Physics of Laser-Wakefield Accelerators (LWFA)
PProceedings of the 2019 CERN–Accelerator–School course on
High Gradient Wakefield Accelerators, Sesimbra, (Portugal)
Physics of Laser-Wakefield Accelerators (LWFA)
Johannes Wenz, Stefan Karsch
Centre for Advanced Laser Applications, Ludwig-Maximilians-Universität München, Munich,Germany
Abstract
Intense ultrashort laser pulses propagating through an underdense plasma areable to drive relativistic plasma waves, creating accelerating structures withextreme gradients. These structures represent a new type of compact sourcesfor generating ultrarelativistic, ultrashort electron beams. This chapter coversthe theoretical background behind the process of LWFA. Starting from the ba-sic description of electromagnetic waves and their interaction with particles,the main aspects of the LWFA are presented. These include the excitation ofplasma waves, description of the acceleration phase and injection mechanisms.These considerations are concluded by a discussion of the fundamental limitson the energy gain and scaling laws.
Keywords
High-intensity lasers; realtivistic laser-plasma interaction; laser-wakefield ac-celeration. Introduction
The development of high power laser systems has enabled new approaches for generating relativisticelectron beams. In a Laser-Plasma Accelerator (LPA) a transient accelerating field configuration isformed when a high intensity laser pulse propagates through a plasma. Its ponderomotive force, causedby its strong intensity gradients, deflects the electrons from their equilibrium position while the ionsremain at rest. Their space-charge force causes the electrons to oscillate and constitute an electron den-sity wave with strong longitudinal electric fields. Since the phase velocity of this plasma wave matchesthe group velocity of the laser pulse in the plasma, trapped electrons stay in phase with this wave foran extended period of time and can be effectively accelerated to ultra-relativistic energies. This wavestructure (termed “wakefield”) exhibits up to four orders of magnitude higher acceleration gradients thanthose achieved in conventional radio-frequency accelerators, thus drastically reducing the accelerationdistance to reach a given energy. Not only the reduction in size, and consequently, cost of LPAs arepromising, but also their unique properties regarding such important quantities as electron bunch dura-tion, current density and emittance. These and other key features have raised global interest in exploringand developing laser-based sources as a powerful tool for medical, industrial and scientific applications.The concept of a LPA in the LWFA regime was proposed over 35 years ago by [1]. In their pioneeringwork they predicted that in a plasma, the strong transverse oscillating fields of an intense laser beamcan be efficiently “rectified” into longitudinal fields, capable of electron acceleration to
GeV energieswithin centimeters of plasma. The first LPA experiments were performed with rather long laser pulses ofnanoseconds to picoseconds by the plasma beat-wave accelerator scheme (PBWA) by [2] and in the self-modulated LWFA regime by [3]. With the emergence of ultrashort high-intensity laser systems, the accel-eration gradient could be improved from few
GeV m − up to several hundred GeV m − , accompaniedby a decrease in the transverse electron beam divergence in the so-called direct laser acceleration (DLA)or forced laser-wakefield (FLWF) regime [4, 5]. All of these early LPA schemes had in common an ex-ponential electron energy spectrum with only a fraction of the accelerated charge contained in the highenergy tail.A breakthrough in this respect was achieved in the year 2004, as three groups [6–8] simultaneouslyAvailable online at https://cas.web.cern.ch/previous-schools a r X i v : . [ phy s i c s . acc - ph ] J u l eported on the production of high-quality electron beams. They were characterized by a high charge(several
10 pC ), a quasi-monoenergetic spectrum around
100 MeV and a few- mrad beam divergence.These results marked the transition to a nonlinear regime of electron cavitation, termed “bubble” or“blow-out” regime, that was predicted in simulations by [9]. Two years later, the barrier wasbroken by increasing the acceleration length to a few centimeters in a laser guiding discharge capillary[10–13]. Electrons approaching have been recently measured with more powerful laser systemsand enhanced guiding [14].Further improvements, such as a high shot-to-shot stability in terms of divergence, energy andbeam pointing, were first demonstrated in a steady-state flow gas cell [15]. Control over the injectionprocess has led to a reduction in energy spread and additionally provided a knob for energy tuning.Among others, successful injection schemes are colliding pulse injection [16, 17], density down ramp[18] and shockfront injection [19, 20], as well as related techniques [21–23].Simultaneously, the longitudinal and transverse electron phase space has been investigated. Mea-surements have confirmed the few-femtosecond duration of typical LWFA bunches, up to date unrivaledby conventional accelerators [24–26]. The transverse emittance has been determined to below 1 π mmmrad [27] and few-fs-exposure shadowgrams of the wake have provided more insight into the accelera-tion process [24, 28]. A good introduction to experiments and theory of LWFA can be found in [29] andinto X-ray applications of LWFA bunches in [30].Current research focuses on overcoming the dephasing limit on the maximum electron energygain. Staging of LWFAs has been demonstrated, where an electron bunch from one LWFA has beensuccessfully coupled into and gained energy in a second LWFA [31]. Likewise, driver/witness type ex-periments as performed earlier with beams from conventional accelerators [32] are currently investigated.Here, an LWFA bunch is used as the driver of the plasma wave, thus overcoming the dephasing limits.Such experiments have shown that LWFA bunches are able to drive a plasma wave and can be effectivelydecelerated in millimeter size plasma targets [33, 34]. Light-Matter Interaction
Electromagnetic phenomena, such as generation and propagation of electric ( (cid:126)E ( (cid:126)x, t ) ) and magnetic( (cid:126)B ( (cid:126)x, t ) ) fields and their interaction with each other as well as with charges and currents, are describedby Maxwell’s equations. Their differential form is given by [35] ∇ · (cid:126)E = ρ(cid:15) , (1a) ∇ · (cid:126)B = 0 , (1b) ∇ × (cid:126)E = − ∂ (cid:126)B∂t , (1c) ∇ × (cid:126)B = µ (cid:15) ∂ (cid:126)E∂t + µ (cid:126)j, (1d)where ρ and (cid:126)j are the charge and current density of the medium and (cid:15) and µ the vacuum permittivityand permeability, respectively. Sometimes it is more convenient to express these equations by a scalarpotential φ and a vector potential (cid:126)A , which can be obtained from Maxwell’s equations : (cid:126)E = −∇ φ − ∂∂t (cid:126)A, (cid:126)B = ∇ × (cid:126)A. (2)These electromagnetic potentials are not uniquely defined, and different solutions can lead to the sameelectric and magnetic fields by a gauge transformation, which in principle amounts to specifying a valuefor the term ∇ · (cid:126)A . In the Coulomb gauge ( ∇ · (cid:126)A = 0 ), the two inhomogeneous Eqs. (1a) and (1d) yield with the use of the vector identities (cid:126) ∇ · ( ∇ × (cid:126)a ) = 0 and ∇ × ( ∇ φ ) = 0 in Eqs. (1b) and (1c) ∇ φ = − ρ(cid:15) , ∇ (cid:126)A − µ (cid:15) ∂ (cid:126)A∂t = − µ (cid:126)j + µ (cid:15) ∇ ∂φ∂t . The scalar potential represents a solution for Poisson’s equation and, once it is found, it can be usedto solve the equation for the vector potential. In vacuum, in the absence of charges and currents, onepossible solution is represented by a plane wave: φ ( (cid:126)x, t ) = 0 , (cid:126)A ( (cid:126)x, t ) = − (cid:126)A L sin( ω L t − (cid:126)k(cid:126)x + ϕ L ) . (3)The electric and magnetic fields (cid:126)E ( (cid:126)x, t ) and (cid:126)B ( (cid:126)x, t ) are derived from Eq. (2) to (cid:126)E ( (cid:126)x, t ) = (cid:126)E L ( (cid:126)x, t ) cos( ω L t − (cid:126)k(cid:126)x + ϕ L ) , (cid:126)B ( (cid:126)x, t ) = (cid:126)B L ( (cid:126)x, t ) cos( ω L t − (cid:126)k(cid:126)x + ϕ L ) . (4)Here, ϕ L is an arbitrary phase offset and ω L = 2 πc/λ L the angular frequency given by the wavelength λ L . (cid:126)E L and (cid:126)B L are spatially ( (cid:126)x ) and temporally ( t ) confined envelope functions, which oscillate inphase and are perpendicular to each other, (cid:126)E ⊥ (cid:126)B . The wave vector (cid:126)k and the angular frequency ω L areconnected via the dispersion relation in vacuum (cid:126)k = ω L /c . (5)Moreover, (cid:126)E ⊥ (cid:126)B implies that (cid:126)E ⊥ (cid:126)k , (cid:126)B ⊥ (cid:126)k , (cid:126)E ⊥ (cid:126)A , and the amplitudes are related by | (cid:126)A L | = c/ω L | (cid:126)B L | =1 /ω L | (cid:126)E L | , i.e., the (cid:126)B -field component is c times smaller than the (cid:126)E -field. The spatial profile of a laser pulse can be described by a Gaussian distribution. For a radial symmetricprofile with radius r the electric field of a monochromatic beam near the focal point can be representedas [36] (cid:126)E ( r, z, t ) = E e − r w ( z )2 (cid:124) (cid:123)(cid:122) (cid:125) (cid:13) e − ( t − z/c )2 τ (cid:124) (cid:123)(cid:122) (cid:125) (cid:13) cos ( ω L t − k L z + ϕ L ) (cid:124) (cid:123)(cid:122) (cid:125) (cid:13) (cid:126)e pol + cc., (6)where the first term (cid:13) describes the radial envelope with the transverse 1/e radius of the beam w ( z ) .The second p art (cid:13) represents the temporal envelope with τ as the pulse duration defined at / e -heightof the electric field, cf. Fig. 1(a). (cid:13) is an oscillatory term containing the carrier angular frequency ω L and (cid:126)e pol is the vector describing the polarization of the laser pulse in the ( x , y )-plane, i.e., (cid:126)e pol = (cid:126)e x,y forlinear and (cid:126)e pol = ( (cid:126)e x ± i(cid:126)e y ) / √ for circular polarization.The transverse radius of the beam evolves along the propagation direction with w ( z ) = w (cid:113) z/z R ) , with z R = πw /λ L , (7)where w is the radius at /e -height of the maximum electric field and z R is the Rayleigh length , definedas the distance from the waist w to the z-position at half the intensity. The interaction range defined astwice the Rayleigh length is called the confocal parameter . It is marked as the shaded area in Fig. 1(b).Far away from the focal plane, the corresponding divergence of the beam can be approximated by tan θ ≈ θ ≈ ∆ w ( z ) / ∆ z = 2 λ L / ( πw ) . Here the origin can be considered as a point source, such that θ ≈ D/f ≡ ( F/ − , where D is the diameter of the collimated beam, f is the focal length and F/ isthe f-number of the focusing optic. Equating these two expressions result in the spot size. In terms of3 −
30 0 30 60 − − V / m ) E l e c t r i c o F i e l d o ( I n t en s i t y o ( W / c m ) Electricofield Envelope Intensity (a)
Temporal profile of the oscillating (cid:126)E -field (cyan line)and its Gaussian envelope (dark blue dashed line) as well asthe intensity profile (red line) for a pulse duration t FWHM =28 fs . − − B ea m y W a i s ty i n y R ad i a l y D i r e c t i on y ( µ m ) I n t en s i t y y ( W / c m ) (b) Spatial confinement of the /e -irradiance (blue line)near the focal spot with d FWHM ≈ µ m . The interactionregion is marked in light red, defined by the intensity I >I / . Fig. 1: Spatial and temporal profile of a Gaussian laser pulse at a wavelength of λ L = 800 nm , W = 1 J and t F W HM = 28 f s focused in a F/ focusing geometry. the laser parameters, it is given by w = 4 λ L fπD . Note that the waist of the beam w and the pulse duration τ are related to the measured quantitiesat FWHM of the temporal and spatial intensity profile via the spot size d F W HM = (cid:112) w andthe pulse duration t F W HM = (cid:112) τ . From these quantities and the total laser pulse energy W L the laser peak power P and the laser peak intensity I = 2 P /πw can be determined. For a Gaussianpulse, as given in Eq. (6), they are P =0 . W L t F W HM , I =0 . W L t F W HM d F W HM . (8)On the other hand, the intensity is defined as the cycle-averaged Poynting vector (cid:104) (cid:126)S (cid:105) = (cid:15) c (cid:104) (cid:126)E × (cid:126)B (cid:105) .For a linearly polarized pulse, it depends on the electric field via I L = (cid:104)| (cid:126)S |(cid:105) T = (cid:15) c (cid:104)| (cid:126)E | (cid:105) T = (cid:15) c E L . (9)The peak magnetic and electric field can be expressed in terms of I (Eq. (8)) E = 2 . (cid:114) I [W cm − ]10 λ L [ µ m] × V m − , B = 0 . (cid:114) I [W cm − ]10 λ L [ µ m] × T m − . State-of-the art high-intensity laser systems readily reach intensities beyond W cm − , correspond-ing to field amplitudes of V m − and T . These electric fields are several order of magnitudehigher than the binding fields within atoms, resulting in ionization and plasma formation in the focalvolume. Although the actual beam profile is rather closer to a super-gaussian beam, the use of Gaussian optics simplifies the ex-pressions. For a top-hat beam, the spot size is d FWHM ≈ . λ L F/ . .3 Plane Wave Interaction with Particles With the high peak intensities delivered by current state-of-the-art laser systems, even the leading edgeof the laser pulse is strong enough to ionize matter nanoseconds to picoseconds before the arrival ofthe peak. The intense part of the laser pulse therefore interacts with free charges.
The relativistic motion of an electron (with charge q = − e , mass m e , velocity (cid:126)v and momentum (cid:126)p ) inthe presence of an electromagnetic field is described by the Lorentz force [37] d(cid:126)pdt = − e ( (cid:126)E + (cid:126)v e × (cid:126)B ) , (10)and the energy equation ddt ( γmc ) = − e ( (cid:126)v · (cid:126)E ) . (11)Here, γ is the relativistic factor and (cid:126)p the momentum. They are defined by γ = (cid:115) p m e c = 1 (cid:112) − β , (cid:126)p = m e γ(cid:126)v, (12)where β is the normalized velocity β = | (cid:126)v e | /c . For a subrelativistic case ( (cid:126)v e (cid:28) c and | (cid:126)B | = | (cid:126)E | /c )the cross product can be neglected and the Lorentz equation simplifies for plane waves Eq. (4) to m e d (cid:126)v e dt = − e (cid:126)E cos( ω L t − (cid:126)k(cid:126)x + ϕ L ) . (13)Integration with respect to time yields the quiver velocity v e . In the electric field (cid:126)E the electron canreach a maximum value of v e,max = e | (cid:126)E | /m e ω L .A ponderomotive energy W pond and potential Φ pond can be associated with the quiver velocity byaveraging the kinetic energy over one optical light cycle. W pond = 12 m e (cid:104) (cid:126)v e (cid:105) T = e (cid:126)E m e ω L = e Φ pond . (14)Once the quiver velocity approaches the speed of light c , the assumption | (cid:126)v | (cid:28) c breaks downand the (cid:126)v × (cid:126)B term cannot be neglected anymore. The ratio of the mean quiver energy to the electronrest mass is defined as a so-called normalized vector potential a . It distinguishes the subrelativistic( a (cid:28) ) and the relativistic regime ( a (cid:38) ) a = e | (cid:126)E | m e ω L c = e | (cid:126)A | m e c = 0 . (cid:113) I [10 W / cm ] · λ L [ µ m] . (15)For a TiSa-based laser system operating at a wavelength of λ L = 800 nm , a = 1 is reached at laserintensities of I ≈ . × W / cm .For high laser intensities, both the electric and magnetic fields are responsible for the electronmotion. The electron trajectory in a plane wave consists of a drift in the z-direction with the velocity v D /c = a / (4 + a ) and Fig. 8 motion in the average rest frame of the electron, [37]. The net effectafter the transit of the laser pulse is a translation of the electron in the forward direction, but the electronis at rest again. An acceleration in an infinite plane wave is therefore not possible, a phenomenon knownas Lawson-Woodward-theorem [38, 39]. 5owever, in a focused beam Eq. (6) the electron can gain energy in the following way: The strongradial intensity gradient pushes an electron, initially located on the laser axis, to an off-axis location. Atthis position it experiences weaker field amplitudes, and when the field reverses, the restoring force isreduced. On average the electron is pushed out from high-intensity regions by the ponderomotive force (cid:126)F pond = − e ∇ Φ pond ∝ −∇ ( I L λ L ) . The coupling of the laser to the electrons is mediated mainly bythis quasi-force, which can be derived from the second-order motion of electrons in the high gradientsof the light field [37]. In the linear regime it is convenient to describe F pond via the normalized vectorpotential a averaged over one laser cycle, i.e., for a Gaussian pulse a ( t ) = (cid:104) a (cid:105) T = a exp[ − ( t/τ ) ] itis given by [40] (cid:126)F p = − m e c ∇ a , (cid:126)F p,rel = − m e c ∇(cid:104) γ − (cid:105) ∝ a. (16)In the relativistic regime the ponderomotive force F p,rel is usually defined by the gradient of the cycleaveraged electron gamma factor (cid:104) γ (cid:105) . The associated maximal relativistic ponderomotive potential foran electron initially at rest can be estimated in a linearly polarized wave by Φ p,rel = m e c e a √ . For example, at intensities of . × W cm − ( a = 2 ), the ponderomotive energy is e Φ p,rel ≈ . . Therefore, a direct laser acceleration of electrons with W el >
10 MeV us-ing table top systems is suppressed. However, the corresponding ponderomotive force can excitea plasma wave, which acts as intermediary and transforms the strong transversal fields of the laser intolongitudinal fields suitable for acceleration. The process of longitudinal acceleration is examined below. Laser-Driven Plasma Waves
As discussed in the previous section the leading edge of the laser pulse ionizes the target medium, andthe intense part of the laser interacts with a plasma. Its ponderomotive force can then displace electronsfrom their equilibrium positions and cause macroscopic regions dominated by space charge. In order tounderstand these dynamics, it is necessary to consider the propagation of the electron-magnetic wave ofthe laser in a plasma medium.
As it is impossible to treat each particle individually, the motion of electrons driven by an electromagneticwave in a plasma can be derived from a set of Eqs. (17) - (19). It consists of the Lorentz equation,the continuity equation and Poisson’s equation Lorentz equation: d(cid:126)pdt = (cid:18) ∂∂t + (cid:126)v · ∇ (cid:19) (cid:126)p = − e [ (cid:126)E + (cid:126)v × (cid:126)B ] , (17) Continuity equation: ∂n e ∂t + ∇ ( n e (cid:126)v ) = 0 , (18) Poisson’s equation: ∇ Φ = − ρ(cid:15) = e δn e (cid:15) , (19) Note that for a linearly polarized pulse a has to be replaced by a / . from Eq. (12) for p/m e c = u ⊥ as given in Eq. (33) derived from Maxwell’s Eq. (1a) and (1d) using the vector identity ∇ ( ∇ × (cid:126)a = 0) ) a direct consequence of Maxwell’s equation (1a) and Eq. (2) in the Coulomb gauge ( ∇ A = 0 ). (cid:126)p = m e γ(cid:126)v Eq. (12) is the momentum and δn e = n e − n e, is the local density perturbation, (cid:126)j = − en e (cid:126)v the current density and ρ = − eδn e is the charge density, respectively. If the electrons forming the plasma are expelled from their equilibrium position via the ponderomo-tive force, they will be pulled back by the positive ions and oscillate around their initial position witha characteristic frequency. This oscillation frequency can be derived from Eq. (17) considering smallamplitudes in a cold plasma, where the initial thermal energy of the electrons is ignored. The equation ofmotion for the plasma fluid is given by n e, m e ∂(cid:126)v/∂t = − n e, e (cid:126)E , where the quadratic terms have beenneglected.It can be reformulated with Eq. (1d) and the vector identity ∇ × ( ∇ × (cid:126)a ) = ∇ ( ∇ · (cid:126)a ) − ∇ (cid:126)a to −∇ ( ∇ (cid:126)E ) + ∇ (cid:126)E = µ n e, e m e (cid:126)E + µ (cid:15) ∂ (cid:126)E∂ t . For oscillating electromagnetic plane waves of the type E ∝ e ik L x − ω L t propagating in a uniform medium( ∇ (cid:126)E = 0 ) the above equation results in the dispersion relation of a cold plasma ω L = ω p + c k L with ω p ≡ (cid:115) n e, e m e (cid:15) , (20)where ω p is the so called plasma frequency. For ω L = ω p a critical density can be defined n e,c ≡ (cid:15) m e e ω L , n e,c = 1 . λ L [ µ m] 10 cm − . In plasmas with densities n e >n e,c ( ω p >ω L ), the characteristic time scale of the plasma is shorter thanthe optical period of the light wave and the electrons can follow the field oscillation, shield the inside ofthe medium and the laser is reflected. The medium is referred to as overdense. In contrast, the plasma iscalled underdense for plasma densities n e
8f the pulse chirp. The modulation in frequency follows the local change in the refractive index via [41] ω L ∂ω∂τ = − η ∂η∂ξ . For Gaussian pulses, the gradients at the leading (falling) edge of the laser pulse will be red (blue) shiftedcausing a broadening of the spectrum, which is known as self-phase modulation (SPM). The densitychange δn e ( ξ ) results in a varying local index of refraction. For a positive density slope, as encounteredby the pulse in a typical LWFA experiment, the group velocity at the front of the pulse experiencesa deceleration, while the tail sees an acceleration. Such pulse shortening effects have been observed inexperiments [46, 47] and can be approximated as τ comp = τ − n e, l cn e,c , (26)where l is the propagation length in the plasma and τ and τ comp are the pulse durations before and afterthe interaction with the plasma. The variation of η also modifies the group and phase velocity (21). v ph = cη ≈ c (cid:34) ω p ω L (cid:18) − (cid:104) a (cid:105) δn e n e, − δωω L (cid:19)(cid:35) , v g = cη ≈ c (cid:34) − ω p ω L (cid:18) − (cid:104) a (cid:105) δn e n e, − δωω L (cid:19)(cid:35) . The combined effect of SF and SPM is a focusing and compression of the pulse, boosting its intensity.Since the ponderomotive potential causes a repulsion of electrons from regions of high intensity, the neteffect looks as it the electron inertia would "squeeze" the laser into a smaller, more intense bullet of light.
A laser pulse excites a plasma wave by displacing electrons from their equilibrium position. Inthe Coulomb gauge ( ∇ A = 0 ), the Lorentz Eq. (17) in terms of the electromagnetic potentials reads: (cid:18) ∂∂t + (cid:126)v · ∇ (cid:19) (cid:126)p = e (cid:20) ∂∂t (cid:126)A + ∇ Φ − (cid:126)v × ∇ × (cid:126)A (cid:21) . (27)In addition, using the vector identity ∇ p = 2 [( (cid:126)p · ∇ ) (cid:126)p + (cid:126)p × ( ∇ × (cid:126)p )] and ∇ p = m e c γ ∇ γ (ob-tained from Eq. (12)) yields m e c ∇ γ = ( (cid:126)v · ∇ ) (cid:126)p + (cid:126)v × ( ∇ × (cid:126)p ) . The term on the left represents the driving term, i.e., the ponderomotive force as introduced in Eq. (16).Inserting the above expression into Eq. (27), the equation of motion can be expressed as ∂(cid:126)p∂t = e ∇ Φ + e ∂ (cid:126)A∂t − m e c ∇ γ (28)Here, (cid:126)v × ∇ × ( (cid:126)p − e (cid:126)A ) is set to 0, indicating that the electrons are only driven by the vector potential.This can be understood by realizing that the curl of Eq. (27) leads to ∂ ( (cid:126)p − e (cid:126)A ) /∂t = (cid:126)v × ( ∇ × ( (cid:126)p − e (cid:126)A )) . Therefore, the canonical momentum (cid:126)p − e (cid:126)A = 0 remains zero for all times if there is no initialperturbation of the plasma and the electrons initially have zero canonical momentum ( (cid:126)p − e (cid:126)A = 0 ) [48].This equation states the starting point for exploring the solution of the linearly and nonlinearly drivenplasma waves. Moreover, for the upcoming derivations it is convenient to use normalized quantities β = (cid:126)v/c, (cid:126)a = e (cid:126)Am e c , φ = e Φ m e c , γ = Em e c , (cid:126)u = (cid:126)pm e c . (29)9
10 20 30 40 − − no r m a li z ed un i t s ξ (µm) E z E ξ (µm) r ( µ m ) − − δ n e ( c m − ) − (a) Top:
Normalized plasma potential φ , longitudinal elec-tric field E z /E and density perturbation δn e /n e, on axis( r = 0 ). Bottom: plasma density map δn e ( r, ξ ) /n e, gen-erated by the ponderomotive force of a Gaussian laser focus. ξ (µm) r ( µ m ) deceleratinglongitudinal0 10 20 30 40 − − E z ( G V / m ) − − ξ (µm) r ( µ m ) transverse0 10 20 30 40 − − E r ( G V / m ) − (b) Spatial extend of the longitudinal E z ( r, ξ ) (on the Top )and the radial electric field E r ( r, ξ ) (on the Bottom ).The green area marks a λ p / -phase region of the wakefieldwith an accelerating and transverse focusing field. Fig. 2: 3D linear wakefield quantities in the moving co-ordinate system created in the focus of a laser pulse( a = 0 . , t F W HM = 28 fs and d F W HM = 22 µ m ) for a plasma density of × cm − . and to introduce a coordinate transformation to a frame moving with the laser pulse at a speed v g .The new variables ( ξ, τ ) are given by ξ = z − v g t and τ = t and their partial derivatives are: ∂∂z = ∂∂ξ and ∂∂t = ∂∂τ = − v g ∂∂ξ v g → c ≈ − c ∂dξ . (30) An analytical solution for small laser intensities ( a (cid:28) ) and weakly perturbed plasmas ( δn e (cid:28) n e, )can be derived from the continuity and Poisson Eqs. (18), (19). The solution for the scalar potential φ ,yields for a Gaussian laser envelope a = a exp( − ξ / ( cτ ) ) exp( − r /w ) after the laser transit φ ( r, ξ ) = − f ( r ) sin( k p ξ ) , f ( r ) = a (cid:114) π k p cτ exp (cid:18) − r w (cid:19) exp (cid:18) − ( k p cτ ) (cid:19) . (31)From the definition of the scalar potential φ , Eq. (2) and Poisson’s Eq. (19), the electric field andthe electron density are given as: E z E p, = − k p ∂φ∂ξ , E r E p, = − k p ∂φ∂r , δn e n e, = 1 k p ∂ φ∂ξ . (32) E p, denotes the cold nonrelativistic wavebreaking field (see Eq. (38)). The solution for the trans-verse and longitudinal electric fields as well as the density perturbation excited by a weak Gaussianlaser pulse ( a = 0 . ) are displayed in Fig. 2. The response of the normalized plasma quantitiesis linear (Fig. 2(a)). The wakefield radius is determined by the focal spot size. Figure 2(b) denotesalternating phase regions where the wake is either accelerating/decelerating or focusing/defocusing.LWFA takes place where focusing and accelerating phases coincide, i.e. in the shaded phase inter-vals with a width λ p / . The peak wakefield potential is determined by the laser intensity and pulseduration: φ ∝ a τ . For a given electron density n e, , the wave is driven resonantly ( φ → φ max ) for10 pulse duration of t optF W HM [fs] ≈ . n e, [10 cm − ]) − / , corresponding to a laser pulse length of L F W HM = 0 . λ p . This requires short laser pulses for electron acceleration at a plasma wavelengthof λ p [ µ m] = 33 . n e, [10 cm − ]) − / . The accelerating and focusing fields in this weakly relativisticregime reach gradients of GeV m − at a = 0 . , and further increase with laser intensity. For a → the wave cannot be treated by linear perturbation theory anymore. At high intensities ( a > ), the plasma response becomes highly nonlinear and has to be treated non-perturbatively. The general case of arbitrary pump strengths has no 3D-solution, and reduced modelsonly have limited predictive power. Therefore, phenomenological scaling laws based on 3D particle-in-cell (PIC) simulations [49–51], see also Section 4.5, are employed. However, analytical solutionsexist in the 1-D spatial case [52, 53], that can explain certain aspects of nonlinear plasma waves andhelp understanding their properties. In 1-D space, the 3-D electron momentum can be decomposed intoa longitudinal z -component and a transverse component in the ( x , y )-plane. The equation of motion inboth directions can be derived [37] from Eqs. (10) and (11): d (cid:126)p ⊥ dt = e ( (cid:126)E ⊥ + (cid:126)v z × (cid:126)B z ) = e d (cid:126)A ⊥ dt , dEdt = c dp (cid:107) dt . Integration for an initially resting electron yields for the transverse and longitudinal momentum (cid:126)p ⊥ = e (cid:126)A ⊥ ⇔ (cid:126)u ⊥ = γ (cid:126)β ⊥ = (cid:126)a, (33) E − cp (cid:107) = m e c ⇔ γ − u (cid:107) . (34)As in the linear case, the solution for the wake potential φ for an arbitrary pump can be deduced fromEq. (28), the continuity Eq. (18) and Poisson’s Eq. (19) ∂ φ∂ξ = k p γ p β p (cid:32) − a γ p (1 + φ ) (cid:33) − / − . (35)The above nonlinear ordinary differential equation can now be solved for the potential φ . The electricfield and density perturbation are again obtained from Eq. (32). For a linearly polarized square pulsewith v g → c the scalar potential φ ,max and the peak electric field E ,max are given by [54]: φ ,max ≈ a / , E ,max = E p, a / (cid:112) a / . (36)For an arbitrary pulse envelope a ( ξ ) the potential, the field and the density are obtained numerically.The solution for a Gaussian laser pulse in the linear ( a ≈ . ) and nonlinear ( a ≈ . ) regime aregiven in the top panels of Fig. 3(b). While nearly sinusoidal in the linear case, the non-linear densitydistribution exhibits narrow peaks with a periodicity of λ p,rel > λ p , separated by low-density regions.The electric field has a sawtooth shape and is linear over most of the period. The non-linear plasmawavelength λ p,rel inceases with a . For a squared pulse and γ p (cid:29) , the solution for λ p can be foundanalytically [29] λ p,rel = λ p a √ a for a (cid:28) π (cid:20) a √ a + √ a a (cid:21) for a (cid:29) . (37)11 .4 Wavebreaking The strength of the accelerating field defines the maximum energy gain for a fixed acceleration length. Itdepends on the density perturbation δn e /n e, of the plasma electrons in the density peaks. With increas-ing driver strength, the speed of the background electrons can exceed the phase velocity of the plasmawave, leading to a breakdown of the wakefield structure called longitudinal wavebreaking. The corre-sponding maximum field E wb limits the validity of the fluid plasma model.In the linear regime, E wb can be estimated by assuming that all electrons of the wave ( δn e = n e, )oscillate with ω p . Solving Eq. (1a), ∇ (cid:107) E ≈ − e(cid:15) n e, yields the cold, nonrelativistic wavebreakingfield [55] E wb = E p, = m e cω p e , E p, [GV m − ] = 96 (cid:113) n e, [10 cm − ] . (38)For relativistic fluid velocities nonlinear plasma waves can easily exceed E wb . The analytic solutioncan be found if the electron density becomes singular in Eq. (35) ( ∂ Φ ∂ ξ → ∞ ), i.e., γ ⊥ = √ a = γ p (1 + φ ) . On the other hand the minimum/maximum potential for the partial differential equation(PDE) behind the driving laser ( γ ⊥ = 1 ) is related to the maximum electric field E z,max via [56] φ min/max = 12 (cid:18) E z,max E p, (cid:19) ± β p (cid:32) (cid:18) E z,max E p, (cid:19) (cid:35) − / . (39)Equalizing φ min and φ = 1 /γ p − yields the cold relativistic wake-breaking field [57] E wb,rel = E p, (cid:113) γ p − . (40)Here, as in the nonrelativistic case, the background electrons can outrun the wake’s phase velocity andare injected. This self-injection generally continues once it has started, causing a large energy spread.In three dimensions, the wave also breaks transversely, which generally happens at much lower a thanin this purely 1-D treatment. For high beam quality, it is sometimes desired to avoid self-trapping byoperating below the wavebreaking threshold, and relying on other injection methods (see Section 4.2). Laser Wakefield Acceleration
The fundamental role of the plasma in LWFA is to transform the strong, transversely oscillating laserfields into quasistatic, longitudinally accelerating fields. The longitudinal dynamics of the electrons, i.e.their orbit in phase space, ( u z , ξ ) , by be treated by the Hamiltonian formalism. The 1D Hamiltonian for an electron interaction with a laser field in a plasma wave in its normalizedquantities is given by [56, 57] H (cid:48) ( z, u z ) = (cid:113) u ⊥ + u z − φ ( z − v g t ) . The first root term describes the normalized kinetic energy γ . It can be further simplified by observingEq. (33). The second term represents the potential energy. Here, the time dependence can be eliminatedby a canonical transformation of the Hamiltonian ( z, u z ) → ( ξ, u z ) , giving: H ( ξ, u z ) = (cid:113) a ( ξ ) + u z ( ξ ) + φ ( ξ ) − β p u z ( ξ ) . (41) Using a generating function F ( z, u z ) = u z × ( z − v g t ) the new Hamiltonian reads H = H (cid:48) − c ∂F∂t . no r m a li z ed un i t s λ u z + − m i n ( u f l u i d ) B p ξ / AC (a) Electron orbits in the nonrelativistic regime ( a < )feature a nearly symmetric separatrix with a width of (cid:39) λ p .The maximum energy gain is ∆ u z ≈ γ p ∆ φ = 1 . × . − no r m a li z ed un i t s λ p u z + − m i n ( u f l u i d ) ξ / AB C (b) Electron orbits in the relativistic regime with an asym-metric separatrix. Its width is given by the nonlinear plasmawavelength λ p,rel ∼ . λ p . The maximum energy gain is ∆ u z ≈ . × . Fig. 3: Longitudinal electron phase space trajectories in a laser-driven plasma wave with t F W HM = 28 fs , a = 0 . / √ ( left ) and a = 3 / √ ( right ). Trapped orbits (red solid lines), the separatrix (purple solid line),fluid (solid black line) and untrapped electron orbits (dashed blue lines). The corresponding normalized wakefieldquantities are plotted at the top of each figure. The shaded area marks the momentum gap to be overcome by fluidelectrons to get trapped. The background electron density in both plots is n e, = 5 × cm − , i.e., γ p = 19 . H ( ξ, u z ) = H = const. describes the motion of an electron on a distinct orbit in the plasma wave.Solving Eq. (41) for u z ( ξ ) gives the trajectory of the electron in the phase space u z = β p γ p ( H + φ ) ± γ p (cid:113) γ p ( H + φ ) − γ ⊥ . (42) u z represents an electron orbit of constant total energy for a given set of a ( ξ ) , φ ( ξ ) and H . Examplesare plotted in the lower part of Fig. 3. The lines represent the possible electron trajectories ( ξ, u z )with varying H in a linear ( a < ) and non-linear ( a > ) regime. In general, two types of orbitscan be distinguished: the closed orbits (solid red) describe the trapped and the open orbits (dashedblue) the untrapped electrons. The separatrix, displayed by the thick purple line, denotes the trajectorywhere open and closed orbits meet, i.e., where the radicand of Eq. (42) is equal to zero. It occursat φ ( ξ min ) = φ min , indicated by a thin vertical line. Therefore, the Hamiltonian of the separatrix is H sep = γ ⊥ ( ξ min ) /γ p − φ min . The absolute minimum u z can be derived from Eq. (42) to u z ( ξ min ) = β p γ p . Electrons initially at rest, u ⊥ ( ξ = + ∞ ) = u z ( ξ = + ∞ ) = 0 , are characterized by the Hamilto-nian H fluid = 1 . These fluid electrons, (thick black line) constitute the plasma wave and are pushed bythe ponderomotive force without gaining any substantial momentum from the wave. Similarly, the elec-trons with a too low/high initial momentum (dashed blue lines) are moving on open orbits and are nottrapped by the wake. The latter follow the condition H > H sep .13n contrast, an electron with a sufficient initial momentum u ⊥ ( ξ = + ∞ ) > u z,min and H
GeV -barrier, cf. Fig. 3(b).
As indicated above, the details of the injection process largely define the overall beam quality, i.e.,the energy spread, charge, and divergence. In order to achieve minimal emittance, the electrons shouldbe injected with sufficient initial momentum into a spatial volume that is matched to the beam chargeand background density. This is difficult to achieve with an external (RF) electron source, becauseit requires matching a beam from the RF injector with structure scales of centimeters to the plasmaaccelerator with 10’s of µ m . External injection has been demonstrated into low-density plasmas withlong wavelength [2, 58], but with currect high-intensity Ti:Sa lasers the comparably high densities so farhave only allowed successful injection very recently [59].From an experimentalist’s point of view, self-injection appears to be the most straighforward ap-proach. It is based on wave-breaking, sometimes augmented by the expansion of the cavity size duringself-focusing of the drive laser. This slows down the phase velocity of the first wakefield peak and allowselectrons to be trapped more easily over a considerable distance, leading to electron beams with a largemomentum spread. Therefore, a localized and controlled injection process is key for achieving narrow-band electron bunches. Many schemes have been studied and still are an active area of research. Threeof them (in addition to self-injection) will be described in the following sections:– Density transition injection . A localized electron density down-ramp increases λ p and reducesthe wake’s phase velocity, causing a defined injection.– Colliding pulse injection . A locally confined beat wave of two colliding laser pulses gives back-ground electrons a momentum kick and injects them into the proper phase.–
Ionization injection . Electrons from core levels of heavy atoms are ionized and freed close tothe peak of the laser pulse; thus, they are ‘born’ in a trapped orbit.
In a purely 1-D model, the minimum initial energy for electrons to be trapped is given by W trap = m e c (cid:16)(cid:113) u z,sep (+ ∞ ) − (cid:17) , (43)where u z,sep (+ ∞ ) describes the separatrix orbit in front of the laser pulse ( a = φ = (cid:126)u ⊥ = 0 ) u z,sep (+ ∞ ) = β p γ p H sep − γ p (cid:113) γ p H sep − . W trap (Eq. (43)) on the normalized maximumelectric field E z,max /E p, with E z,max given by Eq. (39). On the top x -axis, the W trap -dependence withrespect to the wave velocity γ p given by Eq. (22) is displayed for various φ min . The threshold decreasesfor larger wakefield amplitudes and slower wake phase velocities, i.e. for high laser intensities and highelectron densities. With increasing a the fluid orbits get closer to the separatrix, and the necessarymomentum for trapping can be significantly reduced. When the fluid and the separatrix orbit overlap,i.e., the longitudinal velocity of the electrons overtakes the plasma wave velocity, the wavebreakinglimit of the plasma is reached, and the background electrons are directly injected. For common electrondensities of × cm − this occurs at rather high values of a ≈ . Within a 1-D description, thiscold wavebreaking limit can be reduced by thermal effects [60], but still remains higher than observedin most experiments.. The situation changes quite dramatically in a 3D-dimensional geometry. At moderate laser intensities( a ∼ − ), when the laser forms a wake without reaching the blow-out regime, (cf. Section 4.3).Background electrons close to the axis can now tunnel through the decelerating field and be injectedwithout a substantial transverse motion [61]. Upon its propagation through the medium, the laserself-focuses and compresses while a increases. This lengthens the first wakefield bucket and reducesthe phase velocity [62, 63], and electrons can catch up with the plasma wave more easily and are injectedat much lower a . Since this happens at an early stage of the acceleration with a small transversemomentum, the generated bunches exhibit good shot-to-shot stability, high energies and few-mraddivergence. The longitudinal injection mechanism is dominat at low plasma density, where self-focusingis weak and the pulse diffracts after the first injection event without causing further (transverse)self-injection. At high laser intensities and electron densities, transverse wavebreaking can occur [3, 64]. The strongponderomotive force kicks electrons radially away from the axis, forming a bubble-shaped electron cav-ity whose focusing field pulls them back onto the axis, forming a density spike. Here, some electrons canbe scattered forward into an accelerating phase [63]. The additional Coulomb repulsion of these injectedelectrons at the rear of the bubble lengthens the cavity and causes further injection. In contrast to lon-gitudinal self-injection, these injected electrons enter the wakefield with a large transverse momentum,resulting in high divergence. As the laser spot size (and peak a ) usually oscillate around the matchedspot size of the plasma [61], transverse injection can happen several times during the laser pulse evolu-tion at positions with the highest a before the bubble eventually becomes beam loaded, cf. Section 4.4.Due to these oscillations injection happens at random locations, resulting in low reproducibility.The existence of a density threshold between longitudinal and transverse self-injection allowsswitching on and off the latter [61]. Below the threshold, the electrons are trapped longitudinally, withlow charge and a stable, sometimes narrowband spectrum. Above, transverse injection dominates by far,generating less stable electron beams with high charge. According to [65], self-injection occurs if α L W L > π(cid:15) m e c e (cid:20) ln (cid:18) n e,c n e (cid:19) − (cid:21) n e,c n e τ ( l ) , (44)where α L is the fraction of the laser energy W L contained within the FWHM of the focal spot and n e isthe plasma density. τ ( l ) is the self-compressed laser pulse duration as given in Eq. (26).The main advantage of self-injection is its experimental simplicity. However, it often lacks stabil-15 E z,max /E E t r ap ( k e V ) slower.Wake.Phase.Velocity g ~ n.. p e,0..-1/2.. larger.Wake.Field.Amplitudes (a) Injection threshold plotted on a semi-logarithmicscale vs maximum electric field for different electron den-sities n e, = (15 , , × cm − , corresponding to γ p = 11 , , (straight lines), as well as versus γ p fordifferent minimum normalized potentials φ min (dash-dottedlines). The injection threshold is lowered for larger wake-fields and slower phase velocities. The dashed lines ( γ p =19 ) give the trapping threshold for the two cases in Fig.3, E a (cid:39) . trap ≈ . and E a (cid:39) . trap ≈
20 keV .
01 d(mm) den s i t y p r o f il e -1.0 -0.5 0 0.5 1.0 1.5 2.0 − − − − −
10 -5 0 ξ (µm) no r m . un i t s u z − no r m . un i t s (b) Shockfront injection mechanism . The laser ( a =1 . ) interacts with a density profile as shown on thetop. In the high density region (cid:13) with × cm − itdrives a plasma wave with λ p, and creates the phase or-bits shown dashed in the bottom plot. After the transi-tion the plasma wavelength lengthens to λ p, (with lower n e = 3 . × cm − ) in (cid:13) . This sudden expansioncauses the untrapped electrons in the wave peaks to rephaseinside the new separatrix (solid purple line). Fig. 4:
Injection threshold for the self-injection ( left ) and illustration of shockfront injection ( right ). ity, as it relies on nonlinear self-focusing and self-compression processes. The generated electron beamsexhibit fluctuating spectra with rather large momentum spread ( − ). In contrast, reproducible low-emitance, low-energy spread beams require externally controlled injec-tion schemes, where in most cases the main LWFA is operated below the threshold for wavebreaking.The injection process itself is triggered locally at a defined position along the propagation direction bytailoring the longitudinal density profile inside the plasma. In negative density transitions ( ∂n e /∂z < )the plasma wave elongates according to λ p ∝ n − / e and the back of the plasma wave slows downmomentarily, thus lowering the injection threshold and causing self-trapping. After the down-gradient,the laser drives a wake in low density which accelerates the injected electrons, but avoids self-trapping.The control over the density profile therefore allows to separately control injection and acceleration.For a long density transition , L tr > λ p , the wave is gradually slowed down. In a 1-D treatment,the phase of the wake during the density transition can be approximated to ϕ = k p ( z )( z − v g t ) inthe leading order. By definition, the local effective oscillation frequency is ω p,e = − ∂ϕ/∂t = v g k p ( z ) and the effective local wave vector is k p,e = ∂ϕ/∂z = k p ( z ) + ( z − v g t ) ∂k p /∂z . Thus, the local phasevelocity in the limit v g → c can be expressed as v p = ω p,e k p,e = v g (cid:20) k p ∂k p ∂z ( z − v g t ) (cid:21) − ≈ c (cid:18) ξ n e dn e dξ (cid:19) − ≈ c (cid:18) − ξ n e dn e dξ (cid:19) . (45)16 continuous decrease in plasma density results in a reduction of the wave velocity, and this effect ismultiplied in subsequent buckets. At some point, the fluid velocity of plasma electrons is reached, andthe wave breaks. At sufficiently large distances behind the driver injection of electrons will alwaysoccur, as long as the wave is not disturbed by other mechanisms. This injection scheme is often re-ferred to as down-ramp injection [66]. For example, in a density ramp with L tr = n e dn e /dz = 4 λ p ,the electrons with an initial velocity v e = c/ ( γ e ≈ . ) will be injected for v e = v p at the position ξ i = (cid:16) cv e − (cid:17) n e dn e /dz = 16 λ p behind the driver. Experimentally this has been demonstrated in the gra-dient at the rear of a gas jet [18, 22] or by using a transverse laser to create a down ramp [21]. However,long electron density ramps have two major drawbacks: a) The prolonged injection process will lead toa large energy spread, and b) the permanent increase in plasma wavelength causes quick dephasing.In sharp density transitions on a scale length L tr < λ p , with a rapid decline from n e, to n e, ,the change of the plasma wavelength ∆ λ p λ p = λ p, − λ p, λ p, and phase velocity ∆ v p v p = v g, − v g, v g, can be derivedas [67] ∆ λ p λ p = 1 − (cid:114) n e, n e, ≈ − α T − , ∆ v p v p ≈ − (cid:18) − n e, n e,c (cid:19) (cid:18) n e, n e,c (cid:19) ≈ α T − n e, n e,c , with the density ratio α T = n e, /n e, and α T (cid:38) . The priciple is shown in Fig. 4(b). As λ p increasesalmost instantly, the velocity of the plasma wave only increases marginally. Consequently, the velocityof the first wakefield peak suddenly stops, before starting to move again with the new phase velocity.A significant fraction of the electrons, constituting the first wakefield peak before the density transition (cid:13) (drawn in gray dashed lines) are now promoted to an accelerating phase after the density transition( < µ m ) (cid:13) , as indicated by the yellow area. The injected electrons are all located at a similar phase ofthe wake with similar initial energy, exposing them to the same accelerating field. Therefore, they willgain similar energy resulting in a quasi mono-energetic beam.This mechanism has first been proposed numerically [68, 69], and been experimentally verifiedto produce stable, quasi-monoenergetic electron bunches [19, 20]. Here, the gas density profile froma supersonic gas nozzle was modified by a razor blade creating a narrow shock front with the desiredsharp drop in the density profile. Other groups have also reported on the production of monoenergeticbeams by using similar approaches, e.g. using wires for the creation of the shock [70, 71].The simple implementation of this scheme also favors potential applications, such as a quasi-monochromatic all-optical Thomson x-ray source [72]. Another possibility to realize a controlled, localized injection relies on two colliding laser pulses.The first (drive) pulse creates a plasma wave below the self-injection threshold, while the second weaker(injection) pulse counterpropagates and beats with the driver at a position depending on the relative de-lay. This beat wave causes a localized and instantaneous heating of background electrons, some of whichare then injected into the driver’s wake [52, 73]. Assuming two counterpropagating, circularly polarizedlaser beams a = a , √ k L z − ω L t ) (cid:126)e x +sin( k L z − ω L t ) (cid:126)e y ) , a = a , √ k L z + ω L t ) (cid:126)e x − sin( k L z + ω L t ) (cid:126)e y ) , the Hamiltonian (Eq. (41)) for the beatwave can be expressed as H beat = (cid:113) u ⊥ + u z = (cid:112) a + a ) + u z , − − − u z ξ (µ m)1050-5-10 ξ (µ m) − − − u z (a) Beat wave injection of a pump pulse with a , = 1 . and an injection pulse with a , = 0 . ( top ) and a , = 0 . ( bottom ) for circular polarized pulses with a duration of
35 fs . In the beat wave (bold green line) the fluid/untrappedelectrons (dotted black/ dashed blue lines) can gain enoughenergy ( W a , =0 . beat ≈
300 keV and W a , =0 . beat ≈
170 keV )to cross the separatrix (purple line) and to be injected ontrapped orbits (red line). The yellow area corresponds tothe injection volume which determines the injected chargeas well as the final energy spread. I n t. ( W / c m − ) no r m .- un i t s u z + − m i n ( u f l u i d ) N N N u z + − m i n ( u f l u i d ) ξ (µm) 20100-10-20-30 ξ (µm) 20100-10-20-30 a(a = 2.25)a (a = 2.65)(a = 2.25)(a = 2.65) (b) Ionization injection . For high- Z gases inner shell elec-trons are ionized at the peak of the laser. ( center: the greendashed lines mark the barrier-suppression ionization (BSI)values for nitrogen). The ionized electrons are born with u z ( ξ ion ) ≈ (dashed black line) and can be trapped by fol-lowing the green orbits in the phase space. By adjusting the in-tensity of the laser pulse the injection region (marked in yel-low)the final energy spread ∆ W can be reduced, as shown onthe top for a = 2 . with ∆ W/W ≈ and on the bot-tom for a = 2 . , ∆ W/W ≈ . Fig. 5: Phase space orbits demonstrating the principle for optical injection ( left ) and ionization injection ( right )in 1D at a plasma density of n e, = 5 × cm − . where ( a + a ) = a , + a , + a , a , cos(2 k L z ) . Note that the Hamiltonian is independent of time.For the initial condition ( u z = 0 for z = 0 ), the beat wave separatrix is u beat,sep = ± (cid:113) a , a , (1 − cos(2 k L z )) , u beat,max = (cid:112) a , a , , u beat,min = − (cid:112) a , a , with the maximum and minimum values u beat,max and u beat,min , respectively. Figure 5(a) displaysthe trapped and untrapped electron orbits in the phase space generated by a pump pulse with a , = 1 . and a , = 0 . , . , respectively. The ponderomotive force of the beat wave F beat ∝ k L a , a , promotes background electrons from the fluid orbits into trapped orbits. The maximum momentum gainis represented by the beat wave separatrix (green solid line). The yellow area marks the accessible phasespace in the wakefield for electrons heated by the beat wave [73] u beat,max ( ξ ) (cid:38) u sep ( ξ ) , u beat,min ( ξ ) (cid:46) u fluid ( ξ ) . The energy gain of the electrons in the beat wave is given by W beat = m e c [(1 + u beat ) / − ≈
300 keV for a , = 0 . and ≈
170 keV for a , = 0 . , respectively. A lower colliding pulse in-tensity a , reduces the injection phase space resulting in a narrower energy spread, but also reducesthe injected charge [16, 17]. Most colliding experiments are performed with linear polarization since thisleads to more efficient heating. A full description of optical injection, including the consequences of18he heating on the dynamics of the plasma wave, the laser pulse evolution and its influence on the wakeformation is beyond the reach of analytical treatment [74], which commonly overestimate, e.g., the in-jected charge, because of their failure to describe the wakefield inhibition during beating [75]. However,optical injection provides good control over the beam parameters: since the beat wave only exists dur-ing the intersection of both laser pulses, the injection is very localized, resulting in quasi-monoenergeticbeams. A variable delay between both pulses allows to select the injection position, acceleration distanceand final energy, while the amplitude and polarization of the colliding beam controls the the number oftrapped electrons and their energy spread (determined by the volume of the phase space and beam-loading) [16, 76]. The disadvantage of optical injection lies in the difficult setup required for splitting,synchronizing and colliding both laser pulses. A direct comparison of optical and shockfront injectioncan be found in [77]. While downramp and optical injection transport electrons across the separatrix boundary for injection(or vice versa), another way of injection relies on "creating" electrons in already trapped orbits. Sincethe Barrier-Suppression Ionization (BSI) threshold of inner-shell electrons in medium- Z atoms (e.g.,Oxygen, Nitrogen) is comparable to the peak laser field, a small amount of such dopant gas in a Heliumor Hydrogen target gas causes ionization events close to the laser peak. An example for N is shownin Fig. 5(b). Created on axis in the wakefield, these electrons have different momentum compared tothe fluid electrons: they are ionized at the position ξ ion , at rest ( u z ( ξ ion ) = 0 ) (while fluid electrons atthis point travel backwards) and close to the laser peak ( a ( ξ ion ) (cid:39) ). The Hamiltonian yields H ion = 1 − φ ( ξ ion ) . If the acceleration field allows the new-born electrons to reach the phase velocity of the wake duringbefore they are overtaken, they are trapped. The trapping condition depends on the BSI threshold a ( ξ ion ) > a OT BI for the individual ionization state and by the separatrix, H ion < H sep . In Figure 5(b)the orbits of the trapped electrons are marked green. The trapping region (marked in yellow) can be ad-justed by the intensity of the laser pulse. It defines the final energy spread ∆ W , as shown for a = 2 . on the top and a = 2 . on the bottom of the figure. Under the further assumption that the electronsare released only at the maximum of the laser pulse with zero momentum, a ( ξ ion ) ≈ , the trappingcondition ( H ion < H sep ) can be approximated to the inequality [78] − /γ p ≤ φ ( ξ ion ) − φ min ≤ φ max − φ min . This sets a minimum for the required laser intensity − γ p ≤ a + a a + 1 . For a squared pulse (Eqs. (36) and (39)) the inequality is satisfied at typical gas densities ( γ p > ) for a > . . As the main consequence, ionization induced trapping requires relativistic laser intensities.Experimental evidence of this scheme is given by [79], where an efficient injection of electrons in He:Argas mixtures at a vector potential well below the self-trapping regime has been observed. Although itssimple implementation and high-charge output, ionization injection yields broad spectra, because: a)Injection is continuous, as long as the trapping condition is fulfilled, and b) Tunnel ionization smearsout the ionization threshold a OT BI . For a linearly polarized laser, the tunnel ionization rate can beapproximated [80, 81] as Γ( | E L | ) = 4 (cid:18) π (cid:19) / Ω (cid:18) W ion W ion,H (cid:19) / (cid:18) E H | E L | (cid:19) / exp (cid:32) − (cid:18) W ion W ion,H (cid:19) / E H | E L | (cid:33) , E L is the laser field and E H = 5 . − the ionization field of hydrogen, Ω = α f c/r B =4 × s − is the characteristic atomic frequency, W ion and W ion,H = 13 . the ionization energy ofthe specific ion and of hydrogen, respectively. The fraction of species which is ionized during a time ∆ t is given by Γ∆ t . For example, the ionization probability for N → N in a the presence of a Gaussianpulse a = a exp( − t /τ ) with t F W HM = (cid:112) τ = 28 fs is ∼ for a = 1 . and increases to for a ∼ . , i.e., the threshold for ionization lies in the range between a = 1 . − . . Accordingly,small fluctuations in laser self-compression and self-focusing have a direct impact on the injected chargeand decrease the overall stability. In principle, by reducing the laser pulse energy, the energy spreadof an electron bunch can be reduced, but so will its charge and energy. The best experimental resultsusing a additive of nitrogen have demonstrated an energy spread of [82]. Localizing the dopantconcentration can further reduce the energy spread, as demonstrated in [23]. The analytical 1D studies presented so far offers a basic understanding of the LWFA process. The lackof analytical 3D models necessitates numerical simulations to capture, e.g., the transverse dynamics orthe laser pulse evolution. Commonly, the particle-in-cell method (PIC) is used to study a more completepicture of the laser, wakefield and electron bunch evolution.Based on such numerical 3D PIC studies, [9] have found that under optimized conditions anda strong driver, a very efficient acceleration is possible. Their scheme relies on the relativistic pon-deromotive force to radially expel all plasma electrons from the laser path. This forms an electroncavity behind the laser pulse, where only the ions stay behind. After a distance of λ p,rel , they pull backthe electrons, whose trajectories cross on axis, forming a density spike. The shape of the electron cavityis approximately spherical, hence the name “bubble” regime, see Fig. 6. In the bubble, the electrondensity is zero, and this regime is also referred to as “complete blow-out”. Experimentally, this regimehas been studied extensively since the first observation by [6].[83], [84] have developed phenomenological laws for the bubble regime via extensive studiesof PIC codes underpinned by simplified analytical studies. They found that a spherical structure witha radius given by r b ≈ √ a /k p is formed. The normalized potential and the electric fields insidethe cavity moving at relativistic speed are φ = k p (cid:0) r − r b (cid:1) , E z E p, = k p ξ, E ⊥ E p, = k p r ⊥ , where r = ξ + r ⊥ = 0 is the radius from cavity center, r ⊥ = x + y is the transverse and ξ the longitudinal position. The radial fields E ⊥ are linear in r ⊥ , and therefore emittance-preserving.The longitudinal fields are almost linear with a gradient | ∂E/∂ξ | = E z,max /r b , with E z,max = r b k p E p, , E z,max [ GV /m ] ≈ (cid:113) n e, [10 cm − ] √ a . (46)The radial and longitudinal forces acting on the trapped electrons are given by (cid:126)F z = − m e ω p ξ (cid:126)e z , (cid:126)F r = − m e ω p r ⊥ (cid:126)e r . (47)In the leading half of the bubble ( < ξ < r b / ), the electron bunch is decelerating, while inthe back of the bubble it accelerates, with a gradient independent of its radial position. The transversefocusing region for electrons extends over the whole bubble, defocusing only occurs around the on-axisdensity peak. The focusing fields are of similar magnitude as the longitudinal fields (
100 GeV m − ). Note that in these 3D wakes the wakefields are electromagnetic in character, not only electrostatic. ig. 6:
3D OSIRIS
PIC simulation of LWFA acceleration
The laser ( λ L = 800 nm , t F W HM = 28 fs ), propagat-ing towards the right, is focused to a spot size of d F W HM = 9 . µ m with an energy of W L = 100 mJ on the leftand W L = 1 .
55 J on the right. The electron density ( n e, = 5 × cm − ) is shown in blue in the ( y =0) -plane. Left:
In the modest relativistic regime ( a ≈ ), the laser pulse (at x ≈ c/ω p ) excites a weakly nonlinearwake (electric field on axis shown in green). The bubble is not completely void of electrons and the wake’s wave-fronts are horse shoe-shaped. No self-injection occurs during this simulation before the laser is depleted. Right:
In the highly nonlinear regime ( a ≈ . ), the simulation parameters correspond to the matched condition d F W HM ≈ √ a /k p = 9 . µ m . The bubble with radius r b cω p = 2 √ a (magenta circle) is fully developed,the electron density drops to zero and the created fields exceed the wavebreaking limit. Near the rear of the bubblethe onset of transverse self-injection is visible. The energy of the injected electrons is color-coded in black-and-red. Injection into the bubble occurs via transverse wave-breaking (cf. Section 4.2.1) at the densityspike, at the location of the highest accelerating fields. Due to the large initial deflection by the pon-derometive force, electrons forming the spike exhibit a large transverse momentum, resulting in a trans-verse emittance. The optimum condition for bubble formation and injection can be found by matchingthe laser intensity, spot size w b , pulse length ( cτ ≤ w b ) and plasma wavelength as shown in Fig. 6 [50] w b (cid:39) r b = 2 √ a /k p . (48)For a > and a focal spot size of w b = d F W HM = 2 √ a /k p , the ion cavity takes the shape ofa perfect sphere. In terms of the critical power P crit (Eq. (25)) this condition yields P (cid:39) ( a / P crit .Exemplary, at n e, = 5 × cm − , a peak power of
42 TW , focused to a > is required.Between < a < , the blow-out cavity deviates slightly from a spherical shape. However,self-focusing and self-compression will match even initially unmatched pulses [50] to drive a bubble.Self-guiding losses are minimized when the initial spot size matches the blow-out radius, as exper-imentally confirmed by [45]. In [85] it was found that relativistic self-focusing has a stabilizing effect,allowing the wakefield to evolve to the correct shape even from fluctuating initial parameters. The bestperformance is achieved for long focal-lengths ( w b (cid:38) λ p ), and a Rayleigh length larger than the densitygradient at the plasma entrace. Then the majority of the pulse energy coalesces into a single optical fil-ament and can be self-guided over distances comparable to the dephasing length. Smaller spots usuallyresult in beam breakup and poor reproducibility. Thus, increasing in the pulse intensity through self-evolution is more effective than a tight focusing geometry. The choice of the spot size (small enough toavoid depletion, large enough to avoid filamentation) is crucial for the production of high-quality beams.21 .4 Limitations on Energy Gain The main limitations for LWFA are the "detrimental Ds": Diffraction, Dephasing, and Depletion.
In any focused beam, diffraction will reduce the laser intensity after a certain distance, but self-focusingmay balance this over many Rayleigh lengths. The general refractive index (cf. Eq. (24)) is given by η = ckω L = (cid:115) − ω p γ ⊥ ω ≈ − n e (cid:0) a (cid:1) n c (49)For the 3D bubble regime, the self-guiding condition can be reformulated to [50] a = (cid:18) n c n e (cid:19) / . (50)If this condition is fulfilled, the laser propagates with a spot size close to w = 2 √ a /k p r . In a realisticscenario, however, a will change by self-compression and pump depletion, while an initially unmatchedlaser will also oscillate in size. Despite the complex dynamics that is best treated by numerical simula-tions, one thing is certain: Once the pulse depletes, it will quickly diffract. Trapped electrons can reach velocities closer to c than the laser group velocity, so they catch up withthe laser and "dephase", i.e. enter the decelerating phase of the wake (cf. point C (cid:13) in Fig. 3). Conse-quently, the maximum energy which can be extracted from the wake depends on the (relativistic) plasmawavelength. The distance which an electron can travel before it crosses the zero field, E z = 0 , in the labframe is called dephasing length L d . In the linear regime, it corresponds to the distance in which a rel-ativistic electron ( v e ≈ c ) phase slips by λ p / with respect to the laser field ( v g = ηc ). Only in thisphase space region the fields are accelerating and simultaneously focusing, cf. green area in Fig. 2. Withthe refractive index η (Eq. (49)) the dephasing length can be derived to ( c − v g ) L d c = λ p ⇒ L d = λ p ω L ω p In the nonlinear regime a (cid:29) , the increase in the plasma wavelength has to be included. For a linearlypolarized, square pulse (Eq.(37)) the dephasing length at an arbitrary intensity is given accordingly by L d ≈ γ p λ p × (cid:40) for a (cid:28) π a for a (cid:29) (51) L d scales inversely with the electron density ∝ n − / e , and can therefore be mitigated by operation atlower densities, albeit at longer acceleration length. For typical densities of n e, ≈ × cm − and a = 2 , the dephasing length is usually L d ∼ − . Using a phenomenological approach, [50]derived for a nonlinear regime a dephasing length of (cf. Table 1) L Dd = 43 √ a k p ω L ω p . (52)In positive density gradients, dephasing can be mitigated by keeping the electrons at a fixed phase, ifthe phase slippage of the electrons is compensated by a matched reduction of λ p [86, 87].22 .4.3 Pump Depletion During its propagation, the laser’s energy is transferred to the plasma wake until it is depleted and accel-eration ends. The characteristic length for pump depletion L pd can be estimated by comparing the energydensity in the wake, u W = (cid:15) E z , contained in the volume V = πw L pd , and the laser energy density, u L = (cid:15) E , contained in the volume V = πw L cτ L . With a = eE /cm e ω L (Eq. (15)) and the relationfor linear wakes E z /E p, ≈ a (Eq. (32)), the pump depletion is L pd = ω L ω p cτ L a ≈ λ p λ L a . In the relativistic case ( a (cid:29) ) for 1D square pulses with cτ L ≈ λ p,rel / , the scaling of E z (36) yields: L pd ≈ γ p λ p × (cid:40) a for a (cid:28) a π for a (cid:29) (53)In the 3D nonlinear regime, [88] estimates the pump depletion from the laser’s head erosion as it drivesthe wake. This picture is confirmed by simulations in [50], which gives the etching velocity as v etch (cid:39) cω p /ω L . This means that the pulse is fully depleted after a length L Dpd = cv etch ct F W HM (cid:39) ω L ω p ct F W HM . (54)Figure 7(a) displays the dephasing and depletion lengths as blue dashed- dotted and blue dashedlines, respectively. Although both would be equal at n e = 3 × W/cm and a ≈ . , indicating op-timum energy transfer, the plot indicates that the laser would not be self-focused and also not matched tothe plasma (white and green lines, respectively). A shifted operation point (e.g., n e = 2 . × W/cm and a ≈ . ) would fulfil these conditions, albeit with depletion now happening before dephasing.Therefore, this plot can help in selecting the best working point for a given set of laser parameters. So far, only single test electrons in the plasma fields have been considered. A highly charged bunch,however, will modify ("beam-load") the electric field of the plasma wave, and therefore the accelera-tion process. Typically, a beam loaded wakefield will manifest itself in a decrease in energy gain anda modified energy spread. In a nonlinear 1D approach, Poisson’s equation has to be amended by a termdescribing the bunch density distribution n b ( ξ ) ∂ φ∂ξ = k p (cid:18) n e n e, − n b n e, (cid:19) , and the solution for the PDE, analogous to the Eq. (35) is given by [56] ∂ φ∂ξ = k p γ p β p (cid:32) − γ γ p (1 + φ ) (cid:33) − / − + k p n b ( ξ ) n e, . (55)The bunch’s self-fields will superimpose and modify the accelerating field. Figure 8 shows thebeam loading effect upon the wakefield. A Gaussian electron bunch ( t b,F W HM = 10 fs ), is placed atthe ξ = − . λ p behind the drive laser (cid:104) a (cid:105) = 1 . . In the top and bottom panel, a purely laser (bunch)driven wake is shown, respectively, while the second and third panel display two cases with n b /n e, = 0 . lectron9Density9(10 cm − ) N o r m a li z ed V e c t o r P o t en t i a l a Leng t h ( mm ) dephasing9lengthdepletion9lenght (a) Optimized beam charge color coded on a logarithmicscale as given by Eq. (56). The charge is roughly constantfor the matched beam parameters around Q opt ≈
350 pC .As example, for a = 4 . the dephasing length (blue dotteddashed line) and the depletion length (blue dotted line) forthe 3D nonlinear regime are displayed (right axis). Electron9Density9W10 cm − H N o r m a li z ed V e c t o r P o t en t i a l a Wd FWHM Ha Wn e H (b) Maximum electron energy gain.
The highest elec-tron energies are given for the lowest electron densities, aslong as the self-guiding condition is satisfied, e.g.: W Lu =510 MeV for n e, = 5 × cm − and a = 4 . .The corresponding maximum spot size for the ATLAS pa-rameters d FWHM = f ( a , W L , t FWHM ) is given by thedashed blue line and top axis. Fig. 7: Electron output parameters in the matched condition k p r b = 2 √ a . The white dashed line representsthe laser self-guiding condition (Eq. (50)), i.e., the area below the white line can be only accessed via externalguiding. The solid green line shows the matched condition ( w k p = 2 √ a ) for a laser with W L =1 . , t F W HM =28 fs assuming a Gaussian pulse with w b = d F W HM . and n b /n e, = 0 . . In both cases, the electric field and wave potential are perturbed by the charge toa varying degree, manifesting itself in a modified peak field and an elongated first wakefield bucket.However, this superposition can also have a positive side-effect. For the case n b /n e, = 0 . shownin the second panel, the accelerating field (blue line), normally decreasing towards smaller values of ξ ,stays constant over the duration of the electron bunch. This means that all electron in the bunch areaccelerated by the same field, irrespective of their phase in the wake. This reduces chirp and energyspread of the accelerated bunch [89]. In addition, via the reduction of the wakefield amplitude, beamloading can terminate self-injection once a high-charge bunch occupies the first bucket [90, 91].The maximum injected charge can be estimated by assuming a sphere with a radius r b = 2 √ a /k p ,carrying the total ionic charge Q tot = r b n e, e ∝ a / n − / e, . The optimum injected charge Q opt hasbeen derived for a matched laser driver by [92] to Q opt = π(cid:15) m e c e (cid:18) k p r b (cid:19) E z E z = r b k p E p, / ========= ⇒ Q opt = πc e (cid:0) m e (cid:15) a /n e, (cid:1) / , (56)which results in Q opt ≈ Q tot / . For typical parameters, a ∼ − , n e = 5 × cm − , the injectedcharge is Q opt ∼
350 pC . In Figure 7(a), Q opt is plotted with respect to a and n e, . The charge canbe increased for higher laser pulse intensities and larger bubble structures, i.e., lower electron densities.The white dashed line represents the self-guiding condition (Eq. (50)), which has to be fulfilled in order24 no r m . un i t s − no r m . un i t s − no r m . un i t s − − − − − − no r m . un i t s ξ / λ p Fig. 8: Effect of beam loading.
Normalized 1D wakefield quantities , calculated with Eq. (55). top : laser driver(linearly polarized, peak a = 2 . , t F W HM = 20 fs ), bottom : electron driver ( n b /n e, = 0 . , τ b = 10 fs ). twocentral plots : mixed driver, laser at ξ = 0 , electron bunch at ξ = − . λ p . Peak bunch density n b /n e, = 0 . and n b /n e, = 0 . , for the second and third panel, respectively. The bunch charge modifies both the potential andthe longitudinal electric field and lengthens the cavity. Electron density for all cases: n e, = 5 × cm − . to sustain the laser intensity over the desired length. The total energy gain ∆ W of an electron determined by the wake’s longitudinal electric field E z ( z ) andthe distance L acc ∆ W = − e (cid:90) L acc E z ( z ) d z. Neglecting diffraction, dephasing in the linear regime and pump depletion in the nonlinear regime limit L acc . In 1D, ∆ W is given by L d or L pd (Eqs. (51) and (53)) and the maximum field (Eqs. (32) and (36)): ∆ W ≈ e E z L acc = (cid:40) e ( E a ) L d = m e c a γ p π/ for a (cid:28) e ( E a / L pd = m e c a γ p for a (cid:29) . Both regimes scale equally with a and n e , and dephasing can be overcome by longitudinal densitytapering. However, operation the nonlinear regime has the advantage of higher accelerating gradients,therefore, shortening the acceleration length and often avoiding the need for guiding. In the nonlinear 3D25 able 1: Scaling rules for LWFA in the linear and nonlinear 1D and 3D regime as given by [29, 50, 93] a a a w w w L d L d L d L pd L pd L pd λ p λ p λ p E z /E p, E z /E p, E z /E p, ∆ W/m e c ∆ W/m e c ∆ W/m e c Linear < πk p πk p ω L ω p cτ L a ω L ω p πk p a πa ω L ω p
1D NL > πk p a k p ω L ω p a k p ω L ω p a k p a / a ω L ω p NL Lu > √ a k p √ a k p ω L ω p cτ L ω L ω p √ a πk p √ a / a ω L ω p NL GP > (cid:113) n c n p √ a k p a cτ L ω L ω p √ a a ω p τ L ω L ω p regime, the energy gain for a strong driver is limited by dephasing and can be approximated via Eq. (46) W Luel = e (cid:104) E z (cid:105) L Dd = 23 a m e c n c n e, . (57)Table 1 summarizes some LWFA-scaling rules from the literature [9, 50]. Both groups found optimumacceleration occurs when the laser pulse duration τ and spot size w b match the plasma wavelength, w b ∼ cτ ∼ √ a /k p , and the bubble resembles a spherical cavity with r b = w , cf. Section 4.3.[9] is based on a similarity theory with the parameter S = n e a n c , valid for S (cid:28) and a > .In this extreme case, the acceleration is limited by pump depletion, and the self-injected electrons possesa quasi-monoenergetic spectrum. Its predictions for important quantities is given in Table 1 labelledunder NL GP. [50] uses a more phenomenological approach, with the evolution of the driver includingetching effects (cf. Section 4.4.4), dephasing and beam loading, and is summarized in Table 1 under"NL Lu". Its validity extends to lower laser intensities ( a > ) and is not restricted to self-injection .It shows good agreement with experimental results [94] and can also be applied for external injectionschemes. However, such scaling laws should be used with care, since they often overestimate experi-mental parameters.In Figure 7(b) the maximum energy gain W Luel (Eq. (57)) is plotted with respect to a and n e, .Again, the area below the white dotted line cannot be accessed without external guiding schemes.The solid green line represents the matched beam condition ( w b = 2 √ a ) for the parameters given inthe caption, while the dashed blue line (top axis) gives the corresponding spot size d F W HM ( a ) . Aslong as the self-guiding takes place, the maximum gain is achieved for low electron densities by avoidingdephasing effects. For example, for the same laser parameters ( . ,
28 fs ) the optimized case ( a = 4 . and n e, = 5 × cm − ) yields an energy gain of W Luel = 510 MeV , while W Luel = 380 MeV at a = 6 . and n e, = 10 × cm − . A comparison reveals that both scalings predit similar energy around several
100 MeV ’s to for n e, /n crit =100 − and P = (10 − T W and cτ L ∼ λ p . The accelerated charge is predicted to reach the nC regime - keeping inmind that [9] assumes a severely loaded wakefield, while [50] assumes an unperturbed wakefield Experimental scalings
A remarkably simple pattern can be found for the experimental scaling of electron peak energy andcharge with laser energy and power. In Figure 9, all combinations between of these quantities are plot-ted. Simple linear relations exist between the bunch energy or charge and the laser energy or power.This allows a straightforward order-of-magnitude perfomance prediction of LWFAs for a given laser.The dataset encompasses many different laser parameters, injection mechanisms, targets setups with orwithout guiding, but the linear behaviour seems to be maintained up to the PW level. The dataset focuseson quasi-monoenergetic spectra, and energies/charges are referring to the spectral peak. M e V / J M e V / T W C / T W C / J Fig. 9: Experimental results for energy and charge : Experimentally, the best results for electron peak energyand charge closely follow extremely simple scaling laws with respect to the laser power and energy. Note thatthese "laws" are no fit to the data, just lines to guide the eye. Data is based on 50+ publications on LWFA duringthe last 15-20 years [95]
In summary, the following scalings seem to hold for the upper end of the point clouds in Fig. 9: O ( E peak ) ≈ MeV / J ≈ MeV / TW O ( Q peak ) ≈ pC / J ≈ pC / TW . Important equations
Table 2:
Important quantities for LWFA expressed in physical and engineering formula
Quantity Definition Engineering Formula
Gaussian Laser Beam Parameters ( a )( a )( a ) Focal Spot w = λ L π fD = (cid:113) d F W HM w e − Ø [ µ m] = f / @ λ L = 0 . µ m Confocal Parameter z R = 2 πw /λ L ∆ z [ µ m] = 2( f / @ λ L = 0 . µ m Peak Power P = 2 (cid:113) ln 2 π W L t FWHM P [TW] = 940 W L [J] t FWHM [fs] P = π d F W HM I P [TW] = 0 . d F W HM [ µ m] I [10
18 Wcm ] Peak Intensity I = (cid:0) π (cid:1) W L t FWHM d FWHM [ µ m] I [10
18 Wcm ] = 83 × W L [J] t FWHM [fs] d FWHM [ µ m] I = π (cid:15) m e c e a λ L I [10
18 Wcm ] = 1 . a λ L [ µ m] Vector Potential a = eπm e c (cid:113) I (cid:15) c λ L a = 0 . (cid:112) I [10 W cm − ] λ L [ µ m] Peak Electric Field E = ea cm e ω L E [10 V / m] = 3 . a λ L [ µ m] Plasma Parameters ( n e ∝ k p )( n e ∝ k p )( n e ∝ k p ) Plasma Wavelength ω p = (cid:113) n e, e m e (cid:15) λ p [ µ m] = . √ n e, [10 cm − ] Wavebreaking Field E p, = m e cω p e E p, [GV m − ] = 96 (cid:112) n e, [10 cm − ] Plasma Gamma Factor γ p = (cid:113) n cr n e γ p = 33 . n e, [10 cm − ] λ L [ µ m] Critical Density n e,c = (cid:15) m e e ω L n e,c [ cm − ] = . × λ L [ µ m] LWFA Parameters in the Bubble Regime ( r b = 2 √ a /k p )( r b = 2 √ a /k p )( r b = 2 √ a /k p ) Dephasing Length L d = π √ a λ L (cid:16) n c n e, (cid:17) / L d [mm] = 7 . √ a (cid:18) λ − / L [ µ m] n e, [10 cm − ] (cid:19) / Electric Field E p = m e cω p e √ a E p [GV m − ] = 96 (cid:112) n e, [10 cm − ] √ a Electron Energy W el = a (cid:16) n c n e, (cid:17) m e c W el [MeV] ≈ a n e, [10 cm − ] λ L [ µ m] Optimum Charge Q opt = πc e (cid:113) m e (cid:15) n e, a Q opt [pC] = 75 (cid:114) a n e, [10 cm − ] Summary
We reviewed the basic physics of LWFA, starting from a description of the laser field of a Gaussianpulse, and the interaction of light with single atoms and electrons for extreme intensities, leading tothe concept of the ponderomotive force as a repulsive net force of intense light on electrons. It leadsto an expulsion of electrons from the laser axis, and the generation of moving plasma wave structuretrailing the pulse as it propagates through a plasma medium. This so-called wakefield consists of plasma-electrons oscillating around their equilibrium position, a charge separation causing strong longitudinallyaccelerating and transversely focusing electric fields. We reviewed the wakefield generation in a 1-Danalytical and 3D-phenomenological methods, and discussed the conditions and methods for trappingelectrons in the accelerating regions of the wave. Finally, we discussed theoretical and experimentalscaling laws for the process. 28 eferences [1] T. Tajima and J. M. Dawson.
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