DESIGN OF AN ULTIMATE STORAGE RING FOR FUTURE LIGHT SOURCE

DESIGN OF AN ULTIMATE STORAGE RING

FOR FUTURE LIGHT SOURCE

Yichao Jing

Submitted to the faculty of the University Graduate School

in partial fulfillment of the requirement

for the degree

Doctor of Philosophy

in the Department of Physics,

Indiana University

August, 2011

iii

Accepted by the Graduate Faculty, Indiana Univeristy, in partial fulfillment of the

requirement for the degree of Doctor of Philosophy.

Shyh-Yuan Lee, Ph.D.

Chen-Yu Liu, Ph.D.

Doctoral

Committee

Paul E. Sokol, Ph.D.

August 1st, 2011 Rex Tayloe, Ph.D.

v

Copyright c©2011 by

Yichao Jing

ALL RIGHTS RESERVED

vii

To my parents.

ix

Acknowledgments

I would like thank all those who have provided precious help during my graduate

study to make this thesis possible.

I would like to give my sincere gratitude to my advisor Prof. Shyh-Yuan Lee, who

led me into accelerator physics. His deep understanding of physics has shown me the

path through my graduate study. I have benefited a lot from his wide vision and

innovative ideas. He is very supportive and always ready to help in every aspect of

life which makes my living at Indiana University very comfortable and enjoyable. I

was so lucky to meet such a knowledgeable mentor, a good friend and a respectable

senior when I first came to a new country and a new environment.

I would like to thank my committee members, Prof. Paul Sokol, Prof. Chen-Yu

Liu and Prof. Rex Tayloe for their great help on my Ph.D. thesis study. I really

want to thank Prof. Paul Sokol for providing me the great opportunity to work for

ALPHA project where I gained a lot of practical experience. I want to thank Prof.

Chen-Yu Liu and Prof. Rex Tayloe for providing many great help and advices during

my Ph.D. thesis work.

I sincerely thank Dr. Kingyuen Ng, who has provided a lot of useful discussion

and deep insights in many research topics during the past few years.

I wish to thank Dr. Xiaoying Pang and Dr. Xin Wang who graduated from our

group recently. They helped me a lot in every aspect and the time we were working

together will always be a precious memory for me.

Also, I am grateful to my colleagues: Tianhuan Luo, Honghuan Liu, Alfonse

Pham, Hung-Chun Chao, Kun Fang, Zhenghao Gu, Ao Liu and Xiaozhe Shen for

their wonderful friendship and creating such a fun environment for me to live and

work.

x

My biggest thank goes to my parents. Without their support and love, none of

my accomplishments would have been possible. I can only express my gratitude by

dedicating this thesis to them.

xi

Yichao Jing

DESIGN OF AN ULTIMATE STORAGE RING FOR

FUTURE LIGHT SOURCE

Electron storage rings are the main sources of very bright photon beams. They are

driving the majority of condensed matter material science and biology experiments

in the world today. There has been remarkable progress in developing these light

sources over the last few decades. Existing third generation light sources continue

to upgrade their capabilities to reach higher quality photon beam while new light

sources are being planned and designed with ever improving performance. Idea of

ultimate storage rings (USR) has recently been proposed to have beam emittance

down to few tens of pico-meters, reaching diffractive limit of hard X-ray. This theses

work is dedicated to designing a storage ring with ultra-small beam emittance using

n-bend achromat (n-BA) structure.

For ultimate storage rings, large natural chromaticities require strong sextupoles

to correct. Strong non-linear effect requires the study of dynamic aperture (DA).

We calculate and optimize the DA to achieve a 1.5 mm by 1.5 mm aperture size.

Other instabilities such as intra-beam scattering (IBS) and microwave instabilities

(MI) are evaluated self-consistently. Possible free electron laser (FEL) scheme has

been proposed to facilitate the implementation of this ultimate storage ring design.

