Tapering studies for Terawatt level X-ray FELs with a superconducting undulator

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by  Claudio Emma

Tapering studies for Terawatt level X-ray FELs with a superconducting undulator
C. Emma,1 J. Wu,2 P. Emma,2 Z. Huang,2 and C. Pellegrini1,2
1University of California, Los Angeles, California 90095, USA
2Stanford Linear Accelerator Center, Menlo Park, California 94025, USA
(Dated: November 5, 2016)
Westudythetaperingoptimizationschemeforashortperiod,lessthantwocm,superconducting
undulator,andshowthatitcangenerate4keVX-raypulseswithpeakpowerinexcessof1terawatt,
using LCLS electron beam parameters. We study the effect of undulator module length relative to
the FEL gain length for continous and step-wise taper profiles. For the optimal section length of
1.5mwestudytheevolutionoftheFELprocessfortwodifferentsuperconductingtechnologiesNbTi
andNb3Sn. Wediscussthemajorfactorslimitingthemaximumoutputpower,particledetrapping
5 aroundthesaturationlocationandtimedependentdetrappingduetogenerationandamplification
1
of sideband modes.
0
2 PACSnumbers: 41.60.Cr,41.60.Ap

n
a
J I. INTRODUCTION
TABLE I: GENESIS Simulation Parameters

4
Parameter Name NbTi Nb3Sn
2 Muchrecentscientificefforthasbeendevotedtostudy-
ing the possibility of using a tapered undulator [1] to Electron Beam:
]
h achieveterawattlevelhardx-raypulsesinthenextgener-
Beam Energy E 7.2 GeV 6.8 GeV
p ationofFreeElectronLasers(FELs)[2][3][4][5][6]. The 0
Energy Spread σ 1.5 MeV 1.5 MeV
- motivation for pursuing this goal comes primarily from E
c thebioimagingcommunitywhereaterawattpowercoher- Peak Current Ipk 4 kA 4 kA
c Normalized Emittances (cid:15) /(cid:15) 0.4/0.4 µ m 0.4/0.4µ m
x,n y,n
a ent x-raysourcewould openthe doors tosingle molecule Average β function (cid:104)β(cid:105) 12 m 12 m
. imaging[7][8][9]. Recentnumericalwork[10][11]shows
s
c that a self seeded hard X-ray FEL with LCLS-II like pa- Undulator:
si rametersina200-mpermanentmagnetundulatorhasthe
y capabilityofreachingTWlevelpulseswiththelongitudi- Undulator Period λw 20 mm 18 mm
h nalandtransversecoherencenecessaryforcoherentX-ray Undulator Parameter (RMS) aw 2.263 2.263
p imaging [12]. Spatial constraints, an increase in tunabil- Magnetic Gap g 7.2 mm 7.2 mm
[ Integrated Quad Field B 4.5 T 4.5 T
ity and a longer machine lifetime have since led towards q

1 consideringusingsuperconductingtechnologyfortheun-
Radiation:
v dulator design rather than permanent magnets [13].
8
In this study we present the results of tapering opti- Photon Energy E 4keV 4keV
8 γ
mization for the cases of two different superconducting Peak radiation power input P 5 MW 5MW
9 seed
5 technologies, NbTi and Nb3Sn and assess the possibil- Seed laser size σr 31 µ m 31 µ m
0 ity of achieving TW levels of power in a 140 m undula- Rayleigh Range ZR 10 m 10 m
. tor with periodic break sections. We study the impact
1
of changing the undulator section length on the perfor- FEL:
0
5 manceforbothNbTiandNb3Sn. WethenpresentGEN-
FEL parameter ρ 7.67 ×10−4 7.43 ×10−4
1 ESIS simulation results for the NbTi case with the opti-
FEL 3-D gain length L3−D 1.25 m 1.16 m
: malchoiceofsectionlength. Finallywediscusstheprob- g
v Fresnel Parameter F 8 8.6
lem of particle detrapping as a major source of perfor- d
i
X mance degradation in tapered FELs, and propose some
r ideas to improve on current tapering designs by increas-
a
ing the trapping and the extraction efficiency in the ta- Nb3Snaredescribedintable1wherethebeamenergyis
pered section of the undulator. chosen in each case to generate 4keV photons.
Thephysicalsystemstudiedisa140mundulatorcom- The tapering optimization method used is the one de-
posed of a 20 m SASE section followed by a a 120 m ta- scribed in Ref. [11] with a transversely parabolic and
pered undulator with sections 1 to 3 m in length. The longitudinally uniform electron beam distribution. The
SASE and tapered sections are separated by a self seed- advantagesofusingatransverselyparabolicbeamasop-
ing chicane which delivers a 5 MW monochromatic seed posed to a Gaussian in a tapered X-FEL are described
at a photon energy of 4keV. We assume the undulator in detail in Ref. [10]. The quadrupole gradient is kept
modules to be separated in all cases by 0.5 m break sec- constant for simplicity and is set to achieve an average
tionswhereweinstallthefocusingquarupolesinaFODO β function of β =12 m throughout the undulator. The
av
configuration. ThesystemparametersforbothNbTiand effect of varying the electron beam size by changing the
2
2.0 Continuoustaper 2.0 Steptaper 0 2.0
NbTi
(cid:64)(cid:68)PTWrad011...505 L123.s5ec(cid:64)m(cid:68)(cid:61)1 (cid:64)(cid:68)PTWrad011...505 (cid:68)(cid:144)(cid:64)(cid:37)(cid:68)KK(cid:45)(cid:45)105 (cid:64)(cid:68)PTW11..05 NPMb3USn
(cid:45)15 0.5
0.0 0.0
0 20 40 60 80 100 120 0 20 40 60 80 100 120 (cid:45)20 0.0
z(cid:64)m(cid:68) z(cid:64)m(cid:68) 0 20 40 60 80 100 120 0 20 40 60 80 100 120
1.05 1.00 z(cid:64)m(cid:68) z(cid:72)m(cid:76)
1.00
0.98
FIG. 2: Time independent GENESIS simulation results for
(cid:144)aaww,000..9905 Σ(cid:144)Σee,00.96 the optimized taper profile and evolution of the FEL radia-
0.94 tionpowerforNbTiandNb3Snsuperconductingundulators.
0.85
0.92 We also plot for comparison the case of a permanent maget
0.80
0 20 40 60 80 100 120 0 20 40 60 80 100 120 undulator of 2.6 cm period and undulator parameter a =
z(cid:64)m(cid:68) z(cid:64)m(cid:68) w0
1.77 as discussed in the text.
FIG. 1: Power evolution comparison for different undulator
sectionlengthsL separatedbyperiodic0.5mbreaks. The
sec
simulations are time independent and use the Nb3Sn param- overall extraction efficiency. Thus we determine that to
eters listed in table 1 starting after the self seeding chicane obtain a reasonable filling factor and large output power
(z=0). The transverse beam size is normalized to the input a 1.5 m section length is the optimal choice for the un-
beam size σ and the undulator parameter is normalized to dulator design.
e,0
the initial value a .
w0
B. FEL evolution with 1.5 m undulator sections
quadrupole focusing strength will further increase the
output power as discussed in Ref. [11] however this op-
Forouroptimalchoiceofsectionlengthtime,indepen-
tion will not be examined in this work.
dent tapering optimizations produce the magnetic field
profiles displayed in Fig. 2. In both cases the opti-
mal functional form of the tapering law is very close to
II. SIMULATION RESULTS quadratic a (z) ∼ a (cid:2)1−c(z−z )2(cid:3) with varying ta-
w w0 0
per strengths c and a start location around z ∼ 10 m
0
A. Section length study after the self-seeding chicane. The quadratic scaling can
be understood from the 1-D theory of tapered FELs [1].
Inthissectionweinvestigatetheeffectoftheundulator The change in energy for a resonant electron is deter-
section length on the tapered FEL performance. Since mined by a z dependent poderomotive gradient:
the ideal tapering profile is a smoothly varying function
of z we expect that a longer section length limits the
dγ2 eE(z)a (z)
performance since shorter sections better approximate a r =− w sinΨ (1)
continuousmagneticfieldprofile. Furthermorethetrans- dz mc2 R
verse beam envelope oscillations increase for larger sec-
whereforsimplicityweassumetheresonantphaseΨ
tion lengths ∆β2/β2 = (β L)/(β2 −L2) further de- R
av av av to be constant, E(z) is the magnitude of the growing
grading the FEL performance as shown in Fig. 1 for the
radiation field and a (z) is the RMS magnitude of the
continuous tapering case. We find that for the same av- w
tapered magnetic field. The FEL will radiate at a wave-
erage beta function β as the section length becomes
av length λ if the following z dependent resonance condi-
larger than the gain length the maximum output power s
tion is satisfied:
is significantly reduced as illustrated in Fig. 1. Undula-
torsectionsof1-2mshowamarginalimprovementinthe
outputpowerwhengoingfromthecontinuoustothestep γ2(z)= λw (cid:0)1+a (z)2(cid:1) (2)
taper which can be attributed to the additional phase r 2λ w
s
slippage of the electrons with respect to the ponderomo-
tive wave in the step-wise case vs the continuous case. If we assume that the radiative process in the ta-
On the other hand in the 3 m case the maximum power pered section of the undulator is mostly due to coher-
decreasesfrom1.3TWto0.5TW.Thisisaresultofthe entemission, withtheelectricfieldevolvingaccordingto
morepronouncedoscillationsinthebeamβ functionand E(z)∝z,tosatisfysimultaneouslyEq. 1-2themagnetic
the phase mismatch between undulator sections as the fielda (z)mustdecreasequadraticallywithz. Withthe
w
discontinuity in magnetic field value is larger for longer optimal taper profiles for NbTi and Nb3Sn we obtain
sections. The 1 m undulator sections are thus optimal extraction efficiencies of 6.08% and 5.89% respectively,
for minimizing oscillations in the beam size and approx- and output powers over a factor of 75 larger than the
imating a smooth taper profile however the filling factor saturation power in both cases (see Fig. 2). Applying
for such short sections limits the interaction length and the same tapering optimization for a Permanent Magnet
3
(cid:45)8 50 0.90(cid:230) 6
(cid:45)9 40 0.85 5
(cid:56)(cid:64)(cid:72)(cid:76)(cid:45)(cid:68)(cid:60)LogRenz1(cid:45)(cid:45)(cid:45)111210 (cid:89)(cid:64)(cid:68)Degreer123000 (cid:144)NNtrapped0000....67785050 (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) Φ(cid:64)Π(cid:68)Mod2rad1234
(cid:45)13 20 40 60 80 100 120 0 20 40 60 80 100 120 0.60 40 60 80 100 (cid:230)12(cid:230)0 00 20 40 60 80 100 120
z(cid:64)m(cid:68) z(cid:64)m(cid:68) z(cid:64)m(cid:68) z(cid:64)m(cid:68)
FIG.3: Realpartoftheelectronbeamrefractiveindex(left) FIG.4: RadiationPhase(onaxis)andtrappingfractionfrom
and on axis resonant phase (right) for NbTi superconduct- time independent GENESIS simulations of NbTi. The area
ing undulator parameters from time independent GENESIS after 40 m is when the gain is so low that the phase velocity
simulation. of light is almost c thus the radiation phase is constant
Undulator (PMU) with a 26 mm period, beam energy
electrons [1]. From Fig. 3 and 4 we see that as the reso-
of 6.7 GeV and an RMS undulator parameter aw0=1.77 nantphaseincreasesfromzerotoψ =20o fromz=10-50
r
wereporta∼60%reductioninoutputpowercompared
m, the phase shift in the ponderomotive bucket causes
to the superconducting case with PPMU = 1.1 TW at
rad ∼20% of the particles to detrap. This can be mitigated
the undulator exit. As mentioned previously this is one
by introducing phase shifters in the break sections be-
of the motivations for moving from permanent magnets
tween 10-50 m and is an effect which will be examined
to superconducting technology in the design of Terawatt
quantitatively in future studies.
level X-FELs such as LCLS-II. We also note that for the
superconducting case a reduction in the average β func-
tion will further increase the efficiency while this cannot D. Time dependent effects. Sideband growth and
be done in the permanent magnet case. sideband induced detrapping
Ashasbeenpointedoutinpreviouswork[16][11][10],
C. Limits on optimization due to parasitic effects
the performance of tapered hard X-ray FELs is also lim-
itedbyparticledetrappingduetotimedependenteffects.
The FEL process is dominated by the effects of re- Shot-noise fluctuations in the electron beam current in-
fractive guiding and particle detrapping in the tapered ducemodulationsintheradiationbeamlongitudinalpro-
section of the undulator. In Fig. 3 we show the evolu- file. This can cause particle detrapping from the top
tion of the real part of the refractive index [14] (in the and bottom of the ponderomotive bucket as the modu-
resonantparticleapproximation)andtheresonantphase lated radiation field slips through the beam and the par-
given by the expressions: ticles experience fluctuations in the bucket height. Fur-
thermore, the resonant interaction between the FEL pri-
ω2 r2 a (cid:28)cosΨ (cid:29) marywavek0 andtheelectronsynchrotronmotiondrives
(cid:60)(n−1)= p0 b,0 w [JJ] R (3) a sideband instability which amplifies parasitic modes
ω2 r2 2|a | γ
s b s R at wavenumbers k0 ± Ωs where Ωs is the synchrotron
wavenumber:
|a(cid:48) (z)|
sinΨ (z)=χ w (4)
R E(z) eE(z)a (z)
Ω2(z)= w (5)
√ s m c2k
whereχ=(2∗me∗c2/e)(λ /2λ )(1/ 2[JJ])isacon- e w
w s
stant independent of z and the other symbols have their The growth of the sidebands not only increases the
usual meaning. From these expressions we see that if bandwidth of the FEL signal but also induces further
we want to maintain strong optical guiding, the bunch- particle detrapping [17] [18].
ing must be preserved during the tapered section of the
undulator. However as the radiation field grows the ef- Timedependenteffectsareanalysedbysimulatingthe
fect of optical guiding will decrease as 1/|a | inducing a beam parameters in table I for a longitudinally uniform
s
self-limiting mechanism on the growth of the field [15]. bunchof37.5fs. ThesimulationresultsareshowninFig.
Furthermore, in order to provide the largest ponderomo- 5 for a longer undulator of 160 m. The simulation was
tive gradient to induce the greatest energy loss in the doneforalongerundulatortoemphasizethepeakpower
electrons, it is desirable to increase the resonant phase saturation in the final sections of the tapered undulator.
throughoutthetaperedsection. Againherewefindaself- From Fig. 5 we can see that there is a difference in peak
limiting mechanism which limits how much one should output power (averaged over time) compared with the
increase Ψ , since a larger Ψ results in a smaller pon- time independent results, namely 1.78 TW in the time
R R
deromotive bucket creating a tradeoff between the pon- independentcasecomparedto1.55TWafter120mtime
deromotivepotentialstrengthandthenumberoftrapped dependent. The drop can be attributed to the growth of
4
0.6 section of the undulator [19] [20]. In the final section of
(cid:72)(cid:76)PTW011...505 BunchingFactor00000.....12345 tffirheeelqduu.neIndtcumylaitigsohrtsm(bzea>lpl1od0sus0iembl)teotthotehcveoamsraipatetunirosanattiioennftohoreftshtyhinsecehfferleeocctttrrboinyc
introducing a cubic or quartic term to the taper profile.
0.0 0.0
0 50 100 150 0 50 100 150 This option will also be explored in the future.
z(cid:64)m(cid:68) z(cid:64)m(cid:68)
4.0
4
(cid:45)3(cid:87)(cid:144)(cid:64)(cid:68)kc10sw123 Σ(cid:144)Σradrad,012233.....50505 We have preseInItIe.d aCnOuNmCerLicUalSsItOuNdy of tapering op-
00 50 100 150 1.00 50 100 150 timization for a self seeded hard X-ray FEL with super-
z(cid:64)m(cid:68) z(cid:64)m(cid:68) conductingundulatorparameters. Thisworkhasdemon-
stratednumericallythatpeakpowerlevelsover1TWcan
FIG.5: TimedependentGENESISsimulationresultsforthe
beachievedina120mundulatorwithbreaksectionsand
NbTi case parameters in Table 1. The beam is 37.5 fs long
anoptimizedtaperprofilefortwoseparatesuperconduct-
with a flattop longitudinal profile.
ing technologies: NbTi and Nb3Sn. We demonstrated
a 60% increase in output power for the superconducting
z(cid:61)160m
technologywhencomparedwithapermanentmagnetde-
108 z(cid:61)120m sign. We have also outlined the effects which limit the
z(cid:61)80m growth of the power in the tapered undulator and have
found that particle detrapping is the main obstacle to
(cid:64)(cid:68)a.u. 106 achieving larger extraction efficiencies. The two causes
(cid:72)Λ(cid:76)P of particle detrapping we have discussed are due to an
104 increase of the resonant phase after the exponential gain
regime and sideband induced detrapping due to time de-
pendenteffects. Tomitigatethesetwoeffectswepropose
100
to study applying phase shifters at the location of ini-
(cid:45)10 (cid:45)5 0 5 10 tialsaturationandintroducingafasterterminthetaper
(cid:68)Λ(cid:144)Λ(cid:64)(cid:42)10(cid:45)3(cid:68) profile to reduce the sideband growth by changing the
synchrotron frequency faster in the final sections of the
FIG. 6: Growth of the FEL synchrotron sidebands at differ-
undulator.
ent locations in the undulator. The growth of the sidebands
intensifiedinthelatersectionoftheundulatorwherethesyn-
chrotron frequency is changing less rapidly (see Fig. 5).
ACKNOWLEDGMENT
thesidebandsshownin aplotofthespectrumatvarious The authors would like to thank Dr. G. Marcus and
z locations (see Fig. 6). This growth can be mitigated J. Duris for useful discussions. This work was supported
bychangingthesynchrotronfrequencyalongthetapered by U.S. D.O.E. under Grant No. de-sc0009983.
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