Review of coupled betatron motion parametrizations and applications to strongly coupled lattices Marion Vanwelde1C edric Hernalsteens2 1yS. Alex Bogacz3Shinji Machida4and Nicolas Pauly1

2025-04-29 0 0 2.78MB 33 页 10玖币

侵权投诉

Review of coupled betatron motion parametrizations and applications to strongly

coupled lattices

Marion Vanwelde,1, ∗C´edric Hernalsteens,2, 1, †S. Alex Bogacz,3Shinji Machida,4and Nicolas Pauly1

1Service de M´etrologie Nucl´eaire (CP165/84), Universit´e libre de Bruxelles,

Avenue Franklin Roosevelt 50, 1050 Brussels, Belgium

2CERN, European Organization for Nuclear Research,

Esplanade des Particules 1, 1211 Meyrin, Switzerland

3Thomas Jeﬀerson National Accelerator Facility, Newport News, Virginia, U.S.A.

4STFC Rutherford Appleton Laboratory, Harwell Campus, Didcot, OX11 0QX, United Kingdom

(Dated: November 1, 2022)

The coupling of transverse motion is a natural occurrence in particle accelerators, either in the

form of a residual coupling arising from imperfections or originating by design from strong systematic

coupling ﬁelds. While the ﬁrst can be treated perturbatively, the latter requires a robust approach

adapted to strongly coupled optics and a parametrization of the linear optics must be performed to

explore beam dynamics in such peculiar lattices. This paper reviews the main concepts commonly

put forth to describe coupled optics and clariﬁes the proposed parametrization formalisms. The links

between the generalized Twiss parameters used by the diﬀerent approaches are formally proven, and

their physical interpretations are highlighted. The analytical methods have been implemented in a

reference Python package and connected with a ray-tracing code to explore strongly coupled lattices

featuring complex 3D ﬁelds. Multiple examples are discussed in detail to highlight the key physical

interpretations of the parametrizations and characteristics of the lattices.

I. INTRODUCTION

The motion of charged particles in a particle accelera-

tor is typically studied using the linear and uncoupled

theory of betatron motion. The Courant-Snyder the-

ory [1] allows the study of unidimensional and uncou-

pled motion by having an elegant parametrization whose

optical parameters have a clear physical meaning. How-

ever, in many machines, coupling between the two trans-

verse degrees of freedom is present. The coupling of the

particle transverse motion has long been considered an

undesirable eﬀect. Coupling was ﬁrst studied mainly

because of imperfections (quadrupole tilt, vertical dis-

placements of sextupoles [2]). This residual coupling, if

not well controlled, can cause undesirable eﬀects such as

vertical emittance increase or impact linear and nonlin-

ear observables such as amplitude detuning [3]. To take

into account the eﬀect of residual coupling, it is possi-

ble to start from the uncoupled theory and consider the

coupling as a perturbation. This perturbation theory is

no longer applicable as soon as the coupling arises from

strong systematic coupling ﬁelds. In this case, the ma-

chine design contains elements that introduce coupling

on purpose. In colliders, it is the case for interaction

regions where large solenoidal ﬁelds and compensation

elements are present. Atypical optics schemes based on

strong coupling insertions have also been proposed to im-

prove the performance of lepton and hadron colliders,

such as the “M¨obius accelerator” [4], planar-to-circular

beam adapters for circular modes operation [5] and round

beam operation for lepton storage rings [6].

∗Email address: marion.vanwelde@ulb.be

†Email address: cedric.hernalsteens@cern.ch

Recently, vertical excursion ﬁxed ﬁeld accelerators

(vFFAs) were revived1, featuring coupling by design.

The detailed linear and nonlinear study of vFFAs con-

stitutes the main motivation for the present work. In

what follows, xis the horizontal coordinate, yis the ver-

tical coordinate, and zis the longitudinal coordinate. In

conventional, horizontal excursion, FFAs, the nonlinear

magnetic ﬁeld respects a scaling condition that allows

having a constant tune for all energies [8–10] and higher

momentum particles move to orbits of larger radius. By

contrast, vFFA ﬁelds fulﬁll another scaling condition:

B=B0ek(y−y0),

where k=1

∂B

∂y is the normalized ﬁeld gradient, y0is the

reference vertical position and B0is the bending ﬁeld at

the reference position. The bending ﬁeld increases expo-

nentially in the vertical direction leading higher energy

particle orbits to have the same radius but to shift ver-

tically. The median plane of vFFA elements is the plane

at x= 0 (vertical plane). Assuming y0= 0, we can write

the three magnetic ﬁeld components with an out-of-plane

1Although ﬁrst introduced in 1955 by Tihiro Ohkawa as electron

cyclotrons, vFFAs received a lot of interest only from 2013 on-

wards, following Ref. [7]

arXiv:2210.11866v2 [physics.acc-ph] 31 Oct 2022

expansion [11]:

Bx(x, y, z) = B0eky

i=0

bxi(z)xi,

By(x, y, z) = B0eky

i=0

byi(z)xi,

Bz(x, y, z) = B0eky

i=0

bzi(z)xi,

where, by taking into account the fringe ﬁeld function

g(z), the coeﬃcients of these equations are given by the

following recurrence relations [11]:

bx0(z) = 0, bx,i+1(z) = −1

i+ 1(kbyi +dbzi

dz ),

by0(z) = g(z), by,i+2(z) = k

i+ 2bx,i+1,

bz0(z) = 1

dz ,bz,i+2(z) = 1

i+ 2

dbx,i+1

dz .

In the (vertical) median plane, the three ﬁeld compo-

nents are:

Bx0(0, y, z) = 0,

By0(0, y, z) = B0eky g(z),

Bz0(0, y, z) = B0

keky dg

dz .

The vFFAs thus present a non-zero longitudinal ﬁeld

component, which arises due to the fringe ﬁelds at the

vFFA ends. It is especially important as the magnet

construction, respecting the scaling law, will induce im-

portant fringe ﬁelds. If we look at the ﬁeld in the ele-

ment body, by neglecting the fringe ﬁeld (g(z) = const.,

g0(z) = 0, Bz= 0), the transverse ﬁeld components can

be expressed as multipolar expansions by rewriting the

exponential in terms of its Taylor series. It is readily seen

that the ﬁrst-order terms of this expansion correspond to

skew quadrupolar components:

Bx(x, y, z)' −B0(kx +k2

2! (2xy) + O(x3))

' −B0kx,

By(x, y, z) = B0(1 + ky +k2

2! (y2−x2) + O(x3))

∼B0+B0ky.

Because of the longitudinal and skew quadrupolar ﬁeld

components, which are the main sources of transverse

motion coupling, vFFAs feature strongly coupled optics.

It is therefore necessary to study vFFA lattices with a

model adapted to strongly coupled optics. The choice of a

given parametrization for such a machine, suitable for the

design, optimization, and operation phases, is key to a

thorough understanding of the peculiar beam dynamics.

All the methods and analyses presented are applicable to

other coupled lattices in full generality and are relevant

for snake [12] and spin rotator designs.

Several parametrizations attempt to describe coupled

optics as elegantly as the Courant-Snyder theory for un-

coupled motion. The most widely known parametriza-

tions are those of Edwards and Teng (ET) [13] and

of Mais and Ripken (MR) [14]. In addition, these

parametrizations were extended and revisited in several

works: Sagan and Rubin [15], Parzen [16], Wolski [17–

19] and Lebedev and Bogacz (LB) in [20]. The exact

formalisms and notations used by these authors diﬀer,

and slightly diﬀerent parametrization choices lead to an

apparently inhomogeneous theory. To clarify the situa-

tion so that a clear picture can be obtained, we provide

interpretations of these parameters and explicit links be-

tween them for the diﬀerent parametrizations.

The general theory and formalism for the study of lin-

ear beam dynamics in 4D transverse phase-space are pre-

sented in Section II and the peculiarities of coupled mo-

tion are highlighted. In Section III, a review of the cou-

pling parametrizations from ET, MR, and their exten-

sions are detailed using uniﬁed approaches and notations.

Physical interpretations regarding lattice functions and

clariﬁcations of the relationships between the quantities

appearing in the diﬀerent parametrizations are provided.

The links between the ET and MR parametrization cat-

egories are discussed in Section IV. The methods are im-

plemented using the Zgoubidoo Python interface [21] for

the Zgoubi code [22] and discussion in Section V where

applications are presented for example lattices and for re-

alistic examples of snakes and spin rotators. The imple-

mentations have been validated by comparing the gen-

eralized lattice functions computed by Zgoubidoo with

those obtained by MAD-X [23] and PTC [24]. Conclu-

sions and recommendations for the study of vFFA lattices

are provided in Section VI.

II. THEORY OF COUPLED LINEAR

BETATRON MOTION

A. Notations

Lowercase bold letters are used to indicate vectors of

geometric coordinates, where prime denotes the diﬀer-

entiation with respect to the independent scoordinate:

()0=d()

ds ,x≡(x, x0, y, y0)T. The vectors of canonical

coordinates will be designated as:

ˆx ≡









.

Bold uppercase letters indicate matrices (for example,

Mwill denote a transfer matrix), and a hat is added

when it comes to the transfer matrice over a full period,

(“one-turn transfer matrices” ˆ

M). No diﬀerence is made

in the notation to denote the transfer matrices expressed

in geometric variables or canonical variables. However,

the identiﬁcation of the variables for each of the matrices

will be made clear from the context. Moreover, a bar is

added on top to indicate symplectic conjugate matrices:

the symplectic conjugate matrix of Mwill be denoted

M. The symplectic conjugate of a symplectic matrix M

is deﬁned as ¯

M=−SMTS=M−1[1, 25], where Sis

the unit symplectic matrix

S=





0100

−1 0 0 0

0001

0 0 −1 0





,

with STS=I,SS =−I, and ST=−S. The hori-

zontal (x) and vertical (y) directions are referred to as

“physical directions” or “physical space” as opposed to

the “eigen-directions” related to the directions of the de-

coupled motion.

B. Geometric coordinates and canonical

coordinates

The relation between geometric coordinates and

canonical variables reads:

x0=px−e

Ax, (1)

y0=py−e

Ay, (2)

where the vector potential Acomponents are related to

the magnetic ﬁeld by B=∇ × A. The components of

the vector potential Acan often be expressed as a series

expansion. When studying the linear motion, this series

can be approximated by its ﬁrst-order terms in x,px,

y, and py, which allows having only quadratic terms in

the expression of the Hamiltonian. For example, for the

longitudinal ﬁeld produced by a solenoid, one obtains the

components Axand Ayof the vector potential as follows

[2, 20]:

Ax=−1

2R1(s)y+O(y3),

Ay=1

2R2(s)x+O(x3),

where R1(s) = R2(s) = e

p0Bs(0,0, s) are constants pro-

portional to the longitudinal component of the magnetic

ﬁeld.

In the case of a scaling vFFA ﬁeld, the three vector

potential components can be written:

Ax(x, y, z) = B0eky

i=0

axi(z)xi,

Ay(x, y, z) = B0eky

i=0

ayi(z)xi,

Az(x, y, z) = B0eky

i=0

azi(z)xi,

where the coeﬃcients are given by the following recur-

rence relations:

ax0(z) = 0, ax1(z) = 0,

ay0(z) = 0, ay1(z) = 1

dz ,

az0(z) = 0, az1(z) = −g(z),

ax,i+1(z) = −k

i+ 1ayi

ay,i+2(z) = −1

(i+ 2)(i+ 1)−kdazi

dz +d2ayi

dz2

i+ 2kax,i+1

az,i+2(z) = k

(i+ 2)(i+ 1)−kazi +dayi

dz 

i+ 2

dax,i+1

dz .

By truncating the series in the ﬁrst order, the trans-

verse components Axand Aybecome

Ax= 0 + O(x2),

Ay=1

2R2(s)x+O(x3),

where, in this case, R1(s) = 0 and R2(s) =

p0Bz(0, y, s).

The expressions (1) and (2) for the transform between

geometric and canonical variables can be rewritten using

R1,2:

x0=px+1

2R1y,

y0=py−1

2R2x.

In the absence of a longitudinal ﬁeld component, the

canonical variables are equal to the geometric variables:

ˆx =x. However, when there is a longitudinal ﬁeld com-

ponent, it must be considered. In a matrix form this

reads ˆx =Ux, where

U=





1 0 0 0

0 1 −R1

0 0 1 0

2001





.

The matrix Ucan also be used to transform from

a transfer matrix expressed in geometric variables to a

transfer matrix expressed in canonical variables:

Ms0→s,canon. =U(s)Ms0→s,geom.U−1(s0).

C. Coupling sources

The linear coupling between the two transverse di-

rections originates from two types of ﬁeld components:

longitudinal or skew quadrupolar. Solenoids induce a

rotation at a frequency equal to the Larmor frequency

θ=−qBs

2γm =−Ωc

2=−ωLarmor (where Bsis the lon-

gitudinal ﬁeld component and Ωcis the cyclotron fre-

quency) [26]. This rotation introduces a coupling be-

tween the vertical and horizontal motions of the parti-

cle. It can be shown that there is a s-dependent rota-

tion that transforms the coordinate system into a frame

where the motion is decoupled (the so-called “Larmor”

frame) [26]. The rotation angle that decouples the mo-

tion is proportional to the integral of the longitudinal

ﬁeld along the trajectory of the particle. In the Larmor

frame, the solenoid is a magnetic element that focuses

in the two transverse directions [2]. The second mag-

netic ﬁeld which induces linear coupling is the ﬁeld pro-

duced by a skew quadrupole. A particle with a horizontal

(resp. vertical) displacement will be aﬀected by a hori-

zontal (resp. vertical) magnetic ﬁeld and will be subject

to a vertical (resp. horizontal) force inducing a vertical

displacement. The vertical motion ultimately becomes

horizontal again. There is an energy exchange between

the two transverse directions, and the motion is coupled

[27].

D. Equations and invariant of motion,

symplecticity and stability

We assume linearized transverse equations of motion

expressed in the moving Frenet-Serret reference frame

(x,y, and s) attached to the reference trajectory. In

the geometric coordinates (x=x, x0, y, y0T), the 2D

coupled linear equations of motion can be written [2, 20]:

x00 + (κ2

x+K)x+ (N−1

2R0

1)y−1

2(R1+R2)y0= 0,

y00 + (κ2

y−K)y+ (N+1

2R0

2)x+1

2(R1+R2)x0= 0,

where the coeﬃcients κx,κy,K, and Nare deﬁned as

follows:

κx=eBy(0,0, s)

κy=−eBx(0,0, s)

K=1

Bρ (∂By

∂x )x=y=0

N=1

2Bρ (∂By

∂y −∂Bx

∂x )x=y=0.

Bx,By,Bsare the ﬁeld components along the closed or-

bit. κxand κyare the curvature of the design orbit in

the horizontal and vertical directions, Kis related to the

normal component of the ﬁeld gradient, while Nis linked

to the skew component of the ﬁeld gradient [20]. Finally,

R1and R2(see Section II B) are related to the longitudi-

nal ﬁeld component. In the equations of motion, only the

last two terms of the left-hand side reﬂect the coupling

between the two transverse directions. Without these

terms, the equations of motion are Hill’s equations with-

out coupling (see Appendix A). The equations of motion

are obtained from the Hamiltonian for a charged parti-

cle of charge eand mass min an electromagnetic ﬁeld

expressed in Cartesian coordinates [28]:

H(r, π, t) = cpm2c2+ (π−eA(r, t))2) + eφ(r, t),

where r= (x, y, z), π= (Px, Py, Pz) contains the

three canonical conjugate momentum for the coordinates

(x, y, z), Ais the vector potential and φis the scalar po-

tential. To derive the equations of motion expressed in

canonical variables, ensuring conservative solutions [2],

a transformation to the coordinates in the Frenet-Serret

frame is performed and a change of independent variable

from time tto path length along the reference trajectory

sis performed. The Hamiltonian becomes

H(x, px, y, py;s) = −(1 + κxx+κyy)s1−px−(1 −δ)e

Ax2

−py−(1 −δ)e

Ay2

+ (1 −δ)e

As,

where δ=P−P0

P0and px,pyare the canonical momentum

normalized by the total reference momentum P0. More-

over, because the transverse momentum components are

much smaller than the total reference momentum P0,

it is possible to expend the Hamiltonian in a power se-

ries. For linear motion, the Hamiltonian is truncated to

a quadratic form. Considering in addition the nominal

energy only (δ= 0) and expressing the vector potential

components in terms of κx,κy,K,N,R1and R2, the Hamiltonian becomes [20]:

H=p2

x+p2

2+ (κ2

x+K+R2

4)x2

2+ (κ2

y−K+R2

4)y2

2+Nxy +1

2(R1ypx−R2xpy). (3)

The coupling terms are readily apparent, with skew

quadrupolar ﬁelds gradient Ncoupling the motion

through the xy term and longitudinal ﬁelds coupling the

motion through the ypxand xpyterms. The Hamiltonian

equations of motion can be written in a 4 ×4 matrix for-

malism using the bilinear form [13, 20, 25, 29]

H=1

2ˆxTHˆx,

where His a real and symmetric matrix:

H=





κ2

x+K+R2

40N−R2

0 1 R1

NR1

2κ2

y−K+R2

−R2

20 0 1





.