CONTENTS xiii

Contents

Acceptance iii

Acknowledgments ix

Abstract xi

1 Introduction 1

1.1 Motivation of study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Introduction to Accelerator Physics . . . . . . . . . . . . . . . . . . . 3

1.2.1 Frenet-Serret coordinates and Hill’s equation . . . . . . . . . . 3

1.2.2 Floquet theorem and betatron oscillation . . . . . . . . . . . . 6

1.2.3 Synchrotron motion . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Free Electron Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.1 Angular-Modulated Harmonic Generation (AMHG) and Echo

Enable Harmonic Generation (EEHG) . . . . . . . . . . . . . 11

1.3.2 Optics Free FEL Oscillator (OFFELO) . . . . . . . . . . . . . 15

2 Linear Lattice for 10 pm Storage Ring 19

2.1 10 pm storage ring and n-BA structure . . . . . . . . . . . . . . . . . 20

2.1.1 Theoretical Minimum Emittance (TME) . . . . . . . . . . . . 21

2.1.2 Effort in shortening the circumference . . . . . . . . . . . . . . 25

xiv CONTENTS

2.2 Combined function magnet lattics . . . . . . . . . . . . . . . . . . . . 28

3 Nonlinear Lattice and Dynamic Aperture (DA) optimization 33

3.1 Positive chromaticities and sextupole correction. . . . . . . . . . . . . 34

3.2 Dynamic aperture and tune shift with amplitude . . . . . . . . . . . . 36

3.3 Injection issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4 Intra-beam Scattering (IBS) and Microwave Instability (MI) 43

4.1 Single bunch collective instability . . . . . . . . . . . . . . . . . . . . 46

4.2 Intra-beam scattering (IBS) and its effect . . . . . . . . . . . . . . . . 50

4.2.1 Comparison between microwave instability and IBS effect . . . 55

4.3 SASE FEL performance study under microwave instability . . . . . . 56

5 Conclusions 63

Appendix 65

A Undulator theory and laser study 65

A.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

A.2 Elastic photon-electron collision . . . . . . . . . . . . . . . . . . . . . 65

A.3 Klein-Nishina formula . . . . . . . . . . . . . . . . . . . . . . . . . . 68

A.3.1 Total cross section and differential cross section . . . . . . . . 68

A.4 Laser-Beam Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 69

A.5 Laser induced damping . . . . . . . . . . . . . . . . . . . . . . . . . . 74

B Multipole effect on higher order momentum compaction factor 81

B.1 Multipoles’ effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

B.1.1 Momentum compaction factor . . . . . . . . . . . . . . . . . . 82

B.1.2 Higher order dispersion . . . . . . . . . . . . . . . . . . . . . . 90

CONTENTS xv

B.1.3 Multipole effect using Hamiltonian expansion . . . . . . . . . 93

Bibliography 95

xvi CONTENTS

LIST OF TABLES xvii

List of Tables

2.1 Parameters for 10pm storage ring. . . . . . . . . . . . . . . . . . . . . 23

2.2 Parameters for 10pm storage ring with 25BA. . . . . . . . . . . . . . 27

xviii LIST OF TABLES

LIST OF FIGURES xix

List of Figures

1.1 Frenet-Serret coordinate system. . . . . . . . . . . . . . . . . . . . . . 4

1.2 Schematic drawing of SASE FEL and undulators. . . . . . . . . . . . 10

1.3 Oscillator FEL involves a shorter undulator and two reflective mirrors

to confine the optical wave. Laser reaches saturation in N turns. . . . 12

1.4 Initial beam longitudinal phase space (a) with vertical axis the particle

momentum. Different colors are used to depict different regions in ini-

tial particle distribution. Energy modulation is observed after modula-

tor(b). Microbunching forms after chicane with condition R56δ = λ/4(c). 13

1.5 Particles over microbunch when passing through a large dispersive chi-

cane with R56δ = λ(a) and R56δ = 2λ(b). . . . . . . . . . . . . . . . . 15

1.6 Particles form very fine energy strips after the large dispersive chi-

cane(a).Particles further experience energy modulation in second mod-

ulator(b). At the end of 2nd chicane, a density modulation with ul-

trashort period is formed(c). Microbunching is observed in current

distribution at the end of 2nd chicane. . . . . . . . . . . . . . . . . . 16

1.7 Optics Free FEL Oscillator requires two circulating beams. Low energy

beam is the information carrier from modulator to radiator. Radiation

reaches saturation in a few turns. . . . . . . . . . . . . . . . . . . . . 17

xx LIST OF FIGURES

2.1 Plot of TWISS parameters for 11BA structure. Horizontal dispersion

is magnified by 100 times. . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2 Plot of tune space with up to 8th order resonance lines. Red square is

the location for 10 pm storage ring’s tunes. . . . . . . . . . . . . . . . 25

2.3 Plot of TWISS parameters for 25BA structure. Horizontal dispersion

is magnified by 100 times. . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4 TWISS parameters for a superperiod of the combined function magnet

lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.5 Dispersion vs matching quadrupole strength for the combined func-

tion magnet lattice. The different colors represent different drift space

lengths. Longer drift space requires weaker matching quadrupole strength.

