Preliminary simulations of a Laser Wakefield Field Accelerator driven FEL are presented using the 3D unaveraged, broad bandwidth FEL simulation code Puffin. The radius of the matched low emittance electron beam suggests that the FEL interaction will be strongly affected by radiation diffraction. The parameter scaling and comparison between 3D and equivalent 1D simulations appears to confirm the interaction is diffraction dominated. Nevertheless, output powers are predicted to be greater than those of similar unaveraged FEL models. Possible reasons for the discrepancies between the averaged and unaveraged simulation results are discussed

Campbell, L T.

Gruener, F. J.

Maier, A. R.

McNeil, B W. J.

University of Strathclyde Institutional Repository

Strathprints Institutional RepositoryCampbell, L T. and Maier, A. R. and Gruener, F. J. and McNeil, B W. J. (2013) Unaveraged modelling of a LWFA driven FEL. In: FEL 2013. JACoW, pp. 43-46. ISBN 9783954501267 , This version is available at http://strathprints.strath.ac.uk/57083/Strathprints is  designed  to  allow  users  to  access  the  research  output  of  the  University  of Strathclyde. Unless otherwise explicitly stated on the manuscript, Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Please check the manuscript for details of any other licences that may have been applied. You may  not  engage  in  further  distribution  of  the  material  for  any  profitmaking  activities  or  any commercial gain. You may freely distribute both the url (http://strathprints.strath.ac.uk/) and the content of this paper for research or private study, educational, or not-for-profit purposes without prior permission or charge. Any  correspondence  concerning  this  service  should  be  sent  to  Strathprints  administrator: strathprints@strath.ac.ukUNAVERAGED MODELLING OF A LWFA DRIVEN FELL.T. Campbell1,2, A.R. Maier3,4,5, F.J. Gru¨ner3,4,5 and B.W.J. McNeil11SUPA, Department of Physics, University of Strathclyde, Glasgow, UK,2ASTeC, STFC Daresbury Laboratory and Cockcroft Institute, Warrington, United Kingdom3Center for Free-Electron Laser Science, Notkestrasse 85, Hamburg, Germany4Institut fu¨r Experimentalphysik, Universita¨t Hamburg, Hamburg, Germany5Department fu¨r Physik, Ludwig-Maximilians Universita¨t, Garching, GermanyAbstractPreliminary simulations of a Laser Wake Field Accel-erator driven FEL are presented using the 3D unaveraged,broad bandwidth FEL simulation code Puffin. The radiusof the matched low-emittance electron beam suggests thatthe FEL interaction will be strongly affected by radiationdiffraction. Parameter scaling and comparison between3D and equivalent 1D simulations confirm this. The Puf-fin 1D simulations indicate that the energy spread condi-tions for FEL lasing are met, even without any beam phasespace manipulation prior to injection into the undulator.The large diffraction in the Puffin 3D simulations createsboundary problems that will need to be overcome beforefurther progress is made.INTRODUCTIONWith several linac driven X-ray FEL’s currently in opera-tion or construction around the world, there is much interestin the next generation of FEL facilities. The plasma basedLaser Wakefield Accelerators (LWFA’s) are a promisingdriver for future FEL facilities. Due to their large acceler-ation gradients compared to conventional RF-linacs, theircompact size could dramatically reduce facility costs.No plasma accelerator driven FEL has yet reported suc-cessful lasing primarily as the beam energy spreads fromthese accelerators are too large by approximately an orderof magnitude. Some other plasma accelerator schemes ex-ist that may promise an improvement in beam quality, butthese are yet to be realised. A pragmatic approach wastaken in a study [1], which considered a design that usesbeams currently available from LWFA’s to enable modestFEL gains to generate power levels measurably above thespontaneous power to be observed. A cryogenic undula-tor design with a small undulator period and large on-axismagnetic field was used. This design is chosen to max-imise the FEL ρ parameter for a given beam and thus relaxthe energy spread requirement in the FEL σγ/γ . ρ. Achicane to stretch the beam before insertion into the un-dulator was utilized to both increase the beam length withrespect to the cooperation length, and so increase the in-teraction length between radiation and electrons, and to re-duce the localised energy spread. Genesis [2] simulationspredicted a modest gain of ∼ 6 over the spontaneous emis-sion without bunch stretching, and a gain of ∼ 103 whenIn averaged FEL simulation codes where the SlowlyVarying Envelope Approximation (SVEA) is applied, theelectron beam and radiation fields are modelled by a se-ries of phase space ‘slices’ within which periodic boundaryconditions are applied. Modelling the FEL interaction andelectron phase space evolution with beams that are short,have correlated energy spreads (chirps) etc, such as thosegenerated by plasma accelerators, can therefore be prob-lematic. The 3D FEL simulation code Puffin [3] does notperform the Slowly Varying Envelope Approximation oraveraging of the electron or radiation parameters and somay be better suited to such circumstances. Furthermore,the effects of dispersion in short beams, either from thechirp or from energy spread induced from the FEL inter-action, may be more dramatic than equivalent effects in alonger pulse.In the following the 3D parameters and likely conse-quences on the FEL interaction for a plasma acceleratordriven FEL interaction are discussed. Puffin simulationsof the FEL interaction using the parameter set of [1] arethen presented, first in 1D, showing the requirement on thebeam energy spread is satisfied. A 3D simulation is alsopresented. However, the small matched electron beam ra-dius results in a large radiation diffraction which, for thefield sampling size used, cannot be modeled properly.PARAMETERSMuch information can be gained regarding a potentialFEL interaction by calculating the scaled parameters thatdescribe the interaction and comparing against a knownset of limits which the scaled parameters must meet be-fore good FEL lasing action can occur. Several works havederived these criteria over a period as summarised in [4, 5].The physical parameters of the study of [1], in the scaledparameters of Puffin [3], are shown in Table 1. In [1], theelectron beam was matched in one transverse direction tothe natural focusing of the planar undulator. However, thePuffin model assumes a uniform focusing in both trans-verse directions. This results, in the case of the planar wig-gler, in the beam being over-focused when using the samebetatron wavelength. In the following, the beam is focusedto give a matched beam of radius equal to the rms radiusused in [1], giving the same diffraction length. This givesan incorrect betatron wavelength, but it is assumed this willthe stretching was optimized.Proceedings of FEL2013, New York, NY, USA MOPSO08Novel ConceptsISBN 978-3-95450-126-743 Copyrightc ○2013CC-BY-3.0andbytherespectiveauthorshave negligible effect in this case since the betatron wave-length is very large compared to a gain length in both cases.In calculating these parameters, gaussian distributionshave been used throughout for all variables.It is seen that the FEL parameter ρ is sufficiently largethat the energy spread σγ/γr < ρ which indicates that forthese parameters the effect of energy spread alone shouldnot stop an FEL interaction occurring.One can immediately see that the scaled emittance forthe beam ǫ¯ ≡ ǫ/(λr/4π) ≪ 1. To achieve good FEL am-plification ǫ¯ . 1 [3, 6], so that the value here is rather small.Although this small emittance does mean a small effectiveenergy spread due to the betatron motion of the electrons,the Rayleigh range is quite short with respect to the gainlength, as seen from the value of the scaled Rayleigh rangez¯R = zR/lg . This is also confirmed by the relatively smallvalue of the diffraction parameter B [6]. One can there-fore expect to see degradation of the FEL interaction dueto significant radiation diffraction in a gain length.For the parameters of Table 1, the electron bunch is rel-atively short with respect to the cooperation length lc =λr/4πρ ≈ 1.7µm. This reduces the radiation coupling tothe electrons as it slips out of the bunch after only a fewgain lengths. This effect can mitigated and was investi-gated in [1], by ‘stretching’ the chirped electron bunch in achicane prior to injection into the FEL undulator. This alsohas the effect of reducing both the electron energy chirp andthe slice energy spread of the bunch. However, the stretch-ing also reduces the electron current, so increasing the gainlength and therefore the diffraction in the gain length, z¯r.All simulations presented here are for the ‘un-stretched’ pa-rameters of Table 1.The energy chirp of the bunch will cause the bunch todisperse and stretch as it propagates along the undulator.This may affect the FEL process, e.g. by stretching out theelectron microbunching from the resonant wavelength asthe interaction progresses. Short electron bunches may alsogenerate Coherent Spontaneous Emission that may subse-quently be amplified [7]. Such effects cannot be modeled inan averaged simulation code. As a gaussian current profileis used here, it is not expected that CSE will play a signif-icant role in the startup of the interaction. However, thiscan change significantly for differently shaped pulses andif there is wavelength-scale structure in the electron currentdistribution.Presently, Puffin does not model the effects of spacecharge, which may become significant with larger FEL ρ-parameters.1D SIMULATIONSUsing the parameters of Table 1, Puffin was used in1D mode to simulate the FEL interaction of the chirpedbunch. In Fig. 1 the scaled power, relative electron energyand spectral power are plotted in the scaled temporal framez¯2 and frequency ω/ωr respectively. The scaled temporalframe is a window that travels with the radiation, so thatthe ‘head’ of the electron bunch is to the left of the plotTable 1: Parameters used in simulations. The bracketedterm for z¯R andB is the value used in the Puffin simulation(see text.)a¯w 2.3λw 1.5cmγr 600σγ/γr 0.005λr 134nmpulse length σz 0.5µmρ 0.0223k¯β 1.12× 10−2η 2.5× 10−8ǫ¯ 3.12× 10−2z¯R 0.398B 0.71and the bunch slips left to right in the window on propa-gating through the FEL undulator. The Scaled Power asplotted is not the usual time-averaged power, i.e. a powerenvelope, but rather the instantaneous power of an oscilla-tory linearly polarised field (the fast oscillations cannot beresolved in the plot). Note that in the 1D case, |A⊥|2 isthe on-axis intensity so that the Scaled Power, as plotted,is πσ¯2x|A⊥|2, where σ¯x = σ¯y = 0.126 is the rms radiusof the beam in the scaled transverse units x¯, y¯. As the 1Dscaled intensity at saturation for a cold bunch of good qual-ity is |A⊥|2 ∼ 1 [8], the Scaled Power from such a bunchat saturation will be ∼ πσ¯2x ≈ 0.05.Detail of the electron phase space clearly shows that theelectrons are being modulated at the radiation wavelength(of period≈ 0.27 in z¯2) towards the head of the bunch 34 <z¯2 < 39 and are becoming bunched and losing energy tothe radiation around the centre of the bunch 40 < z¯2 < 430 10 20 30 40 5000.51 x 10−3P¯z¯235 40 45 500.911.1γ/γrz¯20 1 2 3 4 5 610−1010−5100P¯nω/ωrFigure 1: Scaled Power (top), electron phase space (mid-dle) and scaled spectral power (bottom) from a 1D simula-tion of the parameters in Table 1. (Note the different scalein z¯2 for the plot of electron phase-space).where the current is maximum.MOPSO08 Proceedings of FEL2013, New York, NY, USAISBN 978-3-95450-126-744Copyrightc ○2013CC-BY-3.0andbytherespectiveauthorsNovel ConceptsDue to the energy chirp, the beam is seen (not shownhere) to stretch as it propagates through the undulator.However, from the discussions above, it is expected, butnot verified here, that 3D diffraction effects will accountfor most of the differences between the 1D and 3D simula-tions presented in the following section.Note that the radiation power appears to have saturatedwith a peak at z¯2 ≈ 35 which has propagated through andbeyond the peak of the electron bunch current located atz¯2 ≈ 42. From the spectrum, most of the power is aboutthe resonant frequency at ω/ωr = 1.3D SIMULATIONSThe same parameters of the previous 1D simulation arenow used in a 3D simulation that includes the diffractioneffects. As noted in the introduction, the method of focus-ing of the electron beam in Puffin here gives a longer beta-tron wavelength than that of a planar undulator, although itis anticipated that the effect of this will be negligible com-pared to the diffraction.Puffin uses a split-step method [3] to simulate diffrac-tion and radiation generation from the electron beam self-consistently. The diffraction step of the integration is nom-inally set to occur every undulator period. However, dueto the relatively large diffraction experienced in this simu-lation it was found necessary to carry out a diffractive step12 times per undulator period.The preliminary 3D results equivalent to the 1D Fig.1 are plotted in Fig. 2. As previously, the scaledinstantaneous power is plotted which here is the nu-merically integrated intensity over the transverse plane:∫|A⊥(x¯, y¯, z¯2, z¯)|2dx¯dy¯. Note that diffraction ‘smearsout’ the fast oscillatory behaviour previously observed inthe 1D simulation - the power at different frequenciesdiffract at different rates.The introduction of the radiation diffraction appears tohave significantly limited the gain, reducing the scaledpower by a factor ∼ 102 from the 1D simulation. Nev-ertheless, this power is ∼ 4 − 5 orders of magnitude overthe spontaneous emission power and significantly greaterthan the factor of ∼ 5 of the averaged simulations of [1].However, the radiation power is at a lower frequencythan the usual resonant FEL frequency, and is broadband.Examination of the intensity in the transverse plane, seeFigures 3 and 4, shows the simulation is completely dom-inated by diffraction. Puffin solves difraction in Fourierspace, so the boundary conditions are cyclic when solv-ing the diffraction. The low frequency emission results in ashorter diffraction length, and the transverse boundaries au-tomatically setup to accomodate a more conventional FELhave proven insufficient to model the extreme diffractionoccuring in this case. Figure 3 shows a reasonable trans-verse intensity distribution at the center of the pulse in thelongitudinal axis, but in Figure 4, further towards the headof the pulse, one observes interference patterns caused byFigure 2: As Fig. 1 but for 3D Puffin simulation with theparameters of Table 1.the radiation propagating through the periodic boundaryconditions.It is unclear at this stage whether the lower frequencypeak of the power spectrum results from errors introducedby the incorrectly modeled diffraction or from other phys-ical effects. Note that the unphysical transverse behaviouroccurs towards the head of the electron bunch, where thereis less charge - there is a guiding effect before the radia-tion propagates away from the higher current region of thebunch and strongly diffracts. It cannot be ruled out thatthese effects are causing large errors in other aspects of thesimulation. Therefore, to model these diffraction effectsproperly, either the area of the transverse field model mustbe significantly, possibly prohibitively, increased. Alterna-tively, absorbing or transparent boundary conditions can beintroduced, however, these cannot conserve total energy inthe simulation.CONCLUSIONSDriven by experimental progress, the simulation of po-tential plasma-based accelerator FELs is an important de-veloping field. The beams from these accelerators can dif-fer substantially from their linac counterparts in terms oftheir peak current, bunch duration, emittance and their cor-related and uncorrelated energy spreads. While averagedFEL simulation codes are able to model linac-driven FELinteractions in excellent agreement with experiment, theymay be less suitable for plasma-based accelerator FELs.The discretisation and averaging of the radiation and elec-tron beam parameters over the resonant radiation wave-length make a mobile electron phase space, coherent spon-taneous and broad bandwidth emission difficult or impos-sible to model correctly.The results presented in this paper are a first step in theunaveraged 3D modeling of a LWFA driven FEL using theProceedings of FEL2013, New York, NY, USA MOPSO08Novel ConceptsISBN 978-3-95450-126-745 Copyrightc ○2013CC-BY-3.0andbytherespectiveauthors−3−2−10123−3 −2 −1 0 1 2 3z¯2 = 42.3181y¯x¯Figure 3: Transverse scaled intensity at z¯2 ≈ 43.5 from 3DPuffin simulation with the parameters of Table 1.−3−2−10123−3 −2 −1 0 1 2 3z¯2 = 40.5544y¯x¯Figure 4: Transverse scaled intensity at z¯2 ≈ 37.6 from 3DPuffin simulation with the parameters of Table 1.Puffin simulation code. The 3D results are seen to be notcorrect due to problems modeling large diffraction effects,and are presented only as ‘work-in-progress’.The 1D simulations indicate that the energy spread con-dition for an FEL is satisfied without the need for a chicaneto stretch the beam so reducing the slice energy spread.The 3D simulation indicates that the system exhibits ex-cessive diffraction for the model parameters. The largediffraction regime is challenging for Puffin to model cor-rectly. While the longitudinal power profile and beamphase space seem reasonable, it is not possible, in light ofthe the diffraction issues, to make any claims for the va-lidity of these aspects. In previous simulations performedwith Puffin, with parameters found in more conventionalFELs where diffraction lengths are closer to being opti-mised, a low frequency filter applied in the transverse planewas sufficient to prevent unphysical reflections from ra-diation propagating to the transverse boundaries. Furthersimulations are needed either using this method, or otherboundary condition methods as discussed above, to solvethis large diffraction regime correctly.To mitigate diffraction effects on the FEL interaction, anunmatched beam of larger radius at the start of the inter-action could be used. An increased beam radius will ofcourse reduce the ρ parameter, so tighten the energy spreadrequirement, and will also introduce a beam radius oscilla-tion. Further research into balancing and optimising theseeffects is required.Further simulations will in future also be performed fordifferent electron beam stretching factors prior to injectioninto the FEL undulator, to check the gain enhancement asobserved in the simulations of [1]. The modeling wouldalso benefit from more realistic current profiles than thegaussian. For the relatively short electron bunches usedhere, CSE may dominate spontaneous emission leading toa quicker FEL start-up [7]. Such a process may also give abetter temporal coherence and greater output intensity .ACKNOWLEDGMENTSThe simulations presented were produced from Archie-WEST at the University of Strathclyde, and Blue Joule atthe Hartree Institute.REFERENCES[1] AR Maier, A Meseck, S Reiche, CB Schroeder, T Segge-brock, and F Gruner, Phys Rev X 2, 031019 (2012)[2] S. Reiche, Nucl. Instr. and Meth. in Phys. Res. A 429, 243-248 (1999)[3] L.T. Campbell and B.W.J. McNeil, Physics of Plasmas 19,093119 (2012)[4] Brian W. J. McNeil and Neil R. Thompson, Nature Photon.4, 814 (2010)[5] Z. Huang and K.-J. Kim, Phys. Rev. Spec. Top. AB 10,034801 (2007)[6] E. L. Saldin, E. A. Schneidmiller, and M. V. Yurkov, New J.Phys. 12, 035010 (2010)[7] B.W.J. McNeil , G.R.M. Robb and D.A. Jaroszynski, OpticsComm. 165, 65 (1999)[8] R. Bonifacio, C. Pellegrini and L.M. Narducci, Opt. Comm.50, 373 (1984)MOPSO08 Proceedings of FEL2013, New York, NY, USAISBN 978-3-95450-126-746Copyrightc ○2013CC-BY-3.0andbytherespectiveauthorsNovel Concepts

Unaveraged modelling of a LWFA driven FEL

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Unaveraged modelling of a LWFA driven FEL

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