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Stabilizing the intensity for a Hamiltonian model of the
FEL
Romain Bachelard, Cristel Chandre, Duccio Fanelli, Xavier Leoncini, Michel
Vittot
To cite this version:
Romain Bachelard, Cristel Chandre, Duccio Fanelli, Xavier Leoncini, Michel Vittot. Stabilizing the
intensity for a Hamiltonian model of the FEL. Nuclear Instruments and Methods in Physics Research
Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, Elsevier, 2008, 593
(1-2), pp.94-97. �10.1016/j.nima.2008.04.050�. �hal-00292542�
R.Ba helard
a
C.Chandrea
D.Fanellib
X.Leon inia
M.Vittota
a
CentredePhysiqueThéorique,CNRSLuminy,Case907,F-13288MarseilleCedex9,Fran e
1
b
Theoreti alPhysi sGroup,S hoolofPhysi sandAstronomy,TheUniversityofMan hester,Man hester,M139PL,UK
Abstra t
Theintensityofanele tromagneti waveintera tingself- onsistentlywithabeamof hargedparti les,asinaFreeEle tronLaser,
displayslargeos illationsduetoanaggrega teofparti les, alledthema ro-parti le.Inthisarti le,weproposeastrategytostabilize
theintensitybydestabilizingthema ro-parti le.Thisstrategyinvolvesthestudyofthelinearstabilityofaspe i periodi orbitof
amean-eldmodel.Asa ontrolparameter-theamplitudeofanexternalwave-isvaried,abifur ationo urinthesystemwhi h
hasdrasti ee tsontheself- onsistentdynami s,andinparti ular,onthema ro-parti le.Weshowhowtoobtainanappropriate
tuningofthe ontrolparameterwhi hisabletostronglyde reasetheos illationsoftheintensitywithoutredu ingitsmean-value.
Keywords: Wave/pa rti le intera tions,Controlof haos,Hamiltonianapproa h
PACS:94.20.wj,05.45.Gg,11.10.Ef
1. Introdu tion
Theampli ationofaradiationeldbyabeamof
par-ti lesandtheradiatedeld,asito ursinaFreeEle tron
Laser, anbemodelled withintheframeworkofa
simpli-edHamiltonian[1℄.The
N + 1
degreeoffreedom Hamil-toniandisplaysakineti part,asso iatedwiththeN
parti- les,andapotentialterma ountingfortheself- onsistentintera tionbetweentheparti lesandthewave.Thus,
mu-tualparti lesintera tionsarenegle ted,whileanee tive
ouplingisindire tlyprovidedthroughthewave.
Thelineartheorypredi ts[1℄fortheamplitudeofthe
ra-diationeldalinearexponentialinstability,andthenalate
os illatingsaturation.Inspe tionoftheasymptoti
phase-spa esuggeststhatabun hofparti lesgetstrappedinthe
resonan eandformsa lumpthatevolvesasasingle
ma ro-parti lelo alized inphasespa e.Theuntrappedparti les
are almost uniformly distributed between two os illating
boundaries,andformtheso- alled haoti sea.
Furthermore,the ma ro-parti le rotates around awell
dened entre in phase-spa eand this pe uliar dynami s
isshowntoberesponsibleforthema ros opi os illations
observedfortheintensity[2,3℄.It anbetherefore
hypoth-1
UnitéMixtedeRe her he(UMR6207)duCNRS,et des
univer-sitésAix-MarseilleI,Aix-MarseilleII etduSudToulon-Var.
Labo-ratoire aliéà laFRUMAM(FR 2291).Laboratoir edeRe her he
ConventionnéduCEA(DSM-06-35).
esizedthat asigni antredu tionin theintensity
u tu-ations anbegainedbyimplementingadedi ated ontrol
strategy,aimedatreshapingthema ro-parti leinspa e.
Thedynami s anbealsoinvestigatedfroma
topologi- alpointofview,bylookingatthephasespa estru tures.
Intheframeworkofasimpliedmeanelddes ription,i.e.
theso- alledtest-parti lepi ture wheretheparti les
pas-sivelyintera twithagivenele tromagneti wave:The
tra-je toriesoftrappedparti les orrespondtoinvarianttori,
whereas unbounded parti les evolve in a haoti region
ofphase-spa e.Then,thema ro-parti le orrespondstoa
densesetofinvarianttori.
Forexample,astati ele tri eld[4,5℄ anbeusedto
in- reasetheaveragewavepower.Whilethe haoti parti les
are simply a eleratedby the external eld, the trapped
onesareresponsiblefortheampli ationoftheradiation
eld.Someshiftintherelativephasebetweentheele trons
andtheponderomotivepotential analsobeimplemented
toimproveharmoni generation.
Inthispaper,weproposetoperturbthesystemwith
ex-ternal ele tromagneti waves. Ourstrategyis to stabilize
theintensityofthewave,by haotizingthepartof
phase-spa eo upiedbythema ro-parti le.Tomodifythe
topol-ogyofphasespa e,anadditionaltestwaveisintrodu ed,
whoseamplitudeplaystheroleofa ontrolparameter.The
residuemethod [6,7,8℄is implemented to identify the
im-portant lo al bifur ationshappening in the systemwhen
0
50
100
150
10
−4
10
−3
10
−2
10
−1
10
0
t
I
−2
0
2
−4
−2
0
2
θ
p
Figure 1. Left : Normalized intensity
I/N
from the dynami s of Hamil tonian(1),withN = 10000
parti lesandH
N
= 0
,P
N
= 10
−
7
.
Right :Snapshotof the
N
parti les att = 800
,withN = 10000
. Thegrey points orre spond tothe haoti parti les, the darkonestotheparti lesinthema ro-parti le.
aspe i periodi orbit.Thoughrstdevelopedina
mean-eldapproa h,ourstrategyprovestoberobustasthe
self- onsisten yofthewaveisrestored.
2. Dynami sofasingleparti le
The dynami s of the wave parti le intera tion, as
en- ountered in the FEL, an bedes ribedby the following
N
-bodyHamiltonian[1℄:HN
({θj
, pj}, φ, I) =
N
X
j=1
p
2
j
2
− 2
r I
N
N
X
j=1
cos (φ + θj
).
(1)It is omposed of a kineti ontribution and an
intera -tiontermbetweentheparti lesandtheradiationeld:the
(θj, pj
)
arethe onjugatephaseandmomentum oftheN
parti les, whereas
(φ, I)
standrespe tivelyfor the onju-gatephaseandintensityoftheradiationeld.Furthermore,therearetwo onservedquantities:
HN
andthetotal mo-mentumPN
= I +
P
j
pj
.We onsiderthedynami sgiven byHamiltonian(1)ona2N
-dimensionalmanifold(dened byHN
= 0
andPN
= ε
whereε
isinnitesimallysmall).Startingfromanegligiblelevel(
I ≪ N
andpj
= 0
),the intensitygrowsexponentiallyandeventuallyrea hesasat-uratedstate hara terizedbylargeos illations,asdepi ted
in Fig. 1.Con erning the parti les dynami s, more than
half ofthem aretrappedbythewave[9℄andformthe
so- alledma ro-parti le(seeFig.1).Theremainingparti les
experien e an errati motion within an os illating water
bag,termed haoti sea,whi hisunboundedin
θ
ontrary tothema ro-parti le.Inordertoknowhowmanyparti leshavearegular
mo-tion,we omputeniteLyapunovexponentsforea h
tra-je tory(theparti lesarethen onsideredasevolvinginan
external eld). The Lyapunov exponents were omputed
overatimeperiodof
T = 300
(on ethe stationarystate rea hed),andatraje toryis onsideredtoberegulariftheLyapunovexponentissmallerthan
0.025
(whileitis typi- allyoforder1
inthe haoti sea).In order to get adeeperinsight into the dynami s, we
onsider the motion of asingle parti le. For large
N
, we assumethatitsinuen e onthewaveisnegligible,thusitanbedes ribedasapassiveparti leinanos illatingeld.
Themotionofthistest-parti leisdes ribedbytheoneand
ahalfdegreeoffreedomHamiltonian:
H1p(θ, p, t) =
p
2
2
− 2
r
I(t)
N
cos (θ + φ(t))
=
p
2
2
− Re(h(t)e
iθ
),
(2)wheretheintera tionterm
h(t)
isderivedfromdedi ated simulationsof theoriginal self- onsistentN
-body Hamil-tonian(1).Inthesaturatedregime,h(t)
ismainlyperiodi . Inparti ular,arenedFourieranalysis showsthat it anbewrittenas:
h(t) = 2
r
I(t)
N
e
iφ(t)
≈ [F + αe
iω
1
t
+ βe
−iω
1
t
]e
iΩt
,
(3)
where
Ω = −0.685
standsforthewavevelo ityandω1
=
1.291
forthefrequen yoftheos illationsof theintensity. As for the amplitudes, the Fourieranalysis provides thefollowing values :
F = 1.5382 − 0.0156i
,α = 0.2696 −
0.0734i
andβ = 0.1206 + 0.0306i
.Hamiltonian(2)resultsfromaperiodi perturbationof
apendulumdes ribedbytheintegrableHamiltonian
H0
H0
=
p
2
2
− |F | cos(θ + Ωt + φF
),
where
F = |F |e
iφ
F
.Thelinearfrequen yofthispendulum
is
p|F | ≈ 1.240
whi h is very lose to the frequen y ofthefor ing
ω1
.Thereforea haoti behaviourisexpe ted whentheperturbation isaddedevenwithsmall valuesoftheparameters
α
andβ
.The Poin aré se tions (strobos opi plot performed at
frequen y
ω1
) ofthe test-parti le (see Fig.2) reveal that thema ro-parti leredu estoasetofinvarianttoriinthismean-eldmodel.Conversely,the haoti seaislledwith
seeminglyerrati traje toriesof parti les, apartfrom the
upperandlowerboundaries,wherethetraje toriesare
sim-ilarto therotational ones of theunperturbed pendulum.
Therotationofthema ro-parti le andtheos illationsof
thewaterbagarevisualizedbytranslating ontinuouslyin
timethestrobos opi plotofphasespa e.
Thema ro-parti leisorganizedarounda entral
(ellip-ti )periodi orbitwith rotationnumber
1
.Theperiodof os illationsof theintensity is the sameas the one of thema ro-parti lewhi h indi atestherole playedbythis
o-herentstru tureintheos illationsofthewave.
Thus, in the test-parti le model, the ma ro-parti le is
formedbyparti leswhi haretrappedontwo-dimensional
invariant tori. This pi ture an be extended to the
self- onsistentmodel,ifone onsiderstheproje tionofatra
je -tory
(φ(t), I(t), {θj(t), pj(t)}j
)
inthe(θ, p)
plane,ea htime it rossesthehyperplaneP
j
sin (φ + θj) = 0
,i.e.dI/dt =
0
.Fromthefulltraje tory,wefollowagivenparti le(an in-dexj
)andplot(θj, pj
)
ea htimethefulltraje tory rosses thePoin arése tion.Thetrappedparti lesappeartobe onnedtodomains
of phase-spa e mu h smaller than the one of the
in-−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
3
4
θ
+
Ω
t [2
π
]
p
−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
3
4
θ + φ [2π]
p
Figure 2. Left : Poin aré se tion of a test-parti le, des ribed by
Hamil tonian(2).Theperiodi orbitwithrotationnumber
1
ismarked bya ross. Right:Poin aré se tionof Hamil tonian(1), whentheparti les interse tthe plane
dI(t)/dt = 0
.Thedierenttraje tories arerepresentedbydierentgreylevels.varianttoriofthetest-parti lemodel,althoughthi ker.It
is worthnoti ing that notonlythese gureshavea
simi-laroveralllayout,butthereisadeeper orresponden ein
thestru tureofthema ro-parti le.Forinstan e,both
g-uresshowsimilarresonantislands attheboundaryofthe
regular region. Sin e we saw that the ma ro-parti le
di-re tlyinuen estheos illationsofthewave,thetest
par-ti leHamiltonian(2)servesasa ornerstoneofour ontrol
strategywhi h onsistsin destabilizingtheregular
stru -tureofthema ro-parti leinordertostabilizetheintensity
of thewave.This strategy fo uses onbreaking up
invari-anttorito reshapethema ro-parti le.In orderto a ton
invarianttori,weusethe entral periodi orbitwhi h,as
wehaveseen,stru turethemotionofthema ro-parti le.
3. Residuemethod
The topology of phase spa e an be investigated by
analysing the linear stability of periodi orbits.
Informa-tion on thenature of these orbits(ellipti , hyperboli or
paraboli )isprovidedusing,e.g.,anindi atorlikeGreene's
residue [6,10℄, a quantity that enables to monitor lo al
hanges of stability in a system subje t to an external
perturbation[7,8℄.
Fromtheintegrationoftheequationsofthetangentow
of the system along a parti ular periodi orbit, one an
dedu etheresidue
R
of thisperiodi orbit.Inparti ular, ifR ∈]0, 1[
, the periodi orbitis alled ellipti (and isin general stable); ifR < 0
orR > 1
it is hyperboli ; andif
R = 0
andR = 1
, it is paraboli while higher orderexpansionsgivethestabilityofsu hperiodi orbits.
Sin etheperiodi orbitanditsstabilitydependontheset
ofparameters
λ
,thefeaturesofthedynami swill hange underappositevariationsofsu hparameters.Generi ally,periodi orbits and their (linear or non-linear) stability
propertiesarerobust tosmall hanges ofparameters,
ex- eptatspe i valueswhenbifur ationso ur.Theresidue
method [7,8℄ dete tstherareeventswhere thelinear
sta-bilityofagivenperiodi orbit hangesthusallowingoneto
al ulatetheappropriatevaluesoftheparametersleading
to thepres ribedbehaviourof thedynami s.As a
onse-quen e,thismethod anyieldredu tionaswellas
enhan e-mentof haos.
−0.05
0
0.05
0
2
4
6
λ
R
0
1
2
3
4
5
6
−3
−2
−1
0
1
2
3
θ
p
0
2
4
6
−3
−2
−1
0
1
2
3
θ
p
Figure3.Upperpanel:Residue urveoftheperiodi orbitofrotation
number
1
,asafun tionofthe ontro lparameterλ
.Lowerleftpanel: Poin arése tionofthe ontrolledHamil tonian(4)ofatest-part i le.Lower rightpanel:Snapshotofthephase-spa eoftheparti lesfor
Hamil tonian(5),with
N = 10000
andλ = λ
c
(sameinitial onditions asforFig.1).4. Destru tionofthema ro-parti le
Theresiduemethod anbeused toenlargethe
ma ro-parti le in the haoti sea [9℄, whi h results in its
stabi-lization:then,theu tuationsoftheintensityofthewave
eventually ollapse.
Nonetheless,it an alsobeusedto redu ethe
aggrega-tionpro ess fortheparti les, bydestroyingthe invariant
toriformingthema ro-parti le:su ha ontrol,aswewill
see, tendsto limittheu tuations in theintensity of the
wave.Here,weimplementthis ontrolwithanextra
test-wave,whoseamplitudeisusedasa ontrolparameter.The
Hamiltonian of the mean-eld model with atest-waveis
hosenas:
H
1p
c
(θ, p, t; λ) = H1p(θ, p, t) − 2λ cos (k(θ − ω1t)),
(4) whereω1
orrespondstotheresonantfrequen yofthe en-tralperiodi orbitofthema ro-parti le,andk = 10
.Then,theamplitude
λ
istunedaround0
,andtheresidueR
ofthe entral periodi orbitO1
istra ked (seeFig.3): when the latter goesabove1
, it means that the entral orbitturnedhyperboli ,andthat haosmighthavelo allyappeared. This o urs for values of
|λ|
larger thanλc
≈
0.07
. An inspe tion of thePoin arése tion onrms this predi tion,asthereisnomoreislandwitha entralperiodiorbitofperiod
2π/ω1
.A tually,nomoreellipti island an bedete ted,apartfromthebordersofthewaterbag:thus,thoughthehyperboli ityof
O1
onlyguaranteeslo al haos, theresonan eisnowfully haoti ,whi h emphasizesthatthe study of a few periodi orbitsmay give quite global
informationonthedynami s.
This ontrolstrategy anthenbegeneralizedtothe
self- onsistent intera tion, by introdu ing a test-wavesimilar
0
50
100
150
200
250
0
0.5
1
1.5
t
I/N
0
0.02
0.04
0.06
0.08
0.1
0
0.2
0.4
0.6
0.8
λ
N
m
/N
∆
I
Figure4.Left:Normalizedintensity
I/N
forHamil tonian(5).Right: RatioN
m
/N
ofparti leswithregulartraje tories,forHamil tonian (5),asafun tionofthe ontro lparameterλ
.∆I
orrespondstothe meanu tuationsofthe intensity.H
N
c
({θj
, pj}, φ, I; λ) =
HN
({θj
, pj}, φ, I)
−λc
X
j
cos (k(θj
− ω1t)),
(5)Though the ontrol dedi ated to the mean-eld model
lostsomeofitsrelevan e,duetothepresen eintheoriginal
model of the feedba k of the ele trons on the wave, the
ontrolleddynami softheparti lesisqualitativelysimilar
totheoneobtainedinthemean-eldframework.Afteran
initialgrowthofthewave,theparti lesorganizethemselves
inawaterbag,butonlyfewofthemstilldisplayaregular
traje tory:from
65%
intheun ontrolledregime,theratio has ollapsed to about6%
forλ = λc
( f Fig.3). As for the wave,theintensity rapidly stabilizes,after the initialgrowth.Therelevan eofa ontrolbasedonamodi ation
ofthema ro-parti leisthus onrmed.Thisisinagreement
withtheexperimentalresultsofDimonte[11℄,whoobserved
thatone oulddestroytheos illationsoftheintensitywith
unstabletest-waves.
Finally, let us note that ontrolling with a weaker
test-wave (
λ ≤ 0.07
) only partially haotizesthe ma ro-parti le :the intensityof thewavestillstabilizes, thoughnot as mu h as for
λ = λc
(see Fig.4). Then, a strongertest-wave does not provide a better ontrol, due to the
reationofnewresonan eislandsinthetest-parti lephase
spa eforlarger
λ
.Con lusion
Weproposedinthispaperamethodtostabilizethe
in-tensityofawaveampliedbyabeamofparti les.Thisis
a hievedbydestroyingthe oherentstru turesofthe
parti- lesdynami s.Bystudyingamean-eldversionofthe
orig-inal Hamiltoniansetting andputting forwardan analysis
ofthelinearstabilityoftheperiodi orbit,wewereableto
enhan ethedegreeofmixingofthesystem:Regular
traje -toriesareturnedinto haoti onesastheee tofaproperly
tuned test-wave,whi his externallyimposed.Theresults
arethentranslatedintotherelevant
N
-bodyself onsistent frameworkallowingusto on ludeupontherobustnessoftheproposed ontrolstrategy.
A knowledgements
This work is supported by Euratom/CEA ( ontra t
EUR344-88-1FUAF)andGDRn
◦
2489DYCOEC.We
a -knowledgeusefuldis ussionswithG.DeNinno,Y.Elskens
andtheNonlinearDynami sgroupatCentredePhysique
Théorique.
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