Boundary reaches stability limits. . . . . . . . . . . . . . . . . . . . . 29

2.6 Beta funtion vs matching quadrupole strength for the combined func-

tion magnet lattice. The different colors represent different drift space

lengths. Solution found from Fig.2.5 does match to theoretical value.

Kc = 0.5(1/m2) is too small for this case. Plot’s boundary reaches

stability limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.7 Beta tune space plot for the combined function magnet lattice. The

different colors represent different drift space lengths. Longer drift

space results in a larger ratio between βx and βy thus a change in

the quadrupole strength is not sensitive in changing βy. Thus vertical

betatron tune is not changed much. Plot’s boundary reaches stability

limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1 4000 turn dynamic aperture for 10 pm storage ring. 1.5 mm aperture

is obtained after correcting the large tune shift with amplitude. . . . 37

3.2 Quadratic tune amplitude dependence. . . . . . . . . . . . . . . . . . 38

LIST OF FIGURES xxi

3.3 ICA analysis for a particle in DA with a small initial offset. It ex-

periences mostly betatron oscillation although some of the instabil-

ity induced by nonlinear effect can be observed in temporal wave

function(up-right plot). . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4 ICA analysis for a particle in DA with a large initial offset. This mode

shows frequency spectrum with noisy peaks which indicates the particle

experiences many different resonances at the boundary of DA. . . . . 41

4.1 Schematic drawing of a longitudinal impedance. . . . . . . . . . . . . 45

4.2 Bunch length vs beam current for ALS. Bunch lengthening is observed

due to single bunch microwave instability. . . . . . . . . . . . . . . . 48

4.3 Rms energy spread vs beam current for ALS. Energy spread can be

blown up by a few times under single bunch microwave instability. . . 49

4.4 Bunch length vs beam current for SPEAR3. It has similar performance

as ALS due to the similar parameters of the storage ring. . . . . . . . 50

4.5 Rms energy spread vs beam current for SPEAR3. The calculated FEL

parameter is much lower than rms energy spread so SASE FEL is not

possible. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.6 Bunch length vs beam current for 10 pm storage ring. Bunching factor

is very big that peak current can reach few kA when beam current is

high. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.7 Rms energy spread vs beam current for 10 pm storage ring. FEL

parameter is closer to rms energy spread when beam current is low. . 53





xxii LIST OF FIGURES













A.1 A 2D plot of differential cross section vs photon energy according to

Klein-Nishina formula. At half of the peak energy, the differential cross

section is half of the value of its peak cross section. . . . . . . . . . . 70

A.2 A 3D plot of differential cross section vs photon energy and emission

angle θ according to Klein-Nishina formula. . . . . . . . . . . . . . . 71

A.3 Ratio between damping times induced by laser and dipoles under dif-

ferent laser cross section and laser power. . . . . . . . . . . . . . . . . 77

A.4 Horizontal damping time induced by laser under different laser cross

section and laser power. . . . . . . . . . . . . . . . . . . . . . . . . . 78

A.5 Vertical damping time induced by laser under different laser cross sec-

tion and laser power. . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

A.6 Longitudinal damping time induced by laser under different laser cross

section and laser power. . . . . . . . . . . . . . . . . . . . . . . . . . 80

Introduction 1

Chapter 1

Introduction

1.1 Motivation of study

Storage rings are the main sources of high-brightness photon beams. They are driv-

ing the majority of condensed matter material science and biology experiments in the

world today. There has been remarkable progress in developing these light sources

over the last few decades. Existing third generation light sources continue to upgrade

their capabilities to reach lower emittances and smaller energy spreads, while new

light sources are being planned and designed with ever improving performance [1, 2].

As light sources, storage rings have many attractive features. They provide a wide,

easily tunable energy spectrum from infrared to hard X-ray with high repetition rates

thus high average flux and brightness. The beams are very stable in energy, inten-

sity, position, and size. Storage rings usually have many beamlines which can serve

many experiments simultaneously and reliably. The cost for each user is also consid-

erably low. Besides this combination of properties, storage rings can be designed to

implement other advanced techniques such as free electron laser (FEL), which offers

extremely high peak brightness in much shorter pulse durations but with typically

2 1. Introduction

lower repetition rate. Although FELs are playing more and more prominent role

in biology science and material science which require ultra short pulses with high

instantaneous brightness for tissue tomography, a broad class of X-ray science still

relies on the low peak brightness (to avoid damaging samples) and high photon pulse

repetition rates (to reach sufficient flux) provided by storage rings. Such experiments

simple cannot be conducted using the ultra-high peak brightness from FEL sources.

