• Aucun résultat trouvé

Stabilizing the intensity for a Hamiltonian model of the FEL

N/A
N/A
Protected

Academic year: 2021

Partager "Stabilizing the intensity for a Hamiltonian model of the FEL"

Copied!
5
0
0

Texte intégral

(1)

HAL Id: hal-00292542

https://hal.archives-ouvertes.fr/hal-00292542

Submitted on 1 Jul 2008

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

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�

(2)

R.Ba helard

a

C.Chandre

a

D.Fanelli

b

X.Leon ini

a

M.Vittot

a

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 iatedwiththe

N

parti- les,andapotentialterma ountingfortheself- onsistent

intera 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

(3)

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),with

N = 10000

parti lesand

H

N

= 0

,

P

N

= 10

7

.

Right :Snapshotof the

N

parti les at

t = 800

,with

N = 10000

. Thegrey points orre spond tothe haoti parti les, the darkones

totheparti 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 ofthe

N

parti les, whereas

(φ, I)

standrespe tivelyfor the onju-gatephaseandintensityoftheradiationeld.Furthermore,

therearetwo onservedquantities:

HN

andthetotal mo-mentum

PN

= I +

P

j

pj

.We onsiderthedynami sgiven byHamiltonian(1)ona

2N

-dimensionalmanifold(dened by

HN

= 0

and

PN

= ε

where

ε

isinnitesimallysmall).

Startingfromanegligiblelevel(

I ≪ N

and

pj

= 0

),the intensitygrowsexponentiallyandeventuallyrea hesa

sat-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 onsideredtoberegularifthe

Lyapunovexponentissmallerthan

0.025

(whileitis typi- allyoforder

1

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,thusit

anbedes 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

),

(2)

wheretheintera tionterm

h(t)

isderivedfromdedi ated simulationsof theoriginal self- onsistent

N

-body Hamil-tonian(1).Inthesaturatedregime,

h(t)

ismainlyperiodi . Inparti ular,arenedFourieranalysis showsthat it an

bewrittenas:

h(t) = 2

r

I(t)

N

e

iφ(t)

≈ [F + αe

1

t

+ βe

−iω

1

t

]e

iΩt

,

(3)

where

Ω = −0.685

standsforthewavevelo ityand

ω1

=

1.291

forthefrequen yoftheos illationsof theintensity. As for the amplitudes, the Fourieranalysis provides the

following values :

F = 1.5382 − 0.0156i

,

α = 0.2696 −

0.0734i

and

β = 0.1206 + 0.0306i

.

Hamiltonian(2)resultsfromaperiodi perturbationof

apendulumdes ribedbytheintegrableHamiltonian

H0

H0

=

p

2

2

− |F | cos(θ + Ωt + φF

),

where

F = |F |e

F

.Thelinearfrequen yofthispendulum

is

p|F | ≈ 1.240

whi h is very lose to the frequen y of

thefor ing

ω1

.Thereforea haoti behaviourisexpe ted whentheperturbation isaddedevenwithsmall valuesof

theparameters

α

and

β

.

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 estoasetofinvarianttoriinthis

mean-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 the

ma 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 rossesthehyperplane

P

j

sin (φ + θj) = 0

,i.e.

dI/dt =

0

.Fromthefulltraje tory,wefollowagivenparti le(an in-dex

j

)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

(4)

in-−3

−2

−1

0

1

2

3

−3

−2

−1

0

1

2

3

4

θ

+

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), whenthe

parti 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, if

R ∈]0, 1[

, the periodi orbitis alled ellipti (and isin general stable); if

R < 0

or

R > 1

it is hyperboli ; and

if

R = 0

and

R = 1

, it is paraboli while higher order

expansionsgivethestabilityofsu 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 lparameter

λ

.Lowerleftpanel: Poin arése tionofthe ontrolledHamil tonian(4)ofatest-part i le.

Lower rightpanel:Snapshotofthephase-spa eoftheparti lesfor

Hamil tonian(5),with

N = 10000

and

λ = 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) where

ω1

orrespondstotheresonantfrequen yofthe en-tralperiodi orbitofthema ro-parti le,and

k = 10

.

Then,theamplitude

λ

istunedaround

0

,andtheresidue

R

ofthe entral periodi orbit

O1

istra ked (seeFig.3): when the latter goesabove

1

, it means that the entral orbitturnedhyperboli ,andthat haosmighthavelo ally

appeared. This o urs for values of

|λ|

larger than

λc

0.07

. An inspe tion of thePoin arése tion onrms this predi tion,asthereisnomoreislandwitha entralperiodi

orbitofperiod

2π/ω1

.A tually,nomoreellipti island an bedete ted,apartfromthebordersofthewaterbag:thus,

thoughthehyperboli ityof

O1

onlyguaranteeslo al haos, theresonan eisnowfully haoti ,whi h emphasizesthat

the study of a few periodi orbitsmay give quite global

informationonthedynami s.

This ontrolstrategy anthenbegeneralizedtothe

self- onsistent intera tion, by introdu ing a test-wavesimilar

(5)

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: Ratio

N

m

/N

ofparti leswithregulartraje tories,forHamil tonian (5),asafun tionofthe ontro lparameter

λ

.

∆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 about

6%

for

λ = λc

( f Fig.3). As for the wave,theintensity rapidly stabilizes,after the initial

growth.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, though

not as mu h as for

λ = λc

(see Fig.4). Then, a stronger

test-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 ludeupontherobustnessof

theproposed 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.

Referen es

[1℄R.Bonifa io,etal.,RivistadelNuovoCimento3 ,1(1990)

[2℄J.L.Tennyson,J.D.Meiss,andP.J.Morrison, Physi a D71 ,1

(1994)

[3℄A.Antoniazzi,Y.Elskens,D.FanelliandS.Ruo,Europ.Phys.

J.B50 ,603(2006)

[4℄S.I.Tsunoda,J.H.Malmberg,Phys.Rev.Lett.49 ,546(1982)

[5℄G.J.Morales,Phys.Fluids23(1980)

[6℄J.M.Greene,J.Math.Phys.20 ,1183(1979)

[7℄J.R.Cary,J.D.Hanson,Phys.Fluids29 ,2464(1986)

[8℄R.Ba helard,C.Chandre,X.Leon ini,Chaos16 ,023104(2006)

[9℄R.Ba helard,A.Antoniazzi,C.Chandre,D.Fanelli,X.Leon ini,

M.Vittot,Eur.Phys.J.D,42 ,125(2007)

[10℄R.S.Ma Kay,Nonlinearity5 ,161(1992)

Figure

Figure 1. Left : Normalized intensity I/N from the dynamis of
Figure 2. Left : Poinaré setion of a test-partile, desribed by
Figure 4. Left : Normalized intensity I/N for Hamil tonian (5). Right : Ratio N m /N of partiles with regular trajetories, for Hamil tonian (5), as a funtion of the ontro l parameter λ

Références

Documents relatifs

modifies the ozone distribution and consequently the opacity between 200 and 300 nm, the predicted change in the trace species photodissociation rate and

Information access control programs are based on a security policy

PROPOSITION 9.1.. Suppose that f is P'-linear. Assume that each basic interval is f-covered by some basic interval different from itself and that there is a basic interval JQ such

Geometry with its formal, logical and spatial properties is well suited to be taught in an environment that includes dynamic geometry software (DGSs), geometry automated theorem

Madan and Yor [25] provided a sufficient condition on the marginals, under which the Azéma-Yor embedding stopping times corresponding to each marginal are automatically ordered, so

The wave equation (4) is solved with a time step of 1 µs, and the minimum and acoustic pressure in the domain is obtained, as shown in Figure 2 and Figure 3 for the baffles

We reuse the following ontologies: Semantic Sensor Network 4 for sensors and observations, Geonames 5 to de- fine the location of the systems and the stations based on their

The theory of classical realizability (c.r.) solves this problem for all axioms of ZF, by means of a very rudimentary programming language : the λ c -calculus, that is to say the