In a matrix form the equations of motion become

ˆx0=SHˆx =A(s)ˆx. (4)

From Equation (4), we can show that, for any solutions

ˆx1and ˆx2, the quantity ˆxT

2Sˆx1is a constant of motion

[1]; the so-called Lagrange invariant. Indeed, if ˆx1and

ˆx2are solutions of equation (4), then

ds(ˆxT

2Sˆx1) = dˆxT

ds Sˆx1+ˆxT

2Sdˆx1

=ˆxT

2HTSTSˆx1+ˆxT

2SSHˆx1

= 0

⇒ˆxT

2Sˆx1= constant.

The linear motion described by the quadratic Hamil-

tonian (Eq. (3)) is a succession of linear canonical trans-

formations represented by the transfer matrices M. The

solution of the equations of motion can therefore be writ-

ten in the form ˆx(s) = Ms0→sˆx(s0), where Ms0→sis

the transfer matrix allowing to propagate the coordinates

from s0to s. The transfer matrix must satisfy the fol-

lowing conditions [2]:

dsMs0→s=A(s)Ms0→s,

Ms0→s0=I.

Expressing the particle trajectory with these transfer

matrices is equivalent to integrating the linear diﬀeren-

tial equations over a ﬁnite distance. The particle motion

can thus be described either by the equations of motion

derived from the Hamiltonian (continuous formalism) or

by transfer matrices (discrete formalism) [30]. One of the

advantages of using matrix formalism is that we can ob-

tain the one-turn transfer matrix ˆ

Mby multiplying the

transfer matrices of the sections contained in the period.

The one-turn transfer matrix computed at scan be ob-

tained from the one-turn transfer matrix computed at

s0:

M(s) = Ms0→sˆ

M(s0)M−1

s0→s.

The Jacobian matrix of a canonical transformation is

symplectic [29]. In the case of linear motion, the transfer

matrix is equal to the Jacobian matrix, and therefore

Msi→sj(expressed in terms of canonical variables) is also

symplectic:

MTSM =S. (5)

It is possible to ﬁnd this symplecticity condition from the

Lagrange invariant [31]. ˆxT

2Sˆx1being an invariant of the

motion, we know that ˆxT

2(s)Sˆx1(s) = ˆxT

2(s0)Sˆx1(s0).

By expressing the coordinates as functions of susing the

transfer matrix and the initial coordinates at s0, we get:

ˆxT

2(s)Sˆx1(s) = ˆxT

2(s0)MTSMˆx1(s0)

=ˆxT

2(s0)Sˆx1(s0)

⇒MTSM =S.

The matrix Sbeing anti-symmetric, the symplecticity

condition on the transfer matrix (Eq. (5)) gives (n2−n)

scalar conditions. The transfer matrix Mwill therefore

contain n2−(n2−n)

2=n

2(n+ 1) independent elements

[1, 13, 20, 29]. For a two-dimensional motion, at least 10

independent parameters are needed to parameterize the

matrix.

It is insightful to study the eigenvalues and eigenvec-

tors of the transfer matrix ˆ

M. The expression of the La-

grange invariant allows to ﬁnd conditions on these eigen-

values and eigenvectors. For a 4×4 transfer matrix, there

are 4 eigenvectors ˆvjcorresponding to the eigenvalues λj:

Mˆvj=λjˆvj. By expressing the Lagrange invariant for

two eigenvectors ˆviet ˆvjof the transfer matrix, we obtain

[2]:

ˆvT

i(s)Sˆvj(s)=(ˆ

Mˆvi(s0))TSˆ

Mˆvj(s0)

=λiλjˆvT

i(s0)Sˆvj(s0).

One obtains:

(λiλj= 1 ⇒ˆvT

iSˆvj6= 0

λiλj6= 1 ⇒ˆvT

iSˆvj= 0

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

ReviewofcoupledbetatronmotionparametrizationsandapplicationstostronglycoupledlatticesMarionVanwelde,1,CedricHernalsteens,2,1,yS.AlexBogacz,3ShinjiMachida,4andNicolasPauly11ServicedeMetrologieNucleaire(CP165/84),UniversitelibredeBruxelles,AvenueFranklinRoosevelt50,1050Brussels,Belgium2CERN,Europ...

展开>> 收起<<

Review of coupled betatron motion parametrizations and applications to strongly coupled lattices Marion Vanwelde1C edric Hernalsteens2 1yS. Alex Bogacz3Shinji Machida4and Nicolas Pauly1.pdf

共33页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Review of coupled betatron motion parametrizations and applications to strongly coupled lattices Marion Vanwelde1C edric Hernalsteens2 1yS. Alex Bogacz3Shinji Machida4and Nicolas Pauly1

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: