Ultrashort high energy electron bunches from tunable surface plasma waves driven with laser wavefront rotation
S. Marini, P. S. Kleij, F. Pisani, F. Amiranoff, M. Grech, A. Macchi, M. Raynaud, C. Riconda
UUltrashort high energy electron bunches from tunable surface plasma waves drivenwith laser wavefront rotation
S. Marini,
1, 2
P. S. Kleij,
1, 2, 3
F. Pisani, F. Amiranoff, M. Grech, A. Macchi,
4, 3
M. Raynaud, and C. Riconda ∗ LSI, CEA/DRF/IRAMIS, ´Ecole Polytechnique, Institut Polytechnique de Paris, CNRS, F-91128 Palaiseau, France. LULI, Sorbonne Universit´e, CNRS, ´Ecole Polytechnique,CEA, Institut Polytechnique de Paris, F-75252 Paris, France. Enrico Fermi Department of Physics, University of Pisa, largo Bruno Pontecorvo 3, 56127 Pisa, Italy National Institute of Optics, National Research Council (CNR/INO), 56124 Pisa, Italy.
We propose to use ultra-high intensity laser pulses with wavefront rotation (WFR) to produceshort, ultra-intense surface plasma waves (SPW) on grating targets for electron acceleration. Com-bining a smart grating design with optimal WFR conditions identified through simple analyticalmodeling and particle-in-cell simulation allows to decrease the SPW duration (down to few opticalcycles) and increase its peak amplitude. In the relativistic regime, for Iλ = 3 . × W / cm µ m ,such SPW are found to accelerate high-charge (few 10’s of pC), high-energy (up to 70 MeV) andultra-short (few fs) electron bunches. PACS numbers:
Surface plasmon polaritons, also known as surfaceplasma waves (SPW) in free electron media, are highly-localized electromagnetic field structures with the abil-ity to confine and enhance light in sub-walength re-gions at the interface between two media [1–4]. Theirunique properties have made them ideal candidates forapplications in a broad range of research fields, frombio/chemical sensing [5, 6] to the design of small pho-tonic devices [7, 8].The excitation of SPW by micrometric wavelength( λ = 0 . µ m) femtosecond (fs) laser pulses irradiatingsolid targets has been demonstrated as a strategy to en-hance secondary emission of radiation and particles. Inthe low intensity regime, from few GW/cm to tens ofTW/cm , surface plasmon polaritons have lead to har-monic emission [9–11] and the production of photoelec-tron bunches at energies up to few 100’s eV [12, 13]. Theadvent of table-top, 10’s TW, fs lasers allowed on-targetintensities I λ > ∼ W / cm µ m . In this ultra-high in-tensity (UHI) regime, any target material quickly turnsinto a plasma, and electrons reach relativistic quiver-velocities in the intense laser field. SPW then become ofinterest not only as unexplored nonlinear plasma modes,but also for their capability of accelerating electrons, be-ing waves with a longitudinal electric field componentand slightly subluminal phase speed. Simulations andexperiments have indeed shown that relativistic SPWcan accelerate high-charge, ultra-short electron bunchesalong the target surface [14–25], with energies largely ex-ceeding their quiver-energy and spatio-temporal correla-tion with XUV harmonic emission [26].In a recent paper, Pisani et al. [27] showed throughelectromagnetic simulations in the linear optics (low in- ∗ [email protected] tensity) regime that using wavefront rotation (WFR) onthe driving laser pulse could help generate more intense,shorter SPW. WFR is a technique used on fs lasers toinduce a rotation of the successive laser wavefronts, thusleading to a time-varying incidence angle of the laser im-pinging onto a target. Since SPW on a grating are ex-cited for a well defined value of this angle, using WFR al-lows for the SPW excitation only over a very short time,leading to the generation of near single-cycle SPW; anenhancement of the excited SPW was also found.In this Letter, we demonstrate how these effects can beharnessed in the UHI regime, and WFR can be used todrive tunable, ultrashort, ultra-intense SPW able to gen-erate near single-cycle, highly energetic electron bunches.The optimal WFR conditions are identified using bothanalytical modeling and kinetic (Particle-In-Cell, PIC)simulations. They allow for a significant increase of boththe SPW amplitude and the electron energy by up to 65%with respect to the case without WFR. A careful designof the grating target allows for an additional increase (by25%) of the electron maximum energy. Electron buncheswith several 10’s of MeV energy and 10’s of pC charge arepredicted considering currently available table-top laserparameters.The interaction setup considered throughout this workis depicted in Fig. 1. A UHI laser pulse impingesonto an overdense plasma with density n (cid:29) n c , with n c = (cid:15) m e ω /e the critical density at the laser fre-quency ω = 2 πc/λ , c the vacuum speed of light, (cid:15) the vacuum permittivity, and m e and − e the electronmass and charge. To resonantly excite a SPW at thevacuum-plasma interface, the target surface is partiallymodulated, and the laser incidence angle ( θ ) is chosensuch that sin θ = (cid:112) ( n/n c − / ( n/n c − − λ /d , with d the target periodicity [1]. The resulting SPW is ex-cited at the laser frequency ω = ω , and satisfies thedispersion relation (non-relativistic cold-fluid model [28]) a r X i v : . [ phy s i c s . p l a s m - ph ] J a n d = 2 λ n ≫ n c vacuum plasma ϕ b x f x ′ yz ′ y ′ θ , z x t = 0 t > 0 (cid:1)(cid:2) ° (cid:3)(cid:4) ° (cid:5)(cid:2) ° (cid:4)(cid:6)(cid:2)(cid:1)(cid:1)(cid:6)(cid:2) x f = 50 λ x f = 25 λ θ v s l / c (cid:1)(cid:2) ° (cid:3)(cid:4) ° (cid:5)(cid:2) ° (cid:4)(cid:6)(cid:2)(cid:1)(cid:1)(cid:6)(cid:2) Δ β FIG. 1: Interaction setup: the central laser wavefronts areshown at best focus ( t = 0), and striking the target ( t > x f frombest focus leads to a “sliding focus” effect, the maximum on-target intensity sliding in the y -direction at a velocity v sl . Theupper left insert compares v sl from Eq. (3) (solid lines) withmeasures from PIC simulations (points) for x f = 25 λ (black)and x f = 50 λ (magenta). c k SPW /ω = ( ω p /ω − / ( ω p /ω − k SPW theSPW wavenumber and ω p = (cid:112) e n/ ( (cid:15) m e ) the electronplasma frequency. For n (cid:29) n c , the SPW phase and groupvelocities are slightly subluminal: v φ → c [1 − n c / (2 n )]and v g → c [1 − n c / (2 n )].As shown in Fig. 1, the target is located at a distance x f from the laser best focus, which together with WFRallows for a “sliding focus” effect, i.e. a displacement intime of the pulse intensity peak along the target surface.If the sliding focus velocity v sl is close to the SPW veloc-ity, the latter will be driven more efficiently. To estimate v sl let us recall that at focus, the electric field of a pulsewith WFR can be written as [29]: E ( y (cid:48) , t ) = E f ( t ) F ( y (cid:48) ) exp [ iφ ( y (cid:48) , t )] . (1)Here E is the maximum electric field, f ( t ) and F ( y (cid:48) )are the electric field temporal and transverse (in our 2Dconfiguration) spatial envelope, and the spatio-temporalphase is φ ( y (cid:48) , t ) = ω t (1 − Ω β y (cid:48) /c ) . (2)The linear dependence in y (cid:48) t leads to an instantaneousangle of propagation of light β ( t ) (cid:39) − ( c/ω ) ∂φ/∂y (cid:48) =Ω β t increasing linearly with time, with Ω β the WFR ve-locity. In Fig.1, Ω β > θ , defined as that of the central wavefront, is chosenas the resonant angle for exciting the SPW. Successivewavefronts are then shifted by an angle ∆ β = Ω β λ /c henceforth referred to as the WFR parameter. As a re-sult, each successive wavefront will strike the target at aslightly different location along the y -direction leading tothe apparent sliding velocity of the pulse on the target.For ultrashort pulses and/or the central wavefronts, we obtain a constant sliding velocity: v sl (cid:39) ∆ β x f /λ cos θ + sin θ ∆ β x f /λ c . (3)As shown in the insert of Fig. 1 (for ∆ β = 33mrad),Eq. (3) is found to be in good agreement with measure-ments from PIC simulations [36].The sign and value of the WFR parameter ∆ β affectsthe duration and amplitude of the excited SPW [27]. In-deed, when the sliding velocity is along the direction ofpropagation of the SPW, the excited wave can increaseits amplitude while maintaining a short duration. Addi-tional tunability can be obtained by calculating an op-timal value of the WFR parameter ∆ β opt such that thesliding velocity v sl coincides with the SPW velocity (cid:39) c ;this leads to: ∆ β opt (cid:39) λ x f (1 + sin θ ) . (4)Eq. (4) depends on x f : ∆ β opt decreases when increasingthe distance between the target and best focus. Thisallows to relax the experimental constraint of obtaininglarge WFR velocity [30]. However, there is a trade offsince at larger values of x f the intensity of the laser at thesurface decreases. For the largest value we investigate, x f = 50 λ [where Eq. (4) gives ∆ β opt (cid:39) x f = 25 λ where ∆ β opt (cid:39) Smilei [32] consideringdifferent laser field strength a = eE / ( m e cω ). Firsta non-relativistic laser intensity a = 0 . a = 5 and electron ac-celeration along the target surface are considered . Inboth cases, the general setup of the simulation is givenin Fig. 1 with numerical parameters in [37]. The gratingtarget, of thickness 3 λ , has density n = 100 n c , ion toelectron mass ratio m i / ( m e ) = 1836 and temperature ra-tio T i / ( T e ) = 0 . T e = 50eV. The periodicity of thegrating is d = 2 λ with a groove’s depth h = 0 . λ anda blazed angle φ b = 13 ◦ . A flat surface (at y > λ )follows the grating so that the laser illuminates only thenumber of ripples corresponding to the projected pulsewaist onto the surface. The driving laser is a p -polarizedGaussian pulse with transverse size w ⊥ = 5 . λ , dura-tion [full-width-at-half-maximum (FWHM) in intensity] T = 10 λ /c [38]. The laser pulse impinges onto the grat-ing target at the resonant angle θ = 31 ◦ . The simulationis run up to time t + 20 λ /c , with t the time when thepeak of the pulse reaches the target. Unless specified oth-erwise, all values are taken at the end of the simulation.We first consider a = 0 . z -component of the mag-netic field [39], noted B SPW [or ˆ B SPW = eB SPW / ( m e ω )],is taken as representative of the SPW, all the otherfield components being proportional to it. For n (cid:29) n c ,and in the vacuum side, the linear approximation yields | E x | ∼ c | B SPW | and | E y | ∼ c | B SPW | (cid:112) n c /n .In Fig. 2, we show a snapshot of ˆ B SPW along the targetsurface for x f = 25 λ , (a) ∆ β = 0 and (b) ∆ β = 67mrad.The latter case corresponds to the most intense andshortest SPW found in our simulations, ∆ β opt = 60mrad.With this optimal WFR parameter the SPW peak am-plitude is increased by ∼
65% with respect to the casewithout WFR and its duration, measured as the signalFWHM, is reduced by four from 14 . . λ /c .Panels (c) and (d) show the maximum value of ˆ B SPW and the measured SPW duration as the result of a para-metric scan of ∆ β for x f = 0 (target at focus, greentriangles) and x f = 25 λ (target off-focus, black circles).At focus, WFR has a small impact on the SPW excita-tion: the most intense SPW is obtained for ∆ β = 0, andusing non-zero ∆ β decreases the duration of the SPWbut also its maximum amplitude. Instead, for x f = 25 λ ,∆ β acts as a tuning parameter allowing both to shortenthe SPW and to increase its amplitude. We observe theshortest and most intense SPW for ∆ β (cid:39) β opt (cid:39) β = 53mrad. Interestingly, even thoughthe on-target laser intensity is reduced when increasing x f to 25 λ , a significant increase of the SPW amplitude isstill obtained using the optimal WFR parameter. A para-metric scan considering x f = 50 λ (not shown) leads toan optimal WFR parameter ∆ β (cid:39) β opt = 30mrad from Eq. (4). Finally,as expected positive values of ∆ β , for which the slidingvelocity is along the SPW propagation direction, give amaximal effect. In contrast, for negative ∆ β , the SPW isstill of a shorter duration but with a reduced amplitude,roughly that obtained when placing the target at best
36 72 y/λ -1.01.0 ˆ B S P W / a y i y f p1(a) ∆ y fi = 14 . λ
36 72 y/λ -1.01.0 ˆ B S P W / a y i y f p2(b) ∆ y fi = 3 . λ . . m a x ( ˆ B S P W / a ) (c) p2p1 ∆ y f i / λ (d) p2p1 −
110 0 110∆ β (mrad)0150 m a x ( p y / m e c ) (e) −
110 0 110∆ β (mrad)015 ∆ y f i / λ (f) FIG. 2: SPW magnetic field at the target surface for(a) ∆ β = 0 and (b) ∆ β = 67mrad with a = 0 . x f = 25 λ . (c) Maximum SPW field amplitude and (d) du-ration (FWHM) versus the WFR parameter ∆ β for a = 0 . p y ) and (f) electron bunch duration (FWHM) versus∆ β for a = 5. In panels (c) to (f), x f = 0 (green triangles), x f = 25 λ (black circles). focus.We now turn our attention to the second series of sim-ulations performed in the UHI regime ( a = 5) and elec-tron acceleration. The bottom row in Fig. 2 shows (e) themaximum electron momentum parallel to the surface and(f) the characteristic width [40] of the accelerated elec-tron bunch as a function of ∆ β , considering x f = 0 (greentriangles) and x f = 25 λ (black circles). Both panels ex-hibit very similar features than observed at low intensity.Placing the target at focus ( x f = 0) the accelerated elec-tron bunch maximum energy and duration are marginallyaffected by WFR. In contrast, for x f = 25 λ , WFR sig-nificantly impacts electron acceleration: taking ∆ β > β = 0 and that with the target at x f = 25 λ with∆ β = 67mrad, one find an increase of the maximumelectron momentum by 62% [from max( p y ) (cid:39) m e c to (cid:39) m e c ] and form much shorter bunches when the op-timal (positive) WFR parameter is considered and targetis off-focus. The optimum value ∆ β = 67mrad found forelectron acceleration in this regime is the same as found -300150 p y / m e c t + 10 λ /c (a) ˆ B S P W / a -1.21.2 -300150 p y / m e c t + 20 λ /c (b) ˆ B S P W / a -1.21.2 ∼ λ y/λ -300150 p y / m e c t + 10 λ /c (c) ˆ B S P W / a -1.21.2 0 72 y/λ -300150 p y / m e c t + 20 λ /c (d) ˆ B S P W / a -1.21.2 ∼ λ FIG. 3: Electron phase-space (red) and SPW field amplitude(blue line, right scale) for a = 5 at times, t = t +10 λ /c and t = t + 20 λ /c . (a)-(b): ∆ β = 0, and (c)-(d): ∆ β = 67mrad.The gray line indicates the end of the grating and beginningof the flat region. earlier for efficient, ultrashort SPW excitation at lowerintensity.Figure 3 gives further insights into the accelerationprocess. The electron phase-space and SPW magneticfield at the target surface ( x f = 25 λ ) are shown at twodifferent times, for (a,b) ∆ β = 0 and (c,d) ∆ β = 67mrad(optimal condition). In both cases the duration of theelectron bunch is proportional to the duration of theSPW, the shortest SPW obtained for ∆ β = 67mradleading to the shortest electron bunch. For ∆ β = 0[panels (a) and (b)], the SPW is strongly damped at t = t +20 λ /c : the electron bunch has reached its paral-lel momentum max( p y ) (cid:39) m e c and has a width (mea-sured from the FWHM in momentum) ∆ y fi = 11 λ . Theacceleration process is more efficient using the optimalWFR parameter ∆ β = 67mrad [panels (c) and (d)]. At t = t + 10 λ /c , two periods after the laser has left thesurface, the magnetic field is intense ( ˆ B SPW (cid:39) . a ) andthe most energetic electrons have already reached mo-mentum up to max( p y ) (cid:39) m e c . Ten periods later, anarrow (∆ y fi = 3 λ ) and energetic [max( p y ) (cid:39) m e c ]electron bunch is obtained, while the SPW has been sig-nificantly damped.Similar observations can be drawn from Fig. 4. Inpanel (a), the electron distribution in energy and direc-tion (the angle is defined in the simulation plane withrespect to the x -axis) is shown, demonstrating that themost energetic electrons are accelerated mainly along thetarget’s surface and in the y > i.e. in theSPW direction of propagation). Panel (b) shows the en-ergy distribution of the electron, for different values of∆ β .These results and in particular the increase of the max-imum electron energy (equiv. momentum) are consistent ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ (a) E (MeV)
20 40 60 80 E (MeV) -80 l og [ d N / d E ] (b) ∆ β = 67 m rad∆ β = 0∆ β = − m rad FIG. 4: (a) Electron energy distribution in MeV as a functionof the emission angle φ = tan − ( p y /p x ) for ∆ β = 67mrad and a = 5. (b) Electron energy distribution for ∆ β = 67mrad(red), ∆ β = 0 (blue) and ∆ β = − with what one expects from the increase of the SPW am-plitude by use of the WFR driving pulse. Indeed, an up-per limit of the electron energy gain in the SPW has beenderived in [18] by generalizing the results of wakefield ac-celeration [33, 34], leading ∆ E ∼ χ γ φ max | ˆ B SPW | m e c so that ∆ E is proportional to the SPW field amplitude.Here γ φ = (1 − v φ /c ) − / and χ is a constant of or-der one, reaching at most 4[18]. In our simulations, themagnetic field of the SPW (time-averaged over the waveperiod) reached at most max | ˆ B SPW | ∼ . β = 0and max | ˆ B SPW | ∼ . β = 67mrad. Consideringthat γ φ (cid:39)
10 for n = 100 n c , we then obtain the upperlimit ∆ E (cid:39) m e c for ∆ β = 0 and ∆ β = 67mradfor ∆ E (cid:39) m e c . These predictions overestimate theelectron energy as they assume i) no wave decay overthe distance required for acceleration, ii) optimal elec-tron injection and iii) acceleration exactly parallel to thetarget surface, while it has been observed that electronsare deflected in the perpendicular direction [19].To gain further insight into the acceleration process,we performed a particle tracking of the most energeticelectrons and evaluated the trajectory-averaged value ofthe longitudinal field (cid:104) E y (cid:105) acting on the particle. This al-lows to define an acceleration length l acc = ∆ E / | e (cid:104) E y (cid:105)| .From the particle track we found ∆ E (cid:39) m e c and (cid:104) E y (cid:105) (cid:39) − . m e cω /e for ∆ β = 0, and ∆ E (cid:39) m e c and (cid:104) E y (cid:105) (cid:39) − . m e cω /e for ∆ β = 67mrad. In bothcases this leads to an acceleration length l acc ∼ λ ,consistent with the observed particle trajectories. Thislength largely exceeds the laser spot size, and is close tothe length over which the SPW decreases its amplitudesignificantly (see, e.g. , Fig. 3). This confirms the elec-trons are accelerated by the SPW as it propagates alongthe target surface.In the optimal case, the highest energy particles (inthe range 30 − (cid:39) λ /c [ ∼ λ = 0 . µ m] and total charge (cid:39) /λ (in our 2D simulations). Assuming a bunchwidth (in the z -direction) of the order of the laser pulsewith w ⊥ = 5 . λ , one could expect few cycles electronbunches with a charge of ∼ . [1] H. Raether, Surface plasmons on smooth and rough sur-faces and gratings , Springer-Verlag (1988)[2] W. Barnes, A. Dereux, and T. Ebbesen, Nature ,824–830 (2003)[3] S. A. Maier,
Plasmonics: fundamentals and applications ,Springer-Verlag (2007)[4] J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M.Echenique, Rep. Prog. Phys. , 1 (2007)[5] P. K. Jain, X. Huang, I. H. El-Sayed, and M. A. El-Sayed,Acc. Chem. Res. , 1578 (2008)[6] K. C. Y. Huang, M.-K. Seo, T. Sarmiento, Y. Huo, J. S.Harris, and M. L. Brongersma, Nature Photonics , 244(2014)[7] E. Ozbay, Science , 189 (2006)[8] T. Chung, S.-Y. Lee, E. Y. Song, H. Chun, and B. Lee,Sensors , 10907 (2011)[9] G. S. Agarwal, and S. S. Jha, Solid State Commun. ,499 (1982)[10] J. L. Coutaz, M. Neviere, E. Pic, and R. Reinisch, Phys.Rev. B , 2227 (1985)[11] P. Jatav, and J. Parashar, Phys. of Plasmas , 022301(2019)[12] J. Kupersztych, P. Monchicourt, and M. Raynaud, Phys.Rev. Lett. , 5180 (2001)[13] J. Zawadzka, D. A. Jaroszynski, J. J. Carey, and K.Wynne, Appl. Phys. Lett. , 2130 (2001)[14] M. Raynaud, J. Kupersztych, C. Riconda, J. C. Adamand A. H´eron, Phys. of Plasmas , 092702 (2007)[15] T. Ceccotti, V. Floquet, A. Sgattoni, A. Bigongiari, O.Klimo, M. Raynaud, C. Riconda, A. Heron, et al. , Phys.Rev. Lett. , 185001 (2013)[16] A. Bigongiari, M. Raynaud, C. Riconda, and A. H´eron,Phys. Plasmas , 052701 (2013)[17] Y. Tian, J. Liu, W. Wang, et al. , Phys. Rev. Lett. , 115002 (2012)[18] C. Riconda, M. Raynaud, T. Vialis, and M. Grech, Phys.of Plasmas , 073103 (2015)[19] L. Fedeli, A. Sgattoni, G. Cantono, et al. ., Phys. Rev.Lett. , 015001 (2016)[20] L. Fedeli, A. Sgattoni, G. Cantono, and A. Macchi, Appl.Phys. Lett. , 051103 (2017)[21] G. Cantono, A. Sgattoni, L. Fedeli, et al. , Phys. Plasmas , 031907 (2018)[22] A Macchi, Phys. Plasmas , 031906 (2018)[23] M. Raynaud, A H´eron, and J-C Adam, Plasma Phys.Controlled Fusion , 014021 (2018)[24] X.M. Zhu, R. Prasad, M. Swantusch, B. Aurand, A.A.Andreev, O. Willi, and M. Cerchez, High Power LaserSci. Eng. , 15 (2020)[25] M. Raynaud, A. H´eron and J.-C. Adam, Sci. Reports ,13450 (2020)[26] G. Cantono, L. Fedeli, A. Sgattoni, A. Denoeud, L.Chopineau, F. R´eau, T. Ceccotti, A. Macchi, et al. , Phys.Rev. Lett. , 264803 (2018)[27] F. Pisani, L. Fedeli, and A. Macchi, ACS Photonics ,1068 (2018)[28] P. K. Kaw and J. B. McBride, Phys. of Fluids , 1784(1970)[29] H. Vincenti, and F. Qu´er´e, Phys. Rev. Lett. , 113904(2012)[30] F. Qu´er´e, H. Vincenti, et al. , J. Phys. B: At. Mol. Opt.Phys. , 124004 (2014)[31] P. S. Kleij, MSc, Universit`a di Pisa– Sorbonne Universit´e2019, Study of grating target and rotation wave front forhigh field plasmonics. [32] J. Derouillat, A. Beck, F. P´erez, et al. , Comput. Phys.Commun. , 351 (2018)[33] T. Tajima and J.M. Dawson, Phys. Rev. Lett. , 267(1979)[34] P. Mora, and F. Amiranoff, J. Appl. Phys. , 3476(1989); P. Mora, Phys. Fluids , 1630 (1992)[35] O. Lundh, J. Lim, C. Rechatin, et al. , Nat. Phys. , 219(2011)[36] In PIC simulations, v sl is measured by locating the posi-tion of the maximum laser field amplitude as a functionof time at the target surface and time-averaging over thelaser high-frequency.[37] The simulation box is 39 λ × λ (in the x - y directions),with 9984 × λ / t = 0 . / √
2. Electromagneticfield boundary conditions are injecting/absorbing in x and periodic in y . Particle boundary conditions in x arereflecting (left) or thermalizing (right), and periodic in y . There are 32 macro-particles per species per cell.[38] The laser transverse profile is Gaussian, F ( y (cid:48) ) =exp( − y (cid:48) /w ⊥ ) with w ⊥ = 5 . λ and its time profile iscos : f ( t ) = cos( πt/ (2 T )) for | t | < T (0 otherwise), with T = 10 λ /c .[39] B SPW is collected at t = t + 20 λ /c , on flat surfacefar from the laser-plasma interaction zone. The magneticfield has been filtered,selecting values of k > k SPW .[40] The duration of the electron bunch is estimated from itsspatial width through the relation ∆ τ fi (cid:39) ∆ y fi /c/c