Storage rings will continue to be the important sources for a large user community

for the indefinite future.

While storage rings are a “mature” technology, they have the potential for sig-

nificantly enhanced performance. One can imagine an ultimate storage ring that

produces high- brightness, transversely coherent X-rays while simultaneously serves

dozens of beamlines and thousands of users annually. For such a source to maximize

transverse photon coherence, the beam emittance must be extremely small in both

transverse planes, approaching and even exceeding the wavelength-dependent diffrac-

tion limit. Storage ring sources have achieved diffraction limited emittances for hard

X-rays in the vertical plane by minimizing horizontal-vertical beam coupling, but

horizontal emittance must be reduced by a factor of 100 or more from the lowest

emittance values achieved today to reach that limit. On the other hand, possible

designs for ultimate rings would necessarily have large circumference and large num-

ber of magnets. Given present day’s technology, there is no difficulties in reaching

such low emittances. However, the cost of such big rings would be considerably high

due to the large vacuum system and magnet construction. An ultimate storage ring

would retain all the general strengths of today’s storage rings mentioned above while

delivering high transverse coherence up to the hard X-ray (≈10 keV) regime.Ultimate

storage rings would have brightnesses and coherent flux one or two orders of mag-

nitude higher than the highest performance ring-based light sources in operation or

presently being constructed.

1.2 Introduction to Accelerator Physics 3

In this thesis, the author intends to present the current progress of designing an

ultimate storage ring including linear lattice design and nonlinear properties study,

while instabilities study will also be discussed. The following 4 chapters are orga-

nized in following orders. First chapter gives an introduction of basic accelerator

physics including Hill’s equation, Floquet transformation, betatron motion and syn-

chrotron motion. There will also be an introduction about Free Electron Laser(FEL)

which we want to implement on this ultimate storage ring. In the second chapter,

we report the linear lattice design which achieves Theoretical Minimum Emittance

(TME) and study the possibility of using combined function magnets to shorten the

circumference. In the third chapter, the calculation and optimization of dynamic

aperture (DA) and effort of understanding the resonances in DA are presented. In

the fourth chapter, the effects of beam instabilities as Microwave Instability (MI) and

Intra Beam Scattering (IBS) are analyzed and calculated. Possible FEL performance

is also evaluated. Chapter five will be the conclusion.

1.2 Introduction to Accelerator Physics

1.2.1 Frenet-Serret coordinates and Hill’s equation

In accelerator, a reference orbit (or designed orbit) is formed once bending magnets

are in place. Under perfect conditions, particles will follow the reference orbit when

circulating in the accelerator. In reality, particles have small amplitude oscillation

around the reference orbit which we call betatron oscillation. To discuss particle

motion with respect to the reference orbit, we use Frenet-Serret coordinate system as

is shown in Fig. 1.1. In the Frenet-Serret coordinate system, particle position can be

expressed as

−→r = −→r0 + xx+ zz (1.1)

4 1. Introduction

x

z

s

Reference Orbit

Particle Position

v

r0

→

r

→

Figure 1.1: Frenet-Serret coordinate system.

where x is the radial (horizontal) unit vector, z is the normal (vertical) unit vector

and s is the tangential (longitudinal) unit vector. The three unit vectors x, s and z

form the basis of the curvilinear coordinate system. In this new coordinate system,

particle motion can be described by a new Hamiltonian

H = −(1 +x

ρ)[

(H − eφ)2

c2−m2c2 − (px − eAx)

2 − (pz − eAz)2]1/2 − eAs, (1.2)

where px, pz are transverse momenta, Ax, Az, As are the vector potentials, H = −psis the new Hamiltonian and (x, px, z, pz, t,−H) are the new phase space coordinates.

We have corresponding Hamilton’s equations

t′

= −∂H∂H

,H′

=∂H

∂t; x

′

=∂H

∂px, p

′

x = −∂H∂x

; z′

=∂H

∂pz, p

′

z = −∂H∂z

. (1.3)

where the apostrophe indicates differentiation with respect to s and we use s instead

of t as the new independent variable. If we look into the Hamiltonian described in


Eq.(1.2), the first two terms in the middle parentheses is particle’s total momentum

p squared. Typically, the transverse momenta px and pz are much smaller than total

momentum p. We can expand the Hamiltonian up to second order in px and pz

H ≈ −p(1 +x

ρ) +

1 + x/ρ

2p[(px − eAx)

2 + (pz − eAz)2] − eAs. (1.4)

Applying Hamilton’s equations in transverse directions to Eq.(1.4), we end up

with betatron equations of motion

x′′ − ρ+ x

ρ2= ±Bz

Bρ

p0

p(1 +

x

ρ)2, (1.5)

and

z′′

= ∓Bx

Bρ

p0

p(1 +

x

ρ)2, (1.6)

where the upper and lower signs are correspondent to positive and negative charged

particles respectively, Bρ = p0/e is the beam rigidity for a reference particle. With

the magnetic field expansion

Bz = ∓B0 +B1x, Bx = B1z, (1.7)

with B1 = ∂Bz

∂x, we can get Hill’s equations

x′′

+Kx(s)x = ±∆Bz

Bρ, z

′′

+Kz(s)z = ∓∆Bx

Bρ, (1.8)

with Kx(s) = 1/ρ2 ∓ B1/Bρ and Kz(s) = ±B1/Bρ being the horizontal and vertical

focusing functions. The inhomogeneous term on the equation’s right illustrates the

field imperfections and higher order magnet components. For an ideal accelerator

with pure dipole and quadrupole fields, the Hill’s equations become homogeneous

and focusing functions Kx(s), Kz(s) are periodic with a period of accelerator circum-

ference.

6 1. Introduction

1.2.2 Floquet theorem and betatron oscillation

As we discuss above, the focusing functions satisfies relation Ky(s+C) = Ky(s) with

subscript y denoting both horizontal and vertical directions. Thus the solution of

Hill’s equation can be written in such a form

y = aw(s)ejψ(s), (1.9)

where oscillation amplitude satisfies w(s+C) = w(s) and phase ψ(s+C) = ψ(s)+Φ

with Φ the phase advance in one revolution. If the accelerator has a symmetric

structure and, is composed of superperiods with period of L, then we can impose

stronger requirement and ask for a periodic solution over superperiods. Plugging

Eq.(1.9) into Eq.(1.8) results in the differential equations:

w′′ +K(s)w − 1

w3= 0, (1.10)

ψ′ =1

w2. (1.11)

The Courant-Snyder parameters are related to the amplitude function by:

β = w2, α = −ww′, γ =1 + α2

β. (1.12)

Hence, the betatron function and phase advance can be written in forms of:

1

2β ′′ +K(s)β − 1

β[1 + (

β ′

2)2] = 0, (1.13)

ψ(s) =

∫ s

0

ds

β(s). (1.14)

Thus a general solution of Hill’s equation is:

y(s) = a√

β(s) cos (ψ(s) + χ) (1.15)

where a√β is the oscillation amplitude for a single particle and νy = (ψ(s + C) −

ψ(s))/2π is the betatron tune depicting the number of betatron oscillations in one


revolution. For a beam which is a cluster of particles, a√β is the rms size of the beam

and εu = a2 is the unnormalized rms emittance of the beam which is the phase space

area divided by π. When there is no acceleration, εu is an invariant quantity. When

beam acceleration happens, normalized emittance given by εn = βγεu is invariant.

The phase space coordinates (y, y′) depict the particle’s position and deflection

angle (velocity) and can be transported in an accelerator by transfer map (linear map

is 6D matrix including 4 dimensions in transverse and 2 dimensions in longitudinal

direction) from point to point. For a linear system, the phase space coordinates at

any particular position can be obtained by propagating from any initial position.

y

y′

= M(s, s0)

y0

y′0

, (1.16)

with M(s, s0) the transfer matrix from initial s0 to final s. In any beam transport

line it can be expressed by Courant-Snyder parameters as:

M(s, s0) =

√

β(s) 0

− α(s)√β(s)

1√β(s)

cosφ sin φ

− sin φ cosφ

1√β0

0

α0√β0

√β0

,

where φ = φ(s) − φ0 is the phase advance from initial position to final position.

For one complete revolution, the transfer matrix can be simplified to:

M(s0, s0) =

cos Φ + α0 sin Φ β0 sin Φ

−γ0 sin Φ cos Φ − α0 sin Φ

, (1.17)

where Φ represents the phase advance for one complete revolution.

1.2.3 Synchrotron motion

We have discussed transverse betatron oscillation in an accelerator. Particles also

experience longitudinal oscillation which is called synchrotron oscillation. With lon-

gitudinal electric field provided by rf cavity, particles gain energy or lose energy in

8 1. Introduction

longitudinal direction. We can derive conjugate equations of longitudinal motion:

d

dt(∆E

ω0) =

1

2πeV (sinφ− sinφs), (1.18)

anddφ

dt=hω2

0η

β2E(∆E

ω0), (1.19)

where V is the voltage across rf cavity, φ and φs are the rf phases for off momentum

particle and synchronous particle respectively, h is the harmonic number of rf cavity

and η is the phase slip factor. Thus we can write the Hamiltonian for synchrotron

motion

H =1

2

hηω20

β2E(∆E

ω0

)2 +eV

2π[cosφ− cos φs + (φ− φs) sinφs]. (1.20)

Using definition of fractional momentum spread

δ =∆p

p0=

ω0

β2E

∆E

ω0, (1.21)

we can express the Hamiltonian in a new form

H =1

2hηω0δ

2 +ω0eV

2πβ2E[cosφ− cosφs + (φ− φs) sinφs]. (1.22)

When particle is experiencing slow acceleration, one should use the Hamiltonian given

by Eq.(1.20) for phase space tracking. On the other hand, if there is no acceleration

(storage mode), one should use the Hamiltonian given by Eq.(1.22) for turn by turn

phase space tracking.

Two fixed points (φs, 0) and (π − φs, 0) can be easily found for the Hamiltonian.

Around the stable fixed point (φs, 0), the particle’s motion is elliptical, while being

hyperbolic around unstable fixed point (π − φs, 0). Starting with Hamiltonian given

by Eq.(1.22), for a small amplitude oscillation around stable fixed point, particle’s

motion becomes simple harmonic oscillation

d2

dt2(φ− φs) =

hω0eV η cosφs2πβ2E

(φ− φs). (1.23)

1.3 Free Electron Laser 9

The stability condition requires η cos φs ≤ 0. So below transition energy η < 0, the

synchronous phase should satisfy 0 ≤ cosφs ≤ π/2. On the other hand, when above

transition energy η > 0, the synchronous phase should satisfy π/2 ≤ cos φs ≤ π.

Thus we can define the synchrotron tune (number of synchrotron oscillations per

revolution) to be

Qs =ωsω0

=

√

−heV η cosφs2πβ2E

. (1.24)

Typically synchrotron tune is a small number of the order of 10−3 − 10−1.

As we can see from above equation, when the phase slip factor η goes to zero, the

synchrotron tune becomes zero thus the longitudinal phase space freezes and we call

this isochronous condition. This would have some special applications in FEL when

we need to preserve longitudinal microbunching structure.

1.3 Free Electron Laser

Free Electron Laser (FEL) is a technique first invented in 1976 by John Madey involv-

ing coherent addition of synchrotron radiation emitted by electrons passing through

periodic structure like alternating magnetic fields–undulators as is shown in Fig. 1.2.

It usually requires long undulator before the radiation can reach saturation. During

this process, radiation spectrum starts out to be noise like with all wavelengths and

a single wavelength λ determined by the period of undulator, electron beam energy

and undulator parameter K = 0.94B0[Tesla]λu[cm] as

λ =λu2γ2

(1 +K2

2) (1.25)

will experience a power growth and peak out with very high intensity. Usually the

peak brightness of FEL is at least 10 orders of magnitude higher than its peer from

a 3rd generation light source. This process is also called Self-Amplified Spontaneous

10 1. Introduction

Figure 1.2: Schematic drawing of SASE FEL and undulators.

Emission (SASE). The power growth process requires the FEL parameter, given by

ρFEL =1

2

(

I

IA

λ2wK

2w

2πγ3

1

4πσxσy

)1/3

, (1.26)

larger than the rms energy spread σE . Thus one needs to have a very good qual-

ity beam with high peak current, low sliced emittance and low sliced energy spread

(most important!). Electron beam properties in Linacs are determined by the injec-

tor. According to current technology, injectors can be designed to have ultra small

emittance and low energy spread and bunch current can be made very high with

bunch compