Sympletic tracking methods for insertion devices a Robinson wiggler example Ji LiJ org Feikes Tom Mertens Edward Rial Markus Ries Andreas Sch alicke and Luis Vera Ramirez

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Sympletic tracking methods for insertion devices:

a Robinson wiggler example

Ji Li,∗J¨org Feikes, Tom Mertens, Edward Rial, Markus Ries, Andreas Sch¨alicke, and Luis Vera Ramirez

Helmholtz-Zentrum Berlin f¨ur Materialien und Energie GmbH (HZB),

Albert-Einstein-Straße 15, 12489 Berlin, Germany

(Dated: October 12, 2022)

Modern synchrotron light sources are often characterized with high-brightness synchrotron radia-

tion from insertion devices. Inevitably, insertion devices introduce nonlinear distortion to the beam

motion. Symplectic tracking is crucial to study the impact, especially for the low- and medium-

energy storage rings. This paper uses a Robinson wiggler as an example to illustrate an universally

applicable analytical representation of the magnetic ﬁeld and to summarizes four diﬀerent symplectic

tracking methods.

I. INTRODUCTION

With the aim of high-brightness synchrotron radiation,

the storage rings of modern synchrotron light sources

mostly adopt strong-focusing lattices, which result in

large negative natural chromaticities and need strong sex-

tupoles to correct the chromaticity to suppress the head-

tail instability. Therefore nonlinear distortion is intro-

duced to beam motion by strong sextupole ﬁelds. Fur-

thermore, insertion devices, fringe ﬁelds and imperfec-

tions of magnets are additional sources of nonlinearity.

The nonlinear distortion from the magnets determines

long-term beam stability and has strong impact on oper-

ational performance.

The analysis of long-term beam dynamics in the stor-

age ring is established by symplectic particle tracking.

In general, symplectic tracking can be divided into two

steps. First, an accurate analytical expression of mag-

netic ﬁeld is needed. Second, the symplectic integra-

tion to solve the Hamiltonian equations of the parti-

cle’s motion inside the magnetic ﬁeld is conducted step-

wise element by element for multiple turns. Unlike the

Runge-Kutta integration which is usually not sympletic

and may introduce artiﬁcial damping and antidamping

eﬀect, sympletic integration leads to the canonical trans-

formation of phase space vector and satisﬁes Liouville’s

theorem.

In tracking codes the eﬀect of dipoles and multipoles

are usually modeled with an impulse boundary approxi-

mation, also called hard-edge model, in which the mag-

netic ﬁeld is assumed to be constant within the eﬀective

boundary of the magnet and zero outside. In this model,

only the longitudinal component of the vector potential is

needed to describe the system. Since the coordinates and

their conjugate canonical momenta are not mixed in the

Hamiltonian, the Hamiltonian can be split into drift-kick

combinations [1].

The proposed Robinson Wiggler (RW) for the Metrol-

ogy Light Source (MLS) [2], designed and studied in

∗ji.li@helmholtz-berlin.de

Ref. [3], is used to illustrate symplectic tracking meth-

ods for insertion devices. It consists of a chain of 12

combined-function magnets, shown in Fig. 1, with the

aim to lengthen the bunch by transferring the longitudi-

nal damping to transverse plane. As shown in Fig. 2, the

magnetic ﬁeld in the RW is three-dimensional (3D), hor-

izontally asymmetric and much more complicated than

the impulse boundary model, thus the splitting methods

for dipoles and multipoles are not applicable any more.

FIG. 1. The model of the RW in RADIA [4].

FIG. 2. The vertical magnetic ﬁeld on the midplane of the

RW.

In this paper, the principle of the RW and the necessity

of symplectic tracking is brieﬂy introduced in section II.

Then in section III the basic concepts for symplectic in-

tegration are revisited. In section IV an analytical repre-

sentation is proposed to describe the 3D ﬁeld in the RW

accurately. On this basis, three sympletic integration

methods are introduced to solve the Hamiltonian equa-

tions of motion for electrons in section V. In section VI, a

monomial map approach independent of analytic expres-

sion of the magnetic ﬁeld is introduced to realize faster

tracking. The methods in this paper are universally ap-

plicable to all wigglers and undulators with a straight

reference trajectory.

arXiv:2210.05345v1 [physics.acc-ph] 11 Oct 2022

FIG. 3. Linear optics of the MLS with Robinson wiggler.

II. MOTIVATION: A ROBINSON WIGGLER

FOR THE METROLOGY LIGHT SOURCE

The Metrology Light Source (MLS) is an electron

storage ring owned by the Physikalisch-Technische Bun-

desanstalt (PTB) and operated and designed by the

Helmholtz-Zentrum Berlin f¨ur Materialien und Energie

(HZB). It is dedicated to metrology applications in the

Ultraviolet (UV) and Extreme violet (EUV) spectral

range as well as in the Infrared (IR) and THz region [5].

It can be operated at any energy between 50 MeV and

629 MeV, while the stored current can be varied from

200 mA down to a single electron (= 1 pA). The main

parameters of the major operational mode, standard user

mode, at the MLS are listed in Table I.

TABLE I. Parameters of the standard user mode at the MLS

Parameter Value

Operation Energy 629 MeV

Injection energy 105 MeV

Tunable energy range 50 - 629 MeV

Tunable current range 1 pA - 200 mA

Circumference 48 m

Horizontal/vertical tunes 3.178 / 2.232

Short/long straight 2.5 m / 6 m

Natural emittance 110 nm rad @ 629 MeV

Natural energy 4.4 ×10−4@ 629 MeV

Momentum compaction factor 0.03

Lifetime @ 150 mA, 629 MeV ∼6 h

The MLS is operated in decay mode. The standard

user mode has a beam lifetime of ∼6 hours at 150 mA

and therefor requires 2-3 injections per day. Each injec-

tion interrupts the user operation for approximately 30

minutes and aﬀects the users’ experiments for another

nearly 1 hour due to thermal load changes on the compo-

nents of optical beamlines after the injection. Therefore

a RW, a chain of combined function magnets, was pro-

FIG. 4. On-axis dipole and quadrupole components of the

RW.

posed to be installed in the dispersive straight section in

the storage ring of the MLS to increase the beam lifetime,

noted in the Fig. 3. The major parameters are listed in

Table II.

TABLE II. Parameters of the RW

Parameter Value

wiggler length 1.9 m

number of poles 12

central pole length 110.47 mm

end pole length 82.85 / 27.62 mm

period length 354.78 mm

maxium on-axis By 1 T

According to Eqs (1–5), the vertical magnetic ﬁeld

and its gradient inside the RW shown in Fig. 4 together

with the positive dispersion yields a negative value of

I4, thus negative damping partition D. Therefore the

transverse emittance xcan be reduced by transferring

longitudinal damping to the horizontal plane, while the

bunch is lengthened due to increased energy spread σδ

[3]. With the vertical white noise excitation acting on

the beam to keep the transverse beam size the same as

that in standard user mode, the lifetime is increased to

∼12 hours at 150 mA because of the increased bunch

volume.

I2=I1

ρ2ds, (1)

I4=I(ηx

ρ+ 2ηx

Bρ

∂By

∂x )ds, (2)

D=I4

,(3)

x∝1

1−D,(4)

σδ∝1

1 + D.(5)

The maximum on-axis By(∼1 T) is close to the dipole

strength(∼1.373 T) in the bending magnet. Although

the RW was carefully designed and optimized, the non-

linear distortion of this strong and long-period (∼0.355

m for one period) insertion device to the stored beam in

the low-energy storage ring is of concern and should be

veriﬁed with symplectic tracking.

III. BASIC CONCEPTS FOR SYMPLECTIC

TRACKING

The problem studied in this paper is the motion of a

particle moving through a static magnetic ﬁeld with a

straight reference trajectory. The magnetic ﬁeld is de-

scribed by a vector potential A= (Ax, Ay, Az) in Carte-

sian coordinate system, so the Hamiltonian for the mo-

tion of a particle is:

H=δ

β0

−az−

s(1

β0

+δ)2−(px−ax)2−(py−ay)2−1

β2

0γ2

(6)

where a particle with charge qand the reference momen-

tum P0has velocity β0cand relativistic factor γ0= (1 −

β02)−1

2and the scaled vector potential a= (ax, ay, az) =

q(Ax, Ay, Az)/P0.

The dynamical variables used in beam dynamics are

deﬁned in the following way: the horizontal and vertical

transverse coordinates are xand y, respectively; their

corresponding momenta pxand pyare deﬁned as:

px=γm ˙x+qAx

,(7)

py=γm ˙y+qAy

.(8)

The longitudinal coordinate is usually expressed as z,

however, lis used to be distinguished from the physical

meaning of subscript zin Eq. (6).

l=s

β0

−ct. (9)

where the particle arrives at position s along the reference

trajectory at time t assuming s = 0 at time t = 0 for the

reference particle.

The longitudinal momentum, referred to as the energy

deviation, is written:

δ=E

cP0

−1

β0

.(10)

The three pairs of canonical variables (x, px), (y, py),

(l, δ) should satisfy the Hamiltonian equations Eq. (11)

and Eq. (12) [6].

dqi

ds =∂H

∂pi

,(11)

dpi

ds =−∂H

∂qi

.(12)

where qi=x,y,land pi=px,py,δ, respectively.

The transformation of the particle from the one posi-

tion sto the next s+ ∆s, equivalent to the solutions of

Eq. (11) and Eq. (12), can be represented by a transfer

map Min the six-dimensional phase space of the canon-

ical coordinates of the particle:

X= (x, px, y, py, z, δ)s+∆s,(13a)

~x = (x, px, y, py, z, δ)s,(13b)

X=M~x. (13c)

It is important that transformation preserves the sym-

plectic nature of the dynamics, otherwise use of non-

symplectic transfer maps can lead to artiﬁcial growth or

damping of the beam motion, resulting in inaccurate in-

formation on the long-term stability of the beam motion.

The criterion of symplectic transformation is:

JT·S·J=S.(14)

where the Jis the Jacobian of the transformation from s

to s+ ∆s,

Jij =∂Xi

∂xj

.(15)

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Sympletictrackingmethodsforinsertiondevices:aRobinsonwigglerexampleJiLi,JorgFeikes,TomMertens,EdwardRial,MarkusRies,AndreasSchalicke,andLuisVeraRamirezHelmholtz-ZentrumBerlinfurMaterialienundEnergieGmbH(HZB),Albert-Einstein-Strae15,12489Berlin,Germany(Dated:October12,2022)Modernsynchrotronlight...

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Sympletic tracking methods for insertion devices a Robinson wiggler example Ji LiJ org Feikes Tom Mertens Edward Rial Markus Ries Andreas Sch alicke and Luis Vera Ramirez

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