Mathematical and Numerical Methods for Non-linear Beam Dynamics

0.62 MB·

by  W. Herr

Mathematical and Numerical Methods for Non-linear Beam Dynamics
W.Herr
CERN,Geneva,Switzerland
Abstract
Non-lineareffectsinacceleratorphysicsareimportantforbothsuccessfulop-
erationofacceleratorsandduringthedesignstage. Sincebothoftheseaspects
are closely related, they will be treated together in this overview. Some of
themostimportantaspectsarewelldescribedbymethodsestablishedinother
areasofphysicsandmathematics. Thetreatmentwillbefocusedontheprob-
lemsinacceleratorsusedforparticlephysicsexperiments. Althoughthemain
6 emphasis will be on accelerator physics issues, some of the aspects of more
1
generalinterestwillbediscussed. Inparticular,wedemonstratethatinrecent
0
2 years a framework has been built to handle the complex problems in a con-
n sistentform,technicallysuperiorandconceptuallysimplerthanthetraditional
a techniques. Theneedtounderstandthestabilityofparticlebeamshassubstan-
J
tially contributed to the development of new techniques and is an important
0
source of examples which can be verified experimentally. Unfortunately, the
2
documentation of these developments is often poor or even unpublished, in

]
manycasesonlyavailableaslecturesorconferenceproceedings.
h
p
-
c 1 Introductionandmotivation
c
a 1.1 Single-particledynamics
.
s
Theconceptsdevelopedhereareusedtodescribesingle-particletransversedynamicsinrings,i.e. circu-
c
i lar accelerators or storage rings. In the case of linear betatron motion the theory is rather complete and
s
y thestandardtreatment[1]sufficestodescribethedynamics. Inparallelwiththistheorythewell-known
h
concepts such as closed orbit and Twiss parameters are introduced and emerge automatically from the
p
[ Courant–Snyderformalism[1]. Theformalismandapplicationsarefoundinmanytextbooks(e.g.[2,3]).

1 Inmanynewacceleratorsorstoragerings(e.g.theLargeHadronCollider(LHC))thedescription
v of the machine with a linear formalism becomes insufficient and the linear theory must be extended to
1
treat non-linear effects. The stability and confinement of the particles is not given a priori and should
1
4 ratheremergefromtheanalysis. Non-lineareffectsareamainsourceofperformancelimitationsinsuch
5 machines. Non-linear dynamics has to be used in many applications and the basics are described in the
0
literature [4,5]. A reliable treatment is required and the progress in recent years allows us to evaluate
.
1
theseeffects. Averyusefuloverviewanddetailscanbefoundin[6–8].
0
6 In this article, we restrict the treatment to the beam dynamics in rings, since properties such as
1
stabilityofparticlesrestricttheformofthetoolstodescribethem. However,manyoftheconcepts(e.g.
:
v Lietransformations)canbeverybeneficialfortheanalysisofsingle-passsystems.
i
X
r 2 Newconcepts
a
Thekeytothemoremodernapproachshowninthisarticleistoavoidtheprejudicesaboutthestability
andotherpropertiesofthering. Instead,wemustdescribethemachineintermsoftheobjectsitconsists
ofwithalltheirproperties, includingthenon-linearelements. Theanalysiswillrevealthepropertiesof
theparticlessuchas,e.g.,stability. Inthesimplestcase,theringismadeofindividualmachineelements
such as magnets which have an existence on their own, i.e. the interaction of a particle with a given
element is independent of the motion in the rest of the machine. To successfully study single-particle
dynamics, we must be able to describe the action of the machine element on the particle as well as the
effectofthemachineelementinthepresenceofallothermagnetsinthering[9].
2.1 Map-basedtechniques
In the standard approach to single-particle dynamics in rings, the equations of motion are introduced
together with an ansatz to solve these equations. In the case of linear motion, this ansatz is known as
theCourant–Snydertreatment. However,thistreatmentmustassumethatthemotionofaparticleinthe
ring is stable and confined. For a non-linear system this is a priori not known and the attempt to find a
completedescriptionoftheparticlemotionmustfail.
Inamoremodernapproach,wedonotattempttosolvesuchanoverallequationbutratherconsider
the fundamental objects of an accelerator, i.e. the machine elements themselves. These elements, e.g.
magnetsorotherbeamelements,arethebasicbuildingblocksofthemachine. Allelementshaveawell-
definedactiononaparticlewhichcanbedescribedindependentlyofotherelementsorconceptssuchas
closed orbit or β-functions. Mathematically, they provide a ‘map’ from one face of a building block to
theother,i.e.adescriptionofhowtheparticlesmoveinsideandbetweenelements. Inthiscontext,amap
canbeanythingfromlinearmatricestohigh-orderintegrationroutines.
Thecollectionofallmachineelementsmakesuptheringanditisthecombinationoftheassociated
maps which is necessary for the description and analysis of the physical phenomena in the accelerator
ring.
The most interesting map is the one which describes the motion once around the machine, the
so-calledone-turnmap(OTM).Itcontainsallnecessaryinformationonstabilityandexistenceofclosed
orbitandopticalparameters. Thereaderisassumedtobefamiliarwiththisconceptinthecaseoflinear
beamdynamics[10],whereallmapsarematricesandtheCourant–Snyderanalysisofthecorresponding
OTMproducesthedesiredinformationsuchas,e.g.,closedorbitorTwissparameters.
Itshouldthereforebethegoaltogeneralizethisconcepttonon-lineardynamics. Thecomputation
ofareliableOTMandtheanalysisofitspropertieswillprovideallrelevantinformation.
2.1.1 Basicconcept
Giventhatthenon-linearmapscanberathercomplexobjects, theanalysisoftheOTMshouldbesepa-
ratedfromthecalculationofthemapitself.
Using the coordinate vector (cid:126)z = (x,x(cid:48) = ∂x/∂s,y,y(cid:48) = ∂y/∂s), the map M transforms the
coordinatesz(cid:126) (s )throughamagnetM atpositions tonewcoordinatesz(cid:126) (s )atpositions :
1 1 1 2 2 2
   
x x
 x(cid:48)   x(cid:48) 
z(cid:126)2(s2) =  y  = M◦ y  = M◦z(cid:126)1(s1). (1)
y(cid:48) y(cid:48)
s2 s1
Thesubsequentanalysisofthemapshouldrevealthequantitiesofinterest.
3 Linearmaps
FirstIshallintroducethesimplestformofmaps,i.e. linearmaps,andgeneralizeafterwards.
3.1 Examplesforlinearmaps
Asafirstexample,IconsideradriftspaceoflengthL.
Theformaldescriptionusingthemapcanbewritten
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
x 1 L x
= ◦ . (2)
x(cid:48) 0 1 x(cid:48)
s2 s1
2
Asasecondexample,Ishowthemap(matrix)forafocusingquadrupoleoflengthLandstrengthk:
(cid:32) √ √ (cid:33)
(cid:18) x (cid:19) cosL· k √1 ·sinL· k (cid:18) x (cid:19)
= √ √ k √ ◦ . (3)
x(cid:48) − k·sinL· k cosL· k x(cid:48)
s2 s1
ThemapforaquadrupolewithshortlengthL(i.e. 1 (cid:29) L·k)canbeapproximatedby
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
x 1 0 x
= ◦ . (4)
x(cid:48) −k·L(= 1) 1 x(cid:48)
s2 f s1
TheseareMaps,anddescribethemovementwithinanelement.
3.2 OTMs
Starting from a position s and applying all maps (for N elements) in sequence around a ring with
0
circumferenceC,wegettheOTMforthepositions (foronedimensiononly):
0
(cid:18) (cid:19) (cid:18) (cid:19)
x x
= M ◦M ◦···◦M ◦ (5)
x(cid:48) 1 2 N x(cid:48)
s0+C s0
(cid:18) (cid:19) (cid:18) (cid:19)
x x
=⇒ = M (s )◦ . (6)
x(cid:48) ring 0 x(cid:48)
s0+C s0
Afterthisconcatenation,wehaveobtainedaOTMforthewholering(6). Inthesimplestcaseall
maps are matrices and the concatenation is just the multiplication of linear matrices. Having obtained
thisOTM,itsanalysiswillgiveallrelevantinformation; thereisnoneedforanyassumptions. Tosolve
Hill’s equation using the standard ansatz [10], we need several assumptions such as existence of closed
orbitandconfinementorperiodicity. Havingderivedthemap,wehavetoextracttheinformationneeded.
Tointroducethistopic,wefirsttreatalinearOTM,i.e.amatrix.
3.3 Linearnormalforms
Thekeytotheanalysisisthatmaps(herematrices)canbetransformedinto(Jordan)normalforms. The
originalmapsandthenormalformsareequivalent,butthenormalformandtherequiredtransformation
automaticallyprovidethedataweneed, suchasstability, closedorbit, opticalparameters(Q,Q(cid:48), Twiss
function,etc.),invariantsofthemotion,resonanceanalysis,andsoon.
3.3.1 Linearnormal-formtransformation
Thebasicideaistomakeatransformationtogetasimplerformforthemap.
AssumingthatthemapM propagatesthevariablesfromlocation1tolocation2,wetrytofind
12
transformationsA ,A suchthat
1 2
A M A−1 = R . (7)
1 12 2 12
The map R is a ‘Jordan normal form’ (or at least a very simplified form of the map). For example,
12
R becomes a pure rotation, which is a simple and typical normal form. The map R describes the
12 12
samedynamicsasM ,butallcoordinatesaretransformedbyA andA . ThetransformationsA ,A
12 1 2 1 2
‘analyse’themotion.
WeassumethattheOTM(hereamatrix)M(s)atthepositionsis(seee.g.thearticleontransverse
dynamics[10])
(cid:18) (cid:19)
cos(∆µ)+α(s)sin(∆µ) β(s)sin(∆µ)
M(s) = . (8)
−γ(s)sin(∆µ) cos(∆µ)−α(s)sin(∆µ)
3
Fig.1: Normal-formtransformforalinearOTM(schematic)
This matrix describes the motion on a phase-space ellipse. We can rewrite M such that one part R
becomesapurerotation(acircle),i.e.
ARA−1 = M. (9)
Rememberinglecturesonlinearalgebra,thistransformationisdonebytheevaluationoftheeigenvectors
andeigenvalues.
Startingfrom
M = A◦R(∆µ)◦A−1 or R(∆µ) = A−1◦M ◦A, (10)
wegetforthetransformation(A)andthesimpleform(R)
(cid:32) (cid:112)β(s) 0 (cid:33) (cid:18) cos(∆µ) sin(∆µ) (cid:19)
A = −√α √1 and R = −sin(∆µ) cos(∆µ) . (11)
β β(s)
ThisisjusttheCourant–Snydertransformationtogetβ,α,etc.and∆µisthetune. Thatis,theCourant–
Snyderanalysisisjustanormal-formtransformofthelinearone-turnmatrix. Thistransformationworks
in more than one dimension using the same formalism. The phase-space ellipse is transformed into a
circle,whichistherepresentationofthenormalform,i.e.therotationshowninFig.1.
Fig.2: Normal-formtransformforanon-linearOTM(schematic)
ComparedwiththeCourant–Snyderanalysis,wehavethefollowingpropertiesofthismethod.
i) The rotation angle (tune for the OTM) ∆µ is contained in the normalized map, and we have
stabilityforrealvaluesofthisphaseadvance∆µ.
ii) Opticalfunctions(β,α,etc.) areinthenormalizingmap.
iii) Noneedtomakeanyassumptions,ansatzorapproximation.
In the case of a more complicated OTM(e.g. Fig. 2), the transformation may be more complicated and
wehavetoexpectthattherotationangledependsontheamplitude. Thisisthecasefornon-linearOTMs,
whichwillbetreatedinalatersection. Ithasalreadybeenmentionedheretointroducethenormal-form
transformationasageneralconcept.
4
3.3.2 Normalizedvariables
Pleasenotethat
(cid:18) (cid:19) (cid:18) (cid:19)
x x
n = A−1◦ (12)
x(cid:48) x(cid:48)
n
inthissimplecaseisjustavariabletransformationtonew,normalizedvariables.
3.3.3 Action-anglevariables
It is common practice to define action-angle variables. Once the particles ‘travel’ on a circle, we can
introducenewvariablesforsimplificationandeasierinterpretationoftheresults.
√
x2+x(cid:48)2
i) Radius(say: 2J,withJ = n n)isconstant(invariantofmotion): actionJ.
2
ii) Phaseadvancesbyconstantamount: angleΨ.
The invariant of motion J should be the same at all locations in the ring and is separate from the angle
ψ. Forthepropagationofcoordinatesaroundthering,botharerelevant. Thisseparationoftheinvariant
fromtheangleagainbecomesimportantinalatersection.
3.3.4 Coupling
Theformalismcaneasilybeusedtostudythecaseofcouplingbetweenthetwotransverseplanes,with
themap(intwodimensions)
(cid:18) (cid:19)
M n
T = , (13)
m N
whereM,m,N,nare2×2matrices. Inthecaseofcoupling,m (cid:54)= 0,n (cid:54)= 0.
Wecantrytorewritethisusingthenormal-formalgorithms,asbefore:
(cid:18) (cid:19)
M n
T = = VRV−1, (14)
m N
with
(cid:18) (cid:19) (cid:18) (cid:19)
A 0 γI C
R = and V = . (15)
0 B −Ct γI
Whathaveweobtained? ThematrixRisoursimplerotationintwoplanesandwehaveinaddition:
i) AandB aretheone-turnmatricesforthe‘normalmodes’;
ii) the matrix V transforms from the coordinates (x,x(cid:48),y,y(cid:48)) into the ‘normal-mode’ coordinates
(w,w(cid:48),v,v(cid:48))viatheexpression(x,x(cid:48),y,y(cid:48)) = V(w,w(cid:48),v,v(cid:48));
iii) thematrixC containsthe‘couplingcoefficients’.
3.3.5 Multipleturns
Normalformsareextremelyusefulwhenamapshouldbeappliedk times(e.g.k turns): thebrute-force
methodwouldbetomultiplytheOTMktimes,whichcanbeveryelaborateandexpensiveincomputing
time. Instead, we can first make a transformation into a normal form with a rotation matrix R, which
contains the phase advance per turn µ. For k turns, we have to simply replace the phase advance µ by
thephaseadvancefork turns,whichisofcoursek·µ(16).
Mk(x,x(cid:48)) = ARk A−1(x,x(cid:48)) = ARk(X,X(cid:48)). (16)
Transformingtherotationbackintotheusualformreturnsthek-turnmap.
5
4 Non-linearmaps
In this section, I should like to introduce non-linear maps. I shall start with a linear map and than
generalize. The effect of a (short) quadrupole depends linearly on amplitude as (rewritten from the
matrixform)
     
x x 0
(cid:126)z(s2) =  xy(cid:48)  =  xy(cid:48)  + k1·0xs1 . (17)
y(cid:48) y(cid:48) k ·y
s2 s1 1 s1
Or,inthealreadyknownconventionusingthematrixM,(cid:126)z(s ) = M·(cid:126)z(s ).
2 1
4.1 Symplecticity
A key concept for the application of maps is a requirement called ‘symplecticity’. In a ring, not all
possiblemapsareallowed. ThisrequiresforamatrixMthatitfulfilsthecondition
MT·S ·M = S (18)
with
 
0 1 0 0
 −1 0 0 0 
S =  . (19)
 0 0 0 1 
0 0 −1 0
ThephysicalmeaningofthisrequirementisthatMisareapreservingand
lim Mn = finite =⇒ detM = 1. (20)
n→∞
ItshouldbenotedthatdetM = 1isrequired,butnotsufficient,forMtobesymplectic.
4.2 Taylormaps
Theeffectofa(thin)sextupolewithstrengthk isintheformof(17)
2
     
x x 0
(cid:126)z(s2) =  xy(cid:48)  =  xy(cid:48)  + 21k2·(x02s1 −ys21) . (21)
y(cid:48) y(cid:48) k ·(x ·y )
s2 s1 2 s1 s1
Wecanwriteitnowformallyas
(cid:126)z(s ) = M◦(cid:126)z(s ). (22)
2 1
Here M is not a matrix but a non-linear map. It cannot be written in linear matrix form. We need
somethinglike(forthevectorelementx)
z(cid:126) (s ) = x(s ) = R ·x+R ·x(cid:48)+R ·y+··· .
x 2 2 11 12 13
+T ·x2+T ·xx(cid:48)T ·x(cid:48)2+
111 112 122
(23)
+T ·xy+T ·xy(cid:48)+···
113 114
+U ·x3+U ·x2x(cid:48)+···
1111 1112
andtheequivalentforothervariables. Pleasenotethatx(s )nowdependsalsoonthesecondtransverse
2
variabley.
6
4.2.1 FormsofTaylormaps
Wecanwrite(23)forj = 1,...,4:
4 4 4
(cid:88) (cid:88)(cid:88)
z (s ) = R z (s )+ T z (s )z (s ). (24)
j 2 jk k 1 jkl k 1 l 1
k=1 k=1 l=1
Ifthismapistruncatedafterthesecondorder,wecallitasecond-ordermapandformallywriteitas
A = [R,T]. (25)
2
Higherorderscanbedefinedasneeded:
4 4 4
(cid:88)(cid:88) (cid:88)
A = [R,T,U] =⇒ [R,T]+ U z (s )z (s )z (s ), (26)
3 jklm k 1 l 1 m 1
k=1 l=1m=1
whichistruncatedafterthethirdorder.
Asanexample,thecompletesecond-ordermapfora(thick)sextupolewithlengthLandstrength
K (intwodimensions)is
(cid:16) (cid:17)
x = x +Lx(cid:48) −K L2(x2−y2)+ L3(x x(cid:48) −y y(cid:48))+ L4(x(cid:48)2−y(cid:48)2) ,
2 1 1 4 1 1 6 1 1 1 1 24 1 1
(cid:16) (cid:17)
x(cid:48) = x(cid:48) −K L(x2−y2)+ L2(x x(cid:48) −y y(cid:48))+ L3(x(cid:48)2−y(cid:48)2) ,
2 1 2 1 1 2 1 1 1 1 6 1 1
(cid:16) (cid:17) (27)
y = y +Ly(cid:48) +K L2x y + L3(x y(cid:48) +y x(cid:48))+ L4(x(cid:48)y(cid:48)) ,
2 1 1 4 1 1 6 1 1 1 1 24 1 1
(cid:16) (cid:17)
y(cid:48) = y(cid:48) +K Lx y + L2(x y(cid:48) +y x(cid:48))+ L3(x(cid:48)y(cid:48)) .
2 1 2 1 1 2 1 1 1 1 6 1 1
ThedefinitionofK isnotunique;itcandifferbysomefactor,e.g.
(cid:18)∂2x ∂2x k (cid:19)
= k·x2 versus = ·x2 . (28)
∂t2 ∂t2 2
4.2.2 SymplecticityofTaylormaps
Wehavearguedthatavitalrequirementformapsisthesymplecticity. Thismustholdalsofornon-linear
maps;however:
i) truncatedTaylorexpansionsarenotmatrices;
ii) itistheassociatedJacobianmatrixJ whichmustfulfilthesymplecticitycondition(see(18)and
(19)):
∂zi (cid:18) ∂zx(cid:19)
J = 2 e.g.J = 2 , (29)
ik ∂zk xy ∂zy
1 1
whereJ mustfulfilJt·S ·J = S.
Ingeneral,J (cid:54)= const →foratruncatedTaylormap;thiscanbedifficulttofulfilforallz.
ik
Asanexample,wetakethesextupolemap(forsimplicityinonedimension)[21]
(cid:16) (cid:17)
x = x +Lx(cid:48) −K L2x2+ L3x x(cid:48) + L4x(cid:48)2+O(3) ,
2 1 1 4 1 6 1 1 24 1
(cid:16) (cid:17) (30)
x(cid:48) = x(cid:48) −K Lx2+ L2x x(cid:48) + L3x(cid:48)2+O(3) .
2 1 2 1 2 1 1 6 1
Thenwecancompute
(cid:18) (cid:19)
0 1+∆S
JT·S ·J = (cid:54)= S. (31)
−1−∆S 0
WefindthatJ isnon-symplecticwiththeerror
K
∆S = L4(L2x(cid:48)2+6Lxx(cid:48)+6x2). (32)
72
7
∆x’ ∆x’
∆x’
Fig.3: Movementofaparticlethroughthinelements
5 Thinelements
From Eq. (32), we may get a hint of how to attack this problem. The error ∆S compared to the fully
symplectic map depends on the fourth power of the sextupole length L4. This can be extended to other
elementsaswellandforshortobjectstheerrormaybenegligible. Wecanattemptto‘make’allelements
thin,i.e.shortenoughtohaveanegligiblenon-symplecticity. Forthispurpose,wedefinethinelements,
i.e.anelementwithnolength,becausetheyavoidproblemswithsymplecticityofnon-linearmapssuch
asTaylormaps. Theerrorinequation(32)becomeszero.
5.1 Conceptofthinelements
Realmagnetshaveafinitelength, i.e.theyarethickmagnets. Thefieldandlengthareusedtocompute
their effect, i.e. the map; in the simplest case it becomes a matrix (3). In passing, we have already
mentionedtheequivalentmatrixforaquadrupolewithzerolength(4),sinceitisoftenusedinanalytical
calculations. A severe consequence of thick elements is that they are not always linear elements (even
not dipoles or quadrupoles). For thick, non-linear magnets a closed solution for maps often does not
exist.
Theprocedureweshallapplytoget‘thin’magnetsistoletthelengthgotozero,butkeepthefield
integralfinite(constant)Asconsequences, thindipolesandquadrupolesarelinearelementsandallthin
magnetsaremucheasiertouseandautomaticallysymplectic. Themovementofaparticlethroughthin
elements is shown schematically in Fig. 3. The key is that there is no change of amplitudes x and y at
theelement;onlythemomentax(cid:48) andy(cid:48) receiveanamplitude-dependentdeflection(kick)
x(cid:48) → x(cid:48)+∆x(cid:48) and y(cid:48) → y(cid:48)+∆y(cid:48). (33)
Modelling a magnet with a finite length by a thin magnet, we have to pay a price and must ask under
which conditions this is a good strategy. We should expect that this approximation is valid when the
finitelengthissmallorwhenthelengthdoesnotmatter. Themainquestionstoanswerareasfollows.
i) Whathappenstotheaccuracyandtheimplicationsforthebeamdynamics?
ii) Whatisthebestimplementationofthinmagnetstominimizethesideeffects?
Weshouldstudytheseissueswithsomesimpleexamplesandgeneralizelater.
5.2 Accuracyofthinelements
WestartagainwiththeexactmatrixforathickquadrupolewiththestrengthK andthelengthL.
(cid:32) √ √ (cid:33)
cosL· K √1 ·sinL· K
M = √ √ K √ . (34)
s→s+L
− K ·sinL· K cosL· K
Thismatrixdescribesthemotioninathickquadrupoleexactlyandwecanestimatetheerrorswemake
byourapproximation. Althoughthisdemonstrationisdoneforaquadrupole,thefinalresultisvalidfor
alllinearandnon-linearelements.
8
Fromtheexactmatrix
(cid:32) √ √ (cid:33)
cosL· K √1 ·sinL· K
M = √ √ K √ , (35)
s→s+L
− K ·sinL· K cosL· K
wemakeaTaylorexpansionin‘small’lengthL:
(cid:18) 1 0 (cid:19) (cid:18) 0 1 (cid:19) (cid:18) −K 0 (cid:19)
L0· +L1· +L2· 2 +··· (36)
0 1 −K 0 0 −K
2
andkeepuptothefirst-orderterminL:
(cid:18) (cid:19) (cid:18) (cid:19)
1 0 0 1
M = L0· +L1· , (37)
s→s+L 0 1 −K 0
(cid:18) (cid:19)
1 L
M = +O(L2). (38)
s→s+L −K ·L 1
Wemaketwoobservationsfromtheapproximation(38).
i) ItisprecisetofirstorderO(L1)andtheerroristosecondorderO(L2).
ii) WehavedetM =(cid:54) 1;therefore,itcannotbesymplectic(see18and19).
Toensurethesymplecticity,weneedtomodifythematrix. Wecantrytoaddaterminthematrix:
(cid:18) (cid:19)
1 L
M = +O(L2) (39)
s→s+L −K ·L 1
andweobtain
(cid:18) (cid:19)
1 L
M = . (40)
s→s+L −K ·L 1−KL2
The matrix (40) is still precise to first order O(L1) but ‘symplectified’ by adding the term −KL2. The
added term is inaccurate to second order (because of L2) and therefore the order of accuracy is not
changedbythismodification. Thisprocedurerestoresthesymplecticity,butdoesnotchangethelevelof
accuracy.
Forthenextstep,wekeepuptothesecondorderinLfromtheTaylorexpansion(36):
(cid:18) 1− 1KL2 L (cid:19)
M = 2 +O(L3). (41)
s→s+L −K ·L 1− 1KL2
2
ItisprecisetosecondorderO(L2)andthereforemoreaccuratethan(40),butagainnotsymplectic.
Weproceedasbeforeandthesymplectificationnowlookslike
(cid:18) 1− 1KL2 L−1KL3 (cid:19)
M = 2 4 +O(L3). (42)
s→s+L −K ·L 1− 1KL2
2
TheaccuracyismaintainedtosecondorderO(L2)anditisnowfullysymplectic.
9
K
L
Fig.4: SchematicpictureofaquadrupolewithlengthLandstrengthK (convention)
K L
L
Fig.5: SchematicpictureofaquadrupolewithlengthLandakickwithstrengthK attheendoftheelement
K L
L/2 L/2
Fig.6: SchematicpictureofaquadrupolewithlengthLandakickwithstrengthK inthecentreoftheelement
5.2.1 Physicalmeaningofthesymplectification
Inprinciple,onecouldgoonwithsuchascheme,butwhatisagoodstrategy? Themainissuesforusing
animplementationareaccuracyandspeed. First,wehavealookatthesignificanceofthesymplectifica-
tion, whichlooksratherarbitrary. Assumealinearelement(quadrupole)oflengthLandstrengthK as
illustratedin(Fig.4). Averysimplewaytoapplyathin-lenskickisshownin(Fig.5). Thequadrupole
istreatedasadriftlengthoflengthLfollowedbyathin-lengthkickwiththestrengthK ·L.
Wecancomputethefullmatrixofthedriftandthekickandobtain
(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
1 0 1 L 1 L
= . (43)
−K ·L 1 0 1 −K ·L 1−KL2
We find that this resembles the ‘symplectification’ we applied to the truncated map of order O(L1).
Becauseofthesymmetry,wewouldgetthesameresultwhenweapplythekickbeforethedrift.
Anotheroptionistoapplythethin-lenskickinthecentreoftheelement,asshowninFig.6. This
timeweapplythekickinthecentreoftheelement,precededandfollowedbydriftspacesofthelength
L/2.
Themultiplicationofthethreematricesleadsusto
(cid:18) 1− 1KL2 L−1KL3 (cid:19)
M = 2 4 +O(L3). (44)
s→s+L −K ·L 1− 1KL2
2
ThisisequivalenttothetruncatedandsymplectifiedmapoforderO(L2).
Wecansummarizethetwooptionsasfollows.
i) Onekickattheendorentry: error(inaccuracy)isoffirstorderO(L1).
ii) Onekickinthecentre: error(inaccuracy)isofsecondorderO(L2).
Wefindthatitisveryrelevanthowtoapplythinlensesandtheaimshouldbetobepreciseandfast(and
simpletoimplement,forexampleinacomputerprogram).
10