Radiation reaction in electron-beam interactions with high-intensity lasers
RRadiation reaction in electron-beam interactions with high-intensity lasers
T. G. Blackburn ∗ Department of Physics, University of Gothenburg, Gothenburg SE-41296, Sweden (Dated: March 31, 2020)
Abstract
Charged particles accelerated by electromagnetic fields emit radiation, which must, by the conservationof momentum, exert a recoil on the emitting particle. The force of this recoil, known as radiation reaction,strongly affects the dynamics of ultrarelativistic electrons in intense electromagnetic fields. Such environ-ments are found astrophysically, e.g. in neutron star magnetospheres, and will be created in laser-matterexperiments in the next generation of high-intensity laser facilities. In many of these scenarios, the energyof an individual photon of the radiation can be comparable to the energy of the emitting particle, whichnecessitates modelling not only of radiation reaction, but quantum radiation reaction. The worldwide de-velopment of multi-petawatt laser systems in large-scale facilities, and the expectation that they will createfocussed electromagnetic fields with unprecedented intensities > Wcm − , has motivated renewed in-terest in these effects. In this paper I review theoretical and experimental progress towards understandingradiation reaction, and quantum effects on the same, in high-intensity laser fields that are probed with ultra-relativistic electron beams. In particular, we will discuss how analytical and numerical methods give insightinto new kinds of radiation-reaction-induced dynamics, as well as how the same physics can be explored inexperiments at currently existing laser facilities. ∗ [email protected] a r X i v : . [ phy s i c s . p l a s m - ph ] M a r ONTENTS
I. Introduction 2II. Theory of radiation reaction 8A. Classical radiation reaction 81. In plane electromagnetic waves 9B. Quantum corrections: suppression and stochasticity 11III. Numerical modelling and simulations 16A. Classical regime 16B. Quantum regime: the ‘semiclassical’ approach 18C. Benchmarking, extensions and open questions 21IV. Experimental geometries, results and prospects 24A. Geometries 24B. ‘All-optical’ colliding beams 27C. Recent results 29V. Summary and outlook 33Acknowledgments 35References 35
I. INTRODUCTION
It is a well-established experimental fact that charged particles, accelerating under the actionof externally imposed electromagnetic fields, emit radiation [1, 2]. The characteristics of thisradiation depend strongly upon the magnitude of the acceleration as well as the shape of theparticle trajectory. For example, if relativistic electrons are made to oscillate transversely by afield configuration that has some characteristic frequency ω , they will emit radiation that hascharacteristic frequency 2 γ ω , where γ is their Lorentz factor. Given ω corresponding to awavelength of one micron and an electron energy of order 100 MeV, this easily approaches the100s of keV or multi-MeV range [3]. 2he total power radiated, as we shall see, increases strongly with γ and the magnitude of theacceleration. We can then ask: as radiation carries energy and momentum, how do we account forthe recoil it must exert on the particle? Equivalently, how do we determine the trajectory whenone electromagnetic force acting on the particle is imposed externally and the other arises fromthe particle itself? That this remains an active and interesting area of research is a testament notonly to the challenges in measuring radiation reaction effects experimentally [4], but also to thedifficulties of the theory itself [5, 6]. The ‘correct’ formulation of radiation reaction within classi-cal electrodynamics has not yet been absolutely established, nor has the complete correspondingtheory in quantum electrodynamics. While these points are undoubtedly of fundamental interest, itis important to note that radiation reaction and quantum effects will be unavoidable in experimentswith high-intensity lasers and therefore these questions are of immense practical interest as well.This is motivated by the fast-paced development of large-scale, multipetawatt laser facili-ties [7]: today’s facilities reach focussed intensities of order 10 Wcm − [8–10], and those up-coming, such as Apollon [11], ELI-Beamlines [12] and Nuclear Physics [13], aim to reach morethan 10 Wcm − , with the added capability of providing multiple laser pulses to the same targetchamber. At these intensities, radiation reaction will be comparable in magnitude to the Lorentzforce, rather than being a small correction, as is familiar from storage rings or synchrotrons. Fur-thermore, significant quantum corrections to radiation reaction are expected [5], which profoundlyalters the nature of particle dynamics in strong fields.The purpose of this review is to introduce the means by which radiation reaction, and quan-tum effects on the same, are understood, how they are incorporated into numerical simulations,and how they can be measured in experiments. While there is now an extensive body of litera-ture considering experimental prospects with future laser systems, our particular focus will be therelevance to today’s high-intensity lasers. It is important to note that much of the same physicscan be explored by probing such a laser with an ultrarelativistic electron beam. Previously suchexperiments demanded a large conventional accelerator [14, 15], but now ‘all-optical’ realizationof the colliding beams geometry is possible thanks to ongoing advances in laser-wakefield accel-eration [16, 17]. Indeed, the first experiments to measure radiation-reaction effects in this config-uration have recently been reported by Cole et al. [18], Poder et al. [19]. This review attempts toprovide the theory context for the interest in their results.Let us begin by introducing the various parameters that determine the importance of radiationemission, radiation reaction, and quantum effects. We work throughout in natural units such that3 IG. 1. Interaction of an electron (initial Lorentz factor γ ) and a circularly polarized electromagneticwave (frequency ω and normalized amplitude a ). In its average rest frame the electron is accelerated on acircular trajectory, with Lorentz factor γ (cid:48) = ( + a ) / , velocity β (cid:48) and radius r (cid:48) . The acceleration leads tothe emission of synchrotron radiation, which has characteristic frequency ω (cid:48) (cid:39) γ (cid:48) / r (cid:48) . the reduced Planck’s constant ¯ h , the speed of light c and the vacuum permittivity ε are all equalto unity: ¯ h = c = ε =
1. In these units the fine-structure constant α = e / ( π ) , where e is theelementary charge.It will be helpful to consider the concrete example shown in fig. 1. Here an electron is ac-celerated by a circularly polarized, monochromatic plane electromagnetic wave. The wave hasangular frequency ω and dimensionless amplitude a = eE / ( m ω ) , where E is the magnitudeof the electric field and m is the electron mass. a is sometimes called the strength parameter orthe normalized vector potential, and it can be shown to be both Lorentz- and gauge-invariant [20].The solution to the equations of motion, where the force is given by the Lorentz force only, can befound in many textbooks (see Gibbon [21] for example), so we will only summarize it here.The electromagnetic field tensor for the wave is eF µν = ma ∑ i f (cid:48) i ( φ )( k µ ε i ν − k ν ε i µ ) , where k is the wavevector, primes denote differentiation with respect to phase φ = k . x , and the ε , areconstant polarization vectors that satisfy ε i = − k . ε i =
0. Then the four-momentum of the4lectron p may be written in terms of the potential eA µ = ma ∑ i f i ( φ ) ε i µ : p µ ( φ ) = p µ + eA µ − (cid:18) eA . p k . p + e A k . p (cid:19) k µ . (1)Translational symmetry guarantees that k . p = k . p . The electron trajectory x µ ( φ ) = (cid:82) ( p µ / k . p ) d φ .Let us say that the electron initially counterpropagates into a circularly polarized, monochro-matic wave, with velocity β and Lorentz factor γ . The electron is accelerated by the wave inthe longitudinal direction, parallel to its wavevector, reaching a steady drift velocity of β d . Trans-forming to the electron’s average rest frame (ARF), as shown in fig. 1, we find that the electronexecutes circular motion with Lorentz factor γ (cid:48) = ( + a ) / , velocity β (cid:48) = a ( + a ) − / andradius r (cid:48) = a / [ γ ( + β ) ω ] . That γ (cid:48) is constant tells us that there is a phase shift of π / v and E , so v · E = E rad , as a fraction f of the electron energy in the ARF γ (cid:48) m , with the result f = E rad / ( γ (cid:48) m ) = π R c /
3. The magnitude of the radiation losses is controlled by the invariant classical radiation reaction parameter [23] R c ≡ α a γ ( + β ) ω m (2) (cid:39) . (cid:18) E
500 MeV (cid:19) (cid:18) I Wcm − (cid:19) (cid:18) λµ m (cid:19) . (3)Here E is the initial energy of the electron, I = E the laser intensity and λ = π / ω its wave-length.If we define ‘significant’ radiation damping to be an energy loss of approximately 10% perperiod [24], we find the threshold to be R c (cid:38) . a γ / (cid:38) × for a laser with a wave-length of 0 . µ m. At this point the force on the electron due to radiative losses must be includedin the equations of motion. We can see this directly by comparing the magnitudes of the radiationreaction and Lorentz forces. Estimating the former as F rad = E rad / ( π r (cid:48) ) and the Lorentz forceas F ext = γ (cid:48) m / r (cid:48) , we have that F rad / F ext (cid:39) R c /
3. For R c (cid:38) radiation-dominatedregime [25–27].We will discuss how the recoil due to radiation emission is included in classical electrodynam-ics in section II A. Before doing so, let us also consider the spectral characteristics of the radiation5mitted by the accelerated electron. In principle the periodicity of the motion, and its infinite dura-tion, means that the frequency spectrum is made up of harmonics of the ARF cyclotron frequency.However, recall that at large γ (cid:48) (cid:39) a , relativistic beaming means that most of the radiation is emit-ted in the forward direction into a cone with half-angle 1 / γ (cid:48) . The length of the overlap betweenthe electron trajectory and this cone defines the formation length l f , which is the characteristicdistance over which radiation is emitted [28, 29]. A straightforward geometrical calculation givesthe ratio between l f and the circumference of the orbit C = π r (cid:48) l f C (cid:39) π a . (4)The invariance of a suggests we could have reached this result in a covariant way; indeed, a fulldetermination of the size of the phase interval that contributes to emission gives the same result,even quantum mechanically [30].The smallness of the formation zone means that the spectrum is broadband, with frequencycomponents up to a characteristic value ω (cid:48) (cid:39) γ (cid:48) / r (cid:48) . Comparing this characteristic frequency tothe cyclotron frequency (in the average rest frame) ω c = / r (cid:48) gives us a measure of the classicalnonlinearity: ω (cid:48) ω c (cid:39) a . (5)At a (cid:29)
1, the radiation is made up of very high harmonics and is therefore well-separated from thebackground. The ratio between the frequency ω (cid:48) and the electron energy in the ARF χ = ω (cid:48) / ( γ (cid:48) m ) is another useful invariant parameter χ ≡ a γ ( + β ) ω m (6) (cid:39) . (cid:18) E
500 MeV (cid:19) (cid:18) I Wcm − (cid:19) / . (7)Restoring factors of ¯ h and c we can show that χ ∝ ¯ h , unlike R c . It therefore parametrizes theimportance of quantum effects on radiation reaction [30, 31], as can be seen by the fact that if χ ∼
1, an individual photon of the radiation can carry off a substantial fraction of the electron’senergy. By setting γ = χ is equal to the ratio of the electric field inthe instantaneous rest frame of the electron to the so-called critical field of QED [32, 33] E cr ≡ m e = . × Vm − , (8)which famously marks the threshold for nonperturbative electron-positron pair creation from thevacuum [34]. 6 .1 1 10 100 10000.010.050.100.501510 a χ Cole et al. ( ) Poder et al. ( ) Wistisen et al. ( ) Wistisen et al. ( ) Bula et al. ( ) ,Burke et al. ( ) γ = a γ = , ω = . e V R c = . R c = . FIG. 2. The importance, and type, of radiation reaction effects can be parametrized by a , the normalizedintensity of the laser field or classical nonlinearity parameter, and χ , the quantum nonlinearity parameter.Classical radiation damping becomes strong when R c = α a χ > .
01 (light blue) and dominates when R c > . χ > .
1. Electron-positron pair creation and QED cascades are important for χ >
1. Experiments that have explored quantumeffects with intense lasers are shown by open circles [14, 15, 18, 19]. Two recent experiments with leptonbeams and aligned crystals are shown by triangles [35, 36]; here the perpendicular component of the leptonmomentum p ⊥ is used to define an equivalent classical nonlinearity parameter a (cid:39) p ⊥ / m . The two parameters R c and χ allow us to characterize the importance of classical and quantumradiation reaction respectively. We show these as functions of a and χ , the classical and quantumnonlinearity parameters, in fig. 2. It is evident that, as a increases, it requires less and less electronenergy to enter the radiation-dominated regime. Indeed, if the acceleration is provided entirely bythe laser so that γ (cid:39) a , radiation reaction becomes dominant at about the same a that quantumeffects become important, assuming that ω corresponds to a wavelength of 0 . µ m. However, for a (cid:46)
50 as is accessible with existing lasers [8–10], it is not possible to probe radiation reactionvia direct illumination of a plasma. Instead, the experiments illustrated in fig. 2 have used pre-accelerated electrons to explore the strong-field regime, thereby boosting both R c and χ . (Note7hat, as R c is defined on a per-cycle basis, it would be possible for classical radiation reactioneffects to be large in long laser pulses while remaining below the threshold for quantum effects.)The next generation of laser facilities will reach a in excess of 100, perhaps even 1000 [11–13].The plasma dynamics explored in such experiments will be strongly affected by radiation reactionand quantum effects. II. THEORY OF RADIATION REACTIONA. Classical radiation reaction
In classical electrodynamics, radiation reaction is the response of a charged particle to the fieldof its own radiation [37, 38]. The first equation of motion to include both the external and self-induced electromagnetic forces in a manifestly covariant and self-consistent way was obtainedby Dirac [39]. This solution starts from the coupled Maxwell’s and Lorentz equations and fea-tures a mass renormalization that is needed to eliminate divergences associated with a point-likecharge [40, 41]. The result is generally referred to as the Lorentz-Abraham-Dirac (LAD) equation.For an electron with four-velocity u , charge − e and mass m it readsd u µ d τ = − em F µν u ν + e π m (cid:18) d u µ d τ + u µ d u ν d τ d u ν d τ (cid:19) (9)where τ is the proper time. Here F µν is the field tensor for the externally applied electromagneticfield, so it is the second term that accounts for the self-force. Although the LAD equation isan exact solution of the Maxwell-Lorentz system, using it directly turns out to be problematic.The momentum derivative d u µ d τ in the RR term leads to so-called runaway solutions, in which theelectron energy increases exponentially in the absence of external fields, and to pre-acceleration ,in which the momentum changes in advance of a change in the applied field [42–44]. These issueshave prompted searches for alternative classical theories of radiation reaction [45–49] that havemore satisfactory properties (see the review by [6] for details).The most widely used classical theory is that proposed by [50]. They realized that if the second(RR) term in eq. (9) were much smaller than the first in the instantaneous rest frame of the charge,it would be possible to reduce the order of the LAD equation by substituting d u d τ → em F µν u ν in theRR term. The result, called the Landau-Lifshitz equation, is first-order in the electron momentum8nd free from the pathological solutions of the LAD equation [6]:d u µ d τ = − em F µν u ν + e π m (cid:104) − me ( ∂ α F µν ) u ν u α + F µν F να u α + ( F να u α ) u µ (cid:3) . (10)The following two conditions for the characteristic length scale L over which the field varies andits magnitude E must be fulfilled in the instantaneous rest frame for the order reduction procedureto be valid: L (cid:29) λ C and E (cid:28) E cr / α , where λ C = / m is the Compton length. Note that both ofthese are automatically fulfilled in the realm of classical electrodynamics [5], as quantum effectscan only be neglected when L (cid:29) λ C and E (cid:28) E cr . The former condition ensures that the electronwavefunction is well-localized and the latter means recoil at the level of the individual photon isnegligible [5]. One reason to favour the Landau-Lifshitz equation is that all physical solutions ofthe LAD equation are solutions of the Landau-Lifshitz equation [51].Once the trajectories are determined, the self-consistent radiation is obtained from the Li´enard-Wiechert potentials, which give the electric and magnetic fields of a charge in arbitrary mo-tion [52]. The spectral intensity of the radiation from an ensemble of N e electrons, the energyradiated per unit frequency ω and solid angle Ω , is given in the far field byd E d ω d Ω = αω π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N e ∑ k = (cid:90) ∞ − ∞ n × ( n × v k ) e i ω ( t − n · r k ) d t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (11)where n the observation direction, and r k and v k are the position and velocity of the k th particle attime t [52].
1. In plane electromagnetic waves
Among the other useful properties of eq. (10) is that it can be solved exactly if the external fieldis a plane electromagnetic wave [53]. Taking this field to be eF µν = ma ∑ i f (cid:48) i ( φ )( k µ ε i ν − k ν ε i µ ) ,using the same definitions as in section I, eq. (10) is most conveniently expressed in terms of the lightfront momentum u − ≡ k . p / ( m ω ) , scaled perpendicular momenta ˜ u x , y ≡ u x , y / u − , and phase φ : d u − d φ = − α a ω m [ f (cid:48) ( φ ) + f (cid:48) ( φ ) ] u − , (12)and d ˜ u i d φ = a f (cid:48) i ( φ ) u − + α a ω f (cid:48)(cid:48) i ( φ ) m . (13)9he remaining component u + is determined by the mass-shell condition u − u + − u x − u y = ω x µ ( φ ) = (cid:82) φ − ∞ ( u µ / u − ) d φ . Equation (12) admits the solution u − = u − + R c I ( φ ) , (14)where u − is the initial lightfront momentum, the classical radiation reaction parameter R c = a u − ω / m as in eq. (2), and I ( φ ) = (cid:82) φ − ∞ [ f (cid:48) ( ψ ) + f (cid:48) ( ψ ) ] d ψ . The choice of notation here re-flects the fact that f (cid:48) ( φ ) is proportional to the electric field and so I ( φ ) is like an integrated energyflux. We use eq. (14) to solve eq. (13), obtaining ˜ u i ( φ ) and then u i = + R c I ( φ ) (cid:20) u i , + a f i ( φ ) + R c H ( φ ) + R c a f (cid:48) i ( φ ) (cid:21) , (15)where u i , is the initial value of the perpendicular momentum component i and H i ( φ ) = (cid:82) φ − ∞ f (cid:48) i ( ψ ) I ( ψ ) d ψ .The electron trajectory in the absence of radiation reaction is obtained by setting α =
0, in whichcase we recover eq. (1) as expected. Note that the lightfront momentum u − is no longer conserved,once radiation reaction is taken into account [54].In section I we estimated that the electron would radiate in a single cycle a fraction 4 π R c / γ (cid:29) u − (cid:39) γ , wecan show this fraction is actually E rad / ( γ m ) = ( π R c / ) / ( + π R c / ) . Here the denominatorrepresents radiation-reaction corrections to the energy loss, guaranteeing that E rad / ( γ m ) <
1. Withthese corrections, the energy emitted, according to the Larmor formula, is equal to the energylost, according to the Landau-Lifshitz equation (see Appendix A of Di Piazza [55] for a directcalculation of momentum conservation).The emission spectrum eq. (11) may also be expressed in terms of an integral over phase. Thenumber of photons scattered per unit (scaled) frequency s = ω / ω and solid angle is [56, 57]d N γ d s d Ω = α s π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N e ∑ k = (cid:90) ∞ − ∞ ε (cid:48) . u k exp ( − isn . ξ k )( u − k ) d φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (16)where the scaled four-position ξ ≡ ω x , and ε (cid:48) and n are the four-polarization and propagationdirection of the scattered photon. Given these relations and the analytically determined trajectory,we can numerically evaluate the number of photons scattered to given frequency and polar angleby integrating eq. (16), summed over polarizations, over all azimuthal angles 0 ≤ ϕ < π .10 . Quantum corrections: suppression and stochasticity We showed in fig. 2 that in many scenarios of interest, reaching the regime where radiationreaction becomes important automatically makes quantum effects important as well. This raisesthe question: what is the quantum picture of radiation reaction? Let us revisit the example westudied classically in section I, that of an electron emitting radiation under acceleration by a strongelectromagnetic wave. One might instinctively liken this scenario to inverse Compton scattering,as energy and momentum are automatically conserved when the electron absorbs a photon (orphotons) from the plane wave and emits another, higher energy photon. However, the recoil isproportional to ¯ h and vanishes in the classical limit; we would then recover Thomson scatteringrather than radiation reaction.The solution is that, in the regime a (cid:29) χ (cid:46)
1, quantum radiation reaction can be iden-tified with the recoil on the electron due its emission of multiple, incoherent photons [23]. Theseconditions express the following: a (cid:29) χ (cid:46) N γ ∝ α ∝ / ¯ h and the momentum change of the electron ∝ ¯ h for each photon, we have that the total momentum change ∝ ¯ h and therefore a classical limit ex-ists [5]. This suggests that one way to determine the ‘correct’ theory of classical radiation reactionis to start with a QED result and take the limit ¯ h →
0. This has been accomplished for both the mo-mentum change [58, 59] and the position [60]. In particular, Ilderton and Torgrimsson [60] wereable to show that, to first order in α , only the LAD, Landau-Lifshitz and Eliezer-Ford-O’Connellformulations of radiation reaction were consistent with QED.In both the classical and quantum regimes, the force of radiation reaction is directed antiparallelto the electron’s instantaneous momentum, and its magnitude depends on the parameter χ . Wedefined this earlier for the particular case of an electron in an electromagnetic wave [see eq. (6)].In a general electromagnetic field F µν , χ = (cid:113) − ( F µν p ν ) mE cr = γ E cr (cid:113) ( E + v × B ) − ( v · E ) , (17)where p = γ m ( , v ) is the electron four-momentum. χ depends on the instantaneous transverseacceleration induced by the external field: in a plane EM wave, where E and B have the same11agnitude and are perpendicular to each other, χ = γ | E | ( − cos θ ) / E cr , where θ is the anglebetween the electron momentum and the laser wavevector, and it is therefore largest in counter-propagation. A curious consequence of eq. (17) is the existence of a radiation-free direction : nomatter the configuration of E and B , there exists a particular v that makes χ vanish [61]. Electronsin extremely strong fields tend to align themselves with this direction, any transverse momentumthey have being rapidly radiated away [61]. As this direction is determined purely by the fields, theself-consistent evolutions of particles and fields is determined by hydrodynamic equations [62].The larger the value of χ , the greater the differences between the quantum and classical predic-tions of radiation emission. Classically there is no upper limit on the frequency spectrum, whereasin the quantum theory there appears a cutoff that guarantees ω < γ m . Besides this cutoff, spin-flip transitions enhance the spectrum at high energy [63]. Let us work in the synchrotron limit,wherein the field may be considered constant over the formation length (i.e. l f (cid:28) λ , using eq. (4)).The classical emission spectrum, the energy radiated per unit frequency ω = x γ m and time by anelectron with quantum parameter χ and Lorentz factor γ , isd P cl d ω = αω √ πγ (cid:20) K / ( ξ ) − (cid:90) ∞ ξ K / ( y ) d y (cid:21) , ξ = x χ . (18)Two quantum corrections emerge when χ is no longer much smaller than one: the non-negligiblerecoil of an individual photon means that the spectrum has a cutoff at x =
1; and the spin con-tribution to the radiation must be included. The former can be included directly by modifying ξ = x / ( χ ) → x / [ χ ( − x )] in eq. (18), which yields the spectrum of a spinless electron (shownin orange in fig. 3). A neat exposition of this simple substitution is given by Lindhard [64] interms of the correspondence principle (see also Sørensen [65]). Then when the spin contributionis added, we obtain the full QED result [30, 31, 66]d P q d ω = αω √ πγ (cid:20)(cid:18) − x + − x (cid:19) K / ( ξ ) − (cid:90) ∞ ξ K / ( y ) d y (cid:21) , ξ = x χ ( − x ) , (19)where we quote the spin-averaged and polarization-summed result. This is shown in blue in fig. 3.The number spectrum d N γ d ω = ω − P q d ω ( χ , γ ) has an integrable singularity ∝ ω − / in the limit ω →
0. The total number of photons N γ = (cid:82) d N γ d ω d ω is finite.The combined effect of these corrections is to reduce the instantaneous power radiated by an12 lassicalclassical ( cuto ff - corrected ) QED0.0 0.5 1.0 1.5 2.00.00.20.40.60.81.0 ω /( γ m ) γ d / d ω × - - - - - χ g ( χ ) FIG. 3. (left) Quantum corrections to the emission spectrum d P / d ω at χ =
1: the classical [eq. (18)] andquantum-corrected spectra [eq. (19)]. (right) These corrections cause the total radiated power to be reducedby a factor g ( χ ) : the full result (blue) and limiting expressions (black, dashed). electron. This reduction is quantified by the factor g ( χ ) = P q / P cl , which takes the form [22, 31] g ( χ ) = √ π (cid:90) ∞ (cid:34) u K / ( u )( + χ u ) + χ u K / ( u )( + χ u ) (cid:35) d u (20) = − √ χ + χ χ (cid:28) Γ ( / ) / χ − / χ (cid:29) K is a modified Bessel function of the second kind and Γ ( / ) (cid:39) . χ < .
05 and χ >
200 respectively.A simple analytical approximation to eq. (20) that is accurate to 2% for arbitrary χ is g ( χ ) (cid:39) [ + . ( + χ ) ln ( + . χ ) + . χ ] − / [66]. The changes to the classical radiation spectrumand the magnitude of g ( χ ) are shown in fig. 3. Note that the total power P q = α m χ g ( χ ) / χ . g ( χ ) is sometimes referred to as the ‘Gaunt factor’ [67],as it is a multiplicative (quantum) correction to a classical result, first derived in the context ofabsorption [68].Figure 3 shows that the radiated power at χ ∼ χ is the ratio between the energies of thetypical photon and the emitting electron. When this approaches unity, even a single emissioncan carry off a large fraction of the electron energy, and the concept of a continuously radiating13 IG. 4. In the classical picture, radiation reaction is a continuous drag force that arises from the emission ofvery many photons that individually have vanishingly low energies (left). In the quantum regime, however,the electrons emits a finite number of photons, any or all of which can exert a significant recoil on theelectron. The probabilistic nature of emission leads to radically altered electron dynamics, with implicationsfor laser-matter interactions beyond the current intensity frontier. From Blackburn [70]. particle breaks down. Instead, electrons lose energy probabilistically , in discrete portions. Theimportance of this discreteness may be estimated by comparing the typical time interval betweenemissions, ∆ t = (cid:104) ω (cid:105) / P , with the timescale of the laser field 1 / ω [69]. Equation (19) yields forthe average photon energy (cid:104) ω (cid:105) (cid:39) . χγ m for χ (cid:28) . γ m for χ (cid:29)
1; the radiated power P = α m χ g ( χ ) /
3. We find ω ∆ t (cid:39) a − χ (cid:28) [ γω / ( a m )] / χ (cid:29) . (22)We expect stochastic effects to be at their most significant when ω ∆ t (cid:38)
1, which implies that thetotal number of emissions in an interaction is relatively small but χ is large.A description of how stochastic energy losses can be modelled follows in section III B. For now,it suffices to interpret eq. (19) as the (energy-weighted) probability distribution of the photonsemitted at a particular instant of time. Even though two electrons may have the same χ and γ , they can emit photons of different energies (or none at all) and thereby experience differentrecoils. Contrast this with the classical picture, in which the continuous energy loss is driven bythe emission of many photons that individually have vanishingly small energies (see fig. 4).Consider, for example, the interaction of a beam of electrons with a plane electromagneticwave, where the Lorentz factors of the electrons are distributed γ ∼ d N e d γ . The distribution is14haracterized by a mean µ ≡ (cid:104) γ (cid:105) and variance σ ≡ (cid:104) γ (cid:105) − µ . Under classical radiation reac-tion, higher energy electrons are guaranteed to radiate more than their lower energy counterparts( P ∝ γ ), with the result that both the mean and the variance of γ decrease over the course of theinteraction [71]. This is still the case if the radiated power is reduced by the Gaunt factor g ( χ ) ,i.e. a ‘modified classical’ model is assumed (see section III A), because radiation losses remaindeterministic [72].Under quantum radiation reaction, radiation losses are inherently probabilistic. While µ willstill decrease (more energetic electrons radiate more energy on average ), the width of the distri-bution σ can actually grow [71, 73]. Ridgers et al. [67] derive the following equations for thetemporal evolution of these quantities, under quantum radiation reaction:d µ d t = − α m (cid:104) χ g ( χ ) (cid:105) (23)d σ d t = − α m (cid:104) ( γ − µ ) χ g ( χ ) (cid:105) + α m √ (cid:104) γ χ g ( χ ) (cid:105) , (24)where (cid:104)· · ·(cid:105) denotes the population average and g ( χ ) = (cid:82) χ d P q / (cid:82) χ d P cl is the second momentof the emission spectrum. Only the first term of eq. (24) is non-zero in the classical limit, and itis guaranteed to be negative. The second term represents stochastic effects and is always positive.Broadly speaking, the latter is dominant if χ is large, the interaction is short, or the initial varianceis small [67, 74]. The evolution of higher order moments, such as the skewness of the distribution,are considered in Niel et al. [74].A distinct consequence of stochasticity is straggling [75], where an electron that radiates less(or no) energy than expected enters regions of phase space that would otherwise be forbidden.Unlike stochastic broadening, which can occur in a static, homogeneous electromagnetic field,straggling requires the field to have some non-trivial spatiotemporal structure. If an unusuallylong interval passes between emissions, an electron may be accelerated to a higher energy orsample the fields at locations other than those along the classical trajectory [76]. In a laser pulsewith a temporal envelope, for example, electrons that traverse the intensity ramp without radiatingreach larger values of χ than would be possible under continuous radiation reaction; this enhanceshigh-energy photon production and electron-positron pair creation [77]. If the laser duration isshort enough, it is probable that the electron passes through the pulse without emitting at all, inso-called quenching of radiation losses [78].The quantum effects we have discussed in this section emerge, in principle, from analytical re-sults including the emission spectrum [eq. (19)]. While further analytical progress can be made in15he quantum regime, using the theory of strong-field QED (see section III B), modelling more re-alistic laser–electron-beam or laser-plasma interactions generally requires numerical simulations.Much effort has been devoted to the development, improvement, benchmarking and deployment ofsuch simulation tools over the last few years. In the following section we review these continuingdevelopments. III. NUMERICAL MODELLING AND SIMULATIONSA. Classical regime
A natural starting point is the modelling of classical radiation reaction effects. In the absenceof quantum corrections, we have all the ingredients we need to formulate a self-consistent pictureof radiation emission and radiation reaction. We showed in section I how using only the Lorentzforce to determine the charge’s motion and therefore its emission led to an inconsistency in energybalance. This is remedied by using either eq. (9) or eq. (10) as the equation of motion, in whichcase the energy carried away in radiation matches that which is lost by the electron.Implementations of classical radiation reaction in plasma simulation codes have largely favouredthe Landau-Lifshitz equation (or a high-energy approximation thereto), as it is first-order in themomentum and the additional computational cost is not large [54, 79–81]. These codes have notonly been used to study radiation reaction effects in laser-plasma interactions [72, 82–86], butalso whether there are observable differences between models of the same [87, 88]. The radiationreaction force proposed by Sokolov [49] has also been implemented in some codes [89, 90], butnote that it is not consistent with the classical limit of QED [60]. It is also possible to solve theLAD equation numerically via integration backward in time [91].Given data on the trajectories of an ensemble of electrons (usually a subset of the all electronsin the simulations), eq. (11) can be used to obtain the far-field spectrum in a simulation whereclassical radiation reaction effects are included [24, 92, 93]. Equation (11) is valid across thefull range of ω ( pace the quantum cutoff at ω = γ m ), including the low-frequency region of thespectrum where collective effects are important: ω < n / e , where n e is the electron number density.This region does not, however, contribute very much to radiation reaction; this is dominated byphotons near the synchrotron critical energy ω c (cid:29) n / e . Thus the spectrum can be divided into coherent and incoherent parts, that are well separated in terms of their energy [69]. In the latter16egion, the order of the summation and integration in eq. (11) can be exchanged, and the totalspectrum determined by summing over the single-particle spectra.In a particle-in-cell code for example, the electromagnetic field is defined on a grid of discretepoints and advanced self-consistently using currents that are deposited onto the same grid [94].Defining the grid spacing to be ∆ , this scheme will directly resolve electromagnetic radiationthat has a frequency less than the Nyquist frequency π / ∆ . Given appropriately high resolution,this accounts for the coherent radiation generated by the collective dynamics of the ensemble ofparticles. The recoil arising from higher frequency components, which cannot be resolved on thegrid, and in any case as a self-interaction is neglected, is accounted for by the radiation reactionforce.Further simplification is possible if the interference of emission from different parts of thetrajectory is negligible. As indicated in section I, at high intensity a (cid:29)
1, the formation lengthof the radiation is much smaller than the timescale of the external field (see eq. (4)). This beingthe case, rather than using eq. (11), we may integrate the local emission spectrum eq. (18) overthe particle trajectory, assuming that, at high γ , the radiation is emitted predominantly in directionparallel to the electron’s instantaneous velocity [56, 95, 96]. The approach is naturally extendedto account for quantum effects, by substituting for the classical synchrotron spectrum eq. (18) theequivalent result in QED, eq. (19).One consequence of doing so is that the radiated power is reduced by the factor g ( χ ) , given ineq. (20). This should be reflected in a reduction in the magnitude of the radiation-reaction force.Consequently, a straightforward, phenomenological way to model quantum radiation reaction is touse a version of eq. (10) where the second term is scaled by g ( χ ) . This ‘modified classical’ modelhas been used in studies of laser–electron-beam [24, 72, 77] and laser-plasma interactions [97,98] as a basis of comparison with a fully stochastic model (shortly to be introduced), as well asin experimental data analysis [19, 35]. It has been shown that this approach yields the correctequation of motion for the average energy of an ensemble of electrons in the quantum regime [67,74]. It is, however, deterministic, and therefore neglects the stochastic effects we discussed insection II B. Sampling at discrete points, i.e. with limited sampling rate, means that only a certain range of frequency compo-nents in a given waveform can be represented. The highest frequency is called the ‘Nyquist frequency’. Modes thatlie are above this are aliased to lower frequencies. . Quantum regime: the ‘semiclassical’ approach In section II B we discussed how ‘quantum radiation reaction’ could be identified with therecoil arising from multiple, incoherent emission of photons. Indeed, if χ (cid:38)
1, any or all of thesephotons can exert a significant momentum change individually. Figure 2 tells us that we generallyrequire a (cid:29) strong-field QED, which separatesthe electromagnetic field into a fixed background, treated exactly, and a fluctuating part, treatedperturbatively [99]; see the reviews by Di Piazza et al. [5], Ritus [30], Heinzl [100] or a tutorialoverview by Seipt [101] which discusses photon emission in particular.Although it is the most general and accurate approach, strong-field QED is seldom used tomodel experimentally relevant configurations of laser-electron interaction [102]. In a scattering-matrix calculation, the object is to obtain the probability of transition between asymptotic freestates; as such, complete information about the spatiotemporal structure of the background fieldis required. Analytical results have only been obtained in field configurations that possess highsymmetry [103], e.g. plane EM waves [30] or static magnetic fields [31]. The assumption that thebackground is fixed also means that back-reaction effects are neglected, even though it is expectedthat QED cascades will cause significant depletion of energy from those background fields [104–106]. Furthermore, the expected number of interactions per initial particle (the multiplicity ) ismuch greater than one in many interaction scenarios. At present, cutting-edge results are those inwhich the final state contains only two additional particles, e.g. double Compton scattering [107–110] and trident pair creation [111–115], due to the complexity of the calculations.The need to overcome these issues has motivated the development of numerical schemes thatcan model quantum processes at high multiplicity in general electromagnetic fields. In this articlewe characterize these schemes as ‘semiclassical’, by virtue of the fact that they factorize a QEDprocess into a chain of first-order processes that occur in vanishingly small regions linked by classically determined trajectories, as illustrated in fig. 5. The rates and spectra for the individualinteractions are calculated for the equivalent interaction in a constant, crossed field, which maybe generalized to an arbitrary field configuration under certain conditions. The first key resultis that, at a (cid:29)
1, the formation length of a photon (or an electron-positron pair) is much smallerthan the length scale over which the background field varies (see section I) and so emission may betreated as occurring instantaneously [30]. The second is that if χ (cid:29) | F | , | g | and F , g (cid:28)
1, where18
QED e − → e − e − e + γγ ∝ e − ( p i ) e − ( p f ) γ ( k ) γ ( k ) e + ( p + ) e − ( p − ) 2 p p P CCF e → eγ [ χ ( t )] 2 p p P CCF e → eγ [ χ ( t )] 2 p p P CCF e → eγ [ χ ( t )] 2 k P CCF γ → e + e − [ χ ( t )] pp p i p f p + p − k k p µ d τ = − em F µν p ν ∂ µ F µν = j ν d k µ d τ = 0 d p µ d τ = em F µν p ν ∂ µ F µν = j ν is simulated asexternal field F µν FIG. 5. A general strong-field QED interaction, featuring the emission and creation of multiple photons andelectron-positron pairs, is simulated ‘semiclassically’ by breaking it down into a chain of first-order pro-cesses (electrons, photons and positrons in blue, orange and red, respectively). Between these pointlike, in-stantaneous events, the particles follow classical trajectories guided by the Lorentz force: ˙ p µ = ± eF µν p ν / m and ˙ X µ = p µ / m , dots denoting differentiation with respect to proper time τ . Modification of the externalfield F µν is driven by the classical currents j µ ( x ) = ± ( e / m ) (cid:82) p µ ( τ ) δ [ x − X ( τ )] d τ . The probability ratesand spectra for the first-order processes are those for a constant, crossed field, and depend on the local valueof the quantum parameter χ ( t ) = (cid:12)(cid:12) F µν [ X ( t )] p ν ( t ) (cid:12)(cid:12) / ( mE cr ) . = ( E − B ) / E and g = E · B / E are the two field invariants, the probability of a QED processis well approximated by its value in a constant, crossed field: P ( χ , F , g ) (cid:39) P ( χ , , ) + O ( F ) + O ( g ) [see Appendix B of Baier et al. [66]]. The combination of the two is called the locally constant,crossed field approximation (LCFA). The first requires the laser intensity to be large, whereas thesecond requires the particle to be ultrarelativistic and the background to be weak (as comparedto the critical field of QED). We will discuss the validity of these approximations, and efforts tobenchmark them, in section III C.Within this framework, the laser-beam (or laser-plasma) interaction is essentially treated clas-sically, and quantum interactions such as high-energy photon emission added by hand. The evo-lution of the electron distribution function F = F ( t , r , p ) , including the classical effect of thebackground field and stochastic photon emission, is given by [67, 116] ∂ F ∂ t + p γ m · ∂ F ∂ r − e (cid:18) E + p × B γ m (cid:19) · ∂ F ∂ p = − F (cid:90) W γ ( p , k (cid:48) ) d k (cid:48) + (cid:90) F ( p (cid:48) ) W γ ( p (cid:48) , p (cid:48) − p ) d p (cid:48) , (25)where W γ ( p , k (cid:48) ) is the probability rate for an electron with momentum p to emit a photon withmomentum k (cid:48) . A direct approach to kinetic equations of this kind is to solve them numeri-cally [71, 117, 118], or reduce them by means of a Fokker-Planck expansion in the limit χ (cid:28) et al. [69], Ridgers et al. [119], so we only summarize it here for photon emission.The electron distribution function is represented by an ensemble of macroparticles, which rep-resent a large number w of real particles ( w is often called the weight ). The trajectory of a macro-electron between discrete emission events is determined solely by the Lorentz force. Each isassigned an optical depth against emission T = − log ( − R ) for pseudorandom 0 ≤ R <
1, whichevolves as d T d t = − W γ , where W γ is the probability rate of emission, until the point where it fallsbelow zero. Emission is deemed to occur instantaneously at this point and T is reset. The en-ergy of the photon ω (cid:48) = | k (cid:48) | is pseudorandomly sampled from the quantum emission spectrum d N γ d ω = ω − P q d ω ( χ , γ ) [see eq. (19)] and the electron recoil determined by the conservation of mo-mentum p = p (cid:48) + k (cid:48) and the assumption that k (cid:48) (cid:107) p if γ (cid:29)
1. If desired, a macrophoton with thesame weight as the emitting macroelectron can be added to the simulation. Electron-positron pair20reation by photons in strong electromagnetic fields is modelled in an analogous way to photonemission [69, 119].Thus there are two distinct descriptions of the electromagnetic field. One component is treatedas a classical field (in a PIC code, this would be discretized on the simulation grid) and the otheras a set of particles. In principle this leads to double-counting; however, as we discussed insection III A, the former lies at much lower frequency than the photons that make up synchrotronemission, and has a distinct origin in the form of externally generated fields (such as a laser pulse)or the collective motion of a plasma. Coherent effects are much less important for the high-frequency components, which justifies describing them as particles [69].
C. Benchmarking, extensions and open questions
The validity of the simulation approach discussed in section III B relies on the assumption thata high-order QED process in a strong electromagnetic background field may be factorized into achain of first-order processes, each of which is well approximated by the equivalent process in aconstant, crossed field. It is generally expected that this reduction works in scenarios where a (cid:29) χ (cid:29) | F | , | g | [30, 66]. However, these asymptotic conditions do not give quantitative boundson the error made by semiclassical simulations. As these are the primary tool by which we predictradiation reaction effects in high-intensity lasers, it is important that they are benchmarked andthat the approximations are examined.One approach is to compare, directly, the predictions of strong-field QED and simulations. Wefocus here on results for single nonlinear Compton scattering [102, 120, 121], the emission of oneand only one photon in the interaction of an electron with an intense, pulsed plane EM wave, byvirtue of its close relation to radiation reaction. It is shown that the condition a (cid:29) a / χ (cid:29) x = ω (cid:48) / ( γ m ) < χ / a , as the formation length for such photons iscomparable in size to the wavelength of the background field. Semiclassical simulations stronglyoverestimate the number of photons emitted in this part of the spectrum because they excludenonlocal effects [120, 121]. Nevertheless, they are much more accurate with respect to the totalenergy loss (and therefore to radiation reaction), because this depends on the power spectrum,to which the low-energy photons do not contribute significantly [102]. This is shown in fig. 6,21 xact QED simulations5 10 15 20 25 300100200300 a Δ p - / m a = a = - - - - k - / p - k - d N γ / d k - FIG. 6. Comparison between exact QED (grey) and simulation results (blue and orange) for single non-linear Compton scattering of an electron in a two-cycle, circularly polarized laser pulse: (left) the lightfrontmomentum loss as a function of laser amplitude a ; and (right) exemplary photon spectra. Adapted fromBlackburn et al. [102]. which compares the predictions of exact QED and semiclassical simulations for an electron with p − / m (cid:39) γ = a andwavelength λ = . µ m. There is remarkably good agreement between the two even for a = et al. [102] is the number of photons absorbedfrom the background field in the process of emitting a high-energy photon. This transfer of energyfrom the background field to the electron is required by momentum conservation. Without emis-sion, there would be no such transfer of energy. This is consistent with the classical picture, inwhich plane waves do no work in the absence of radiation reaction. Strong-field QED calculationsdepend crucially on the fixed nature of the background field; however, for single nonlinear Comp-ton scattering, near-total depletion of the field is predicted at a (cid:38) j · E term in Poynting’s theorem.Quantum effects are manifest in how photon emission (and pair creation), modify those classicalcurrents, as illustrated in fig. 5. In Blackburn et al. [102], the classical work done on the elec-tron is shown to agree well with the number of absorbed photons predicted by exact QED. Thisis consistent with the results of Meuren et al. [125], which indicate that the ‘classical’ dominatesthe ‘quantum’ component of depletion, the latter associated with absorption over the formationlength, if a (cid:29)
1. 22he failure of the semiclassical approach to reproduce the low-energy part of the photonspectrum arises from the localization of emission. Most notably, the number spectrum d N γ d ω = ω − P q d ω ( χ , γ ) [see eq. (19)] diverges as ω − / as ω →
0. This can be partially amelioratedby the use of emission rates that take nonlocal effects into account. Di Piazza et al. [126] sug-gest replacing the LCFA spectrum in the region x (cid:46) χ / a with the equivalent, finite, result for amonochromatic plane wave, which they adapt for use in arbitrary electromagnetic field configura-tions. Ilderton et al. [127] propose an approach based on formal corrections to the LCFA, in whichthe emission rates depend on the field gradients as well as magnitudes.While the studies discussed above have given insight into the limitations of the LCFA, they donot examine the applicability of the factorization shown in fig. 5, as this requires by definition thecalculation of a higher order QED process. At the time of writing, there are no direct comparisonsof semiclassical simulations and strong-field QED for either double Compton scattering (emissionof two photons) or trident pair creation (emission of a photon which decays into an electron-positron pair). Factorization, also called the cascade approximation , has been examined directlywithin strong-field QED for the trident process in a constant crossed field [113] and in a pulsedplane wave [114, 115]. In the latter it is shown that at a =
50 and an electron energy of 5 GeV,the error is approximately one part in a thousand.The dominance of the cascade contribution makes it important to consider whether the propa-gation of the electron between individual tree-level process, as shown in fig. 5, is done accurately.In the standard implementation, this is done by solving a classical equation of motion includingonly the Lorentz force [69, 119]. The evolution of the electron’s spin is usually neglected andemission calculated using unpolarized rates, such as eq. (19). King [109] show that the accuracyof modelling double Compton scattering in a constant crossed field as two sequential emissionswith unpolarized rates is better than a few per cent. There are, however, scenarios, where thespin degree of freedom influences the dynamics to a larger degree. Modelling these interactionswith semiclassical simulations requires spin-resolved emission rates [22, 128] and an equation ofmotion for the electron spin [129, 130]. In a rotating electric field, as found at the magnetic nodeof an electromagnetic standing wave [104], where the spin does not precess between emissions,the asymmetric probability of emission between different spin states leads to rapid, near-completepolarization of the electron population [131, 132]. Similarly, an electron beam interacting witha linearly polarized laser pulse can acquire a polarization of a few per cent [128]. To make thislarger, it is necessary to break the symmetry in the field oscillations, which can be accomplished by23ntroducing a small ellipticity to the pulse [133], or by superposition of a second colour [134–136].A more fundamental limitation on the applicability of the LCFA is that the emission rates arecalculated at tree level only. The importance of loop corrections to the strong-field QED vertexgrows as α χ / in a constant, crossed field [137], leading to speculation that α χ / is the ‘true’expansion parameter of strong-field QED [138]. When χ (cid:39) a / χ (cid:29)
1. In the high-energylimit, radiative corrections grow logarithmically, as in ordinary (i.e., non-strong-field) QED [140,141].The difficulty in probing the regime α χ / (cid:38) γ and so χ [139]. Overcoming this barrier at the desired χ requires theinteraction duration to be very short. The beam-beam geometry proposed by Yakimenko et al. [142] exploits the Lorentz contraction of the Coulomb field of a compressed (100 nm), ultrarel-ativistic (100 GeV) electron beam, which is probed by another beam of the same energy. In thelaser-electron-beam scenario considered by Blackburn et al. [143], collisions at oblique incidenceare proposed for reaching χ (cid:38) χ is reached in thecombined laser-plasma, laser-beam interaction proposed by Baumann and Pukhov [144]. Whileit seems possible to approach the fully nonperturbative regime experimentally, albeit for extremecollision parameters, there is no suitable theory at α χ / (cid:38)
1, and quantitative predictions arelacking in this area.
IV. EXPERIMENTAL GEOMETRIES, RESULTS AND PROSPECTSA. Geometries
It may be appreciated that the radiation-reaction and quantum effects under consideration here,as particle-driven processes, can only become important if electrons or positrons are actually em-bedded within electromagnetic fields of suitable strength. However, the estimates in section I were24
IG. 7. Tightly focussed laser pulses can ponderomotively expel electrons from the region of highest inten-sity, suppressing the onset of radiation-reaction and quantum effects. There are three typical experimentalgeometries that ensure that energetic electrons are embedded in strong EM fields as desired: (top) laser–particle-beam, (centre) laser-plasma, and (bottom) laser-laser. made for a plane EM wave, in which case the electron is guaranteed to interact with the entirewave, including the point of highest intensity. In reality, such intensities are reached by com-pressing the laser energy into ultrashort pulses [145] that are focussed to spot sizes close to thediffraction limit [8–10]. The steep spatiotemporal gradients in intensity that result mean that laserpulses can ponderomotively expel electrons from the focal region, in both vacuum [146, 147] andplasma [16], curtailing the interaction long before the particles experience high a or χ .The literature contains many possible experimental configurations designed to explore or ex-ploit radiation reaction and quantum effects. These configurations can be divided, broadly, intothree categories, based on how they ensure the spatial coincidence between particles and strongfields. Figure 7 illustrates the three categories. In the first ( laser–particle-beam ), the electrons areaccelerated to ultrarelativistic energies before they encounter the laser pulse. The effective ‘massincrease’ makes the beam rigid and so it passes through the entirety of the laser pulse, avoidingsubstantial deflection and ensuring that it is exposed to the strongest electromagnetic fields. Con-25retely, the ponderomotive force is suppressed at high γ : d (cid:104) p (cid:105) / d t = − m ∇ (cid:104) a (cid:105) / ( (cid:104) γ (cid:105) ) , where (cid:104)·(cid:105) denotes a cycle-averaged quantity [148]. It should be noted that it is possible for radiation reactionto amplify this force to the point that it can prevent an arbitrarily energetic electron from penetrat-ing the laser field [149, 150]; however, this requires a (cid:38) χ ∼ a ∼
10 (see fig. 2).In the second ( laser-plasma ), the electrons are electrostatically bound to a population of ions,which are substantially more massive and therefore less mobile. Large-scale displacement of theelectrons away from the laser fields is then suppressed by the emergence of plasma fields. Ifthe plasma is overdense, i.e. opaque to the laser light, then only electrons in a thin layer nearthe surface experience the full laser intensity and are accelerated to relativistic energies. How-ever, the high density of electrons in this region means that a significant fraction of the laserenergy is converted to high-energy radiation, leading to, for example, dense bursts of γ rays andpositrons [151, 152], reduced efficiency of ion acceleration [79] and the generation of long-livedquasistatic magnetic fields [86]. If the target is close to underdense, by contrast, the laser can prop-agate through the plasma bulk and the interaction is volumetric in nature. The combination of laserand induced plasma fields, as well as radiation reaction, leads to confinement and acceleration ofthe electrons, and copious emission of radiation [153–155].Finally, electrons can be trapped in the collision of more than one laser pulse ( laser-laser ),where they interact with an electromagnetic standing, rather than travelling, wave [106]. Ra-diation reaction induces a rich set of dynamics in this configuration [156–159]. The fact thatstanding waves can do work in reaccelerating the particles after they recoil means that, at intensi-ties (cid:38) Wcm − , the emitted photons seed avalanches of electron-positron pair creation [104];this intensity threshold is lowered in suitable multibeam setups [160–162]. The case of optimalfocussing is achieved in a dipole field [163], where the peak a (cid:39) P / [ PW ] [164]. Such ex-treme intensities, at moderate power, are the reason this configuration has been studied as meansof high-energy photon production [165, 166].It is important to note that the distinction between the three categories defined here is notabsolute. Mixing between them occurs in, for example, the interaction of a linearly polarized laserpulse with relativistically underdense plasma: here re-injected electron synchrotron emission , theradiation emission when electrons are pulled backwards into the oncoming laser by a charge-separation field [152], exhibits features of both the ‘laser-plasma’ and ‘laser-beam’ geometries.26urthermore, the exponential growth of particle number in a QED cascade driven by multiplelaser pulses can create an electron-positron plasma of sufficient density to shield the interior fromthe laser pulse [167], leading to a transition between the ‘laser-laser’ and ‘laser-plasma’ categories.Twin-sided illumination of a foil has features of both ab initio [98]. B. ‘All-optical’ colliding beams
This paper focuses on the first of the three configurations discussed in section IV A, laser–particle-beam , for the reason that it allows χ > . a than would berequired in a laser-plasma or laser-laser interaction. As is shown in fig. 2 and by eq. (6), given a500-MeV electron beam, quantum effects on radiation reaction can be reached even at an intensityof 10 Wcm − . The colliding beams geometry therefore represents a promising first step towardsexperimental exploration of the radiation-dominated or nonlinear quantum regimes.Thus far we have not specified the source of ultrarelativistic electrons. The theoretical descrip-tion of the interaction does not depend on the source, of course, but it is of immense practicalimportance. Furthermore, the characteristics of the source (its energy, bandwidth, emittance, etc.)are key determining factors in the viability of measuring radiation-reaction or quantum effects.For example, the fact that electron beam energy spectra are expected to broaden due to stochas-tic effects makes the variance of the spectrum, σ , an attractive signature of the quantum natureof radiation reaction [71, 73]. However, such broadening can occur classically in the interactionof an electron beam with a focussed laser pulse, because components of the beam can encounterdifferent intensities and therefore lose different amounts of energy [168]. Thus a crucial role isplayed by the initial energy spread and size of the incident electron beam [4].In fact, it was pioneering experiments with a conventional, radio-frequency (RF), linear ac-celerator that provided the first demonstration of nonlinear quantum effects in a strong laser field:nonlinearities were measured in Compton scattering [14] and Breit-Wheeler electron-positron paircreation [15] in Experiment 144 at the Stanford Linear Accelerator (SLAC) facility. In this exper-iment, the 46 . . × Wcm − ,duration 1 . a (cid:39) . χ = .
3) [169]. The pair creation mechanismwas reported to be the multiphoton Breit-Wheeler process, as n = γ ray (emitted by the electron in Compton scattering) to overcome the27ass threshold. In total, 106 ±
14 positrons were observed over the series of 22,000 laser shots.The yield was strongly limited because, even though the electron energy was sufficient to reacha quantum parameter χ ∼ .
3, in the regime a (cid:28)
1, the pair creation probability is suppressedas a n , where n , the number of participating photons, was found to be n (cid:39) n = . (cid:46) a (cid:46) ω ∼ n e ∼ cm − , the plasma wavelength ∼ µ m), the electron beams producedin wakefield acceleration are similarly micron-scale, with durations of order 10 fs.Besides the high energy and the small size of the electron beam, we have the intrinsic synchro- More recent analysis, in which the two stages of photon emission and pair creation are treated within a unifiedframework using strong-field QED, thereby including the direct, ‘one-step’, contribution, indicates that the experi-ment did, in fact, observe the onset of nonperturbative effects [111], see also [112]. γ rays [178–180]. Now, with advances in laser technology,a multibeam facility is capable of reaching the radiation-reaction and nonlinear quantum regimes.Recently two such experiments were performed using the Gemini laser at the Rutherford AppletonLaboratory [18, 19], a dual-beam system that delivers twin synchronized pulses of duration 45 fsand energy ∼
10 J, with a peak a (cid:39)
20: we discuss these experiments in detail in section IV C.Upcoming laser facilities, such as Apollon [11] or ELI [12, 13], aim for laser-electron collisionsat even higher intensity: see, for example, Lobet et al. [181] for simulations of dual-beam interac-tions at a (cid:39) et al. [182], employs a single laser pulse as accelerator andtarget: a foil is placed at the end of a gas jet, into which a laser is focussed to drive a wakefield andaccelerate electrons; when the laser pulse reaches the foil, it is reflected from the ionized surfaceback onto the trailing electrons. This guarantees temporal and spatial overlap of the two beams,but precludes the possibility of separately optimizing the two laser pulses; in Ta Phuoc et al. [182]the electron energies (cid:39)
100 MeV and the peak a (cid:39) .
5, so radiation reaction effects were negligi-ble. Simulations of similar single-pulse geometries predict the efficient production of multi-MeVphotons at a >
50 [183] and electron-positron pairs at a (cid:38)
300 [184, 185].
C. Recent results
The Gemini laser of the Central Laser Facility (Rutherford Appleton Laboratory, UK) is apetawatt-class dual-beam system [186], well-suited for the all-optical colliding beams experimentsdiscussed in section IV B. It delivers two, synchronized, linearly polarized laser pulses of duration45 fs, energy 10 J and wavelength 0 . µ m. Available focussing optics include long-focal-lengthmirrors ( F /
20 and F /
40) for laser-wakefield acceleration and, most importantly, a short-focal-length ( F /
2) off-axis parabolic mirror with an F / χ is largest (seeeq. (17)): the more weakly focussed laser that drives the wakefield passes through the hole andis subsequently blocked, avoiding backreflection in the amplifier chains; the accelerated electron29 as jet f /40 f /2 Magnet CsI arraySpectrometerscreenElectron beam γ- ray beamVacuum window FIG. 8. Layout of an all-optical colliding-beams experiment. A hole in the short-focal-length ( F / γ rays they emit in the collision,and the accelerating laser pulse, pass through this hole before being blocked or diagnosed as appropriate.The collision is timed to occur close to the rear of the gas jet (on the right-hand side, as viewed in thefigure), before the electron beam can diverge significantly, which maximizes overlap between the beams.Reproduced from Cole et al. [18]. beam, and any radiation produced in the collision with the tightly focussed laser, can pass throughto reach the diagnostics. Both the experiments that will be described in this section used thisgeometry, which is illustrated in fig. 8, but with different electron acceleration stages.In Cole et al. [18], the accelerating laser pulse was focussed onto the leading edge of a su-personic helium gas jet, producing a ∼
15 mm plasma acceleration stage with peak density n e (cid:39) . × cm − . The use of a gas jet allowed the second laser pulse to be focussed close tothe point where the electron beam emerges from the plasma (at the rear edge), so the collisionbetween electron beam and laser pulse took place when the former was much smaller than thelatter (approximately 1 µ m rather than 20 µ m , which includes the effect of a systematic timedelay between the two). The advantages of using a gas cell, as in Poder et al. [19], are the higherelectron beam energies and significantly better shot-to-shot stability. However, in this case, thesecond laser pulse must be focussed further downstream of the acceleration stage (approximately1 cm), by which point the electron beam has expanded to become comparable in size to the laser.Thus full 3D simulations were required for theoretical modelling of the interaction, whereas 1D30plane-wave) simulations were sufficient in Cole et al. [18].Fluctuations in the pointing and timing of the two lasers, as well as systematic drifts in the lat-ter, mean that the overlap between electron beam and target laser pulse varies from shot to shot. Itis helpful, therefore, to gather as large a dataset as possible (with the second, high-intensity, laserpulse both on and off), in which case high-repetition-rate laser systems are at a clear advantage.However, this is not nearly so important as being able to identify ‘successful’ collisions when theyoccur; even a small set of collisions ( N ∼
10) can provide statistically significant evidence of radi-ation reaction when this is done. This speaks to the importance of measuring both the electron and γ -ray spectra on a shot-to-shot basis; identifying coincidences between the two provides strongerevidence of radiation reaction than could be obtained by either alone.In Cole et al. [18], successful shots were distinguished by measuring the total signal in the γ -raydetector S γ ∝ N e a (cid:104) γ (cid:105) (background-corrected), where N e is the total number of electrons in thebeam, (cid:104) γ (cid:105) their mean squared Lorentz factor, and a an overlap-dependent, effective value for a (the former two can be extracted from the measured electron spectra). Over a sequence of 18 shots(eight beam-on , i.e. with the f / beam-off ), four were measured with a normalizedCsI signal, ˆ S γ = S γ / ( N e (cid:104) γ (cid:105) ) ∝ a , four standard deviations above the background level. Thesefour also had electron beam energies below 500 MeV (as identified by a strong peak feature inthe measured spectra), whereas the ten beam-off shots had a mean energy of 550 ±
20 MeV. Theprobability of measuring four or more beams with energy below 500 MeV in a sample of eight,given this fluctuation alone, is approximately 10%. However, the probability that four beams havethis lower energy and a significantly higher γ -ray signal is the considerably smaller 0.3%.Statistically significant evidence of radiation reaction was obtained by correlating the elec-tron beam energy with the critical energy of the γ -ray spectrum ε crit , a parameter characterizingthe hardness of the spectrum. This was accomplished by fitting the depth-resolved scintillatoroutput to a parametrized spectrum dN γ / d ω ∝ ω − / exp ( − ω / ε crit ) , having first characterized itsresponse to monoenergetic photons in the energy range 2 < ω [ MeV ] <
500 with G
EANT et al. [187]). This choice of parametrization approximately reproducesa synchrotron-like spectral shape, with an exponential rollover at high energy and a scaling like ω − / at low energy. The four successful shots demonstrate a negative correlation between the fi-nal electron energy and ε crit , as is shown in fig. 9; this is consistent with radiation-reaction effects,as the hardest photon spectra should come from electron beams that have lost the most energy. Theprobability to observe this negative correlation and to have electron energy lower than 500 MeV31
00 420 440 460 480 500 520 540 560 final /MeV101520253035404550 c r i t / M e V Quantum modelClassical modelNo RRData
500 1000 1500 2000 2500
Electron energy (MeV) E l e c t r on s pe r M e V pe r % ene r g y s p r ead ( no r m . ) b
500 1000 1500 2000 2500
Electron energy (MeV) E l e c t r on s pe r M e V pe r % ene r g y s p r ead ( no r m . ) c
500 1000 1500 2000 2500
Electron energy (MeV) E l e c t r on s pe r M e V pe r % ene r g y s p r ead ( no r m . ) d Scatt. laser on /E Scatt. laser off G a mm a y i e l d / T o t a l r e f e r en c e bea m ene r g y b c d a Exp. resultsTheor. calculation
FIG. 9. Experimental evidence of radiation reaction: (left) in Cole et al. [18], the post-collision elec-tron beam energies and critical energies of the γ -ray spectra (black points), are consistent with theoreticalsimulations that include radiation reaction, with slightly better agreement for the stochastic model; (right)in Poder et al. [19], the fractional reduction in the total electron beam energy is correlated with the total γ -ray signal, with the best agreement with theory given by the ‘modified classical’ model (see section III A).Details are given in the main text. on all four successful shots is 0.03%, which qualifies, under the usual three-sigma threshold, asevidence of radiation reaction.Simulations of the collision confirmed that the critical energies and electron energy loss wereconsistent with theoretical expectations of radiation reaction. The coloured regions in fig. 9 givethe areas in which 68% (i.e. one sigma) of results would be found for a large ensemble of ‘nu-merical experiments’, given the measured fluctuations in the pre-collision electron energy spectraand the collision a , and under specific models of radiation reaction. The results exclude the ‘noRR’ model, in which the electrons radiate, but do not recoil. They are more consistent with thestochastic, quantum model discussed in section III B than the deterministic, classical model ofLandau-Lifshitz: however, it is important to note that both models are consistent with the data atthe two-sigma level. Subsequent analysis has confirmed that the ‘modified classical’ model dis-cussed in section III A, which includes the quantum suppression factor g ( χ ) , given in eq. (20), butnot the stochasticity of emission, gives practically the same region as the quantum model [188], asis stated in Cole et al. [18]. This is because the electron beam energy effectively parametrizes themean of the spectrum, the evolution of which depends only on g ( χ ) according to eq. (23); to seestochastic effects, we must consider instead the width of the distribution [67, 71, 73]. Given elec-32ron beams with narrower initial energy spectra, it would be possible to identify stochastic effects(or their absence) by correlating the mean and variance of the final electron energy spectra [189].Evidence of radiation reaction was also obtained in the experiment reported by Poder et al. [19], by a complementary form of analysis. The total CsI signal was used to discriminate success-ful collisions: the signal normalized to the total energy in a reference (beam-off) shot S γ / ( N e (cid:104) γ (cid:105) ) was observed to be linearly correlated with the percentage energy loss of the electron beam (ascompared against a reference beam, see fig. 9). Three shots were selected as exemplary cases ofpoor, moderate and strong overlap, according to these two values, with corresponding strength ofradiation-reaction effects. This shots are labelled (b), (c) and (d) respectively in the right-handpanel of fig. 9. The analysis then focussed on comparison of the measured electron energy spectraagainst those predicted by simulations under various models of radiation reaction. These compar-isons showed that the Landau-Lifshitz equation, i.e. classical radiation reaction, overestimated theenergy loss, with a quality of fit of R = .
87. Simulations with the ‘modified classical’ modelimproved the agreement to R = .
96; this was found to give better agreement than the fullystochastic model, in which case R = .
92. This discrepancy was attributed to possible failure ofthe LCFA as the collision a (cid:39)
10. Nevertheless, by considering the detailed shape of electronenergy spectra, it was possible to find clear evidence of radiation reaction, as well as signatures ofquantum corrections.
V. SUMMARY AND OUTLOOK
Let us now consider the relation between the results of these two experiments discussed insection IV C. Both present clear evidence that radiation reaction, in some form, has taken place.The reduction in the electron energies, the total γ -ray signal, and, in Cole et al. [18], the spectralshape of the latter, are all broadly consistent with each other. The differences arise in the compar-ison of different models of radiation reaction, bearing in mind that, in the regime where χ (cid:39) . a (cid:39)
10, quantum corrections are expected to be non-negligible, but not large, and the intensity isnot so large that the LCFA is beyond question. The use of simulations that rely on this approxi-mation is, however, necessary because the number of photons emitted, per electron in the beam,is much larger than unity, and therefore an exact calculation from QED is intractable at present(see section III C). In Cole et al. [18], the shot-to-shot fluctuations in the electron beam energy andalignment, and the fact that the electron spectrum is analyzed by means of a single value rather than33ts complete shape, mean that all three models (classical, modified classical, and quantum) are notdistinguishable from each other at level of two standard deviations. At the one-standard-deviationlevel, the two models that include quantum corrections provide better agreement.Poder et al. [19], with significantly more stable electron beams, are able to confirm that theclassical model is not consistent with the data either. However, the fact that neither the modifiedclassical or quantum models provide a very good fit to the data leaves open the question of whetherit is the failure of the LCFA or, as they state, “incomplete knowledge of the local properties of thelaser field.” Accurate determination of the initial conditions, in both the electron beam and thelaser pulse, will be of unquestioned importance for upcoming experiments that aim to discernthe properties of radiation reaction in strong fields. It will be vital to characterize the uncertain-ties in both the experimental conditions and the theoretical models in our simulations, which areinevitably based on certain approximations.Nevertheless, these results demonstrate the capability of currently available high-intensitylasers to probe new physical regimes, where radiation reaction and quantum processes become theimportant, if not dominant, dynamical effects. These experiments provide vital data in the unex-plored region of parameter space χ (cid:38) . a (cid:29) χ and a . Not only will this make radiation reaction and quantum corrections more distinct, it willalso allow us to measure nonlinear electron-positron pair creation by the γ rays emitted by the col-liding electron beam [117, 118, 181, 191], a strong-field QED process without classical analogue.Such findings will underpin the study of particle and plasma dynamics in strong electromagneticfields for many years to come. 34 CKNOWLEDGMENTS
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