEDGMENTS
We are grateful to C. Rechatin, O. Lundh, J. Faure,M. Ribiere and S. Sebban for fruitful discussions. Weacknowledge the Agence Nationale pour la Recherche,through the COKER Project No. ANR-06-BLAN-0123-01, the European Research Council through thePARIS ERC project (under Contract No. 226424), andLASERLAB-EUROPE/LAPTECH, EC FP7 ContractNo. 228334 for their financial support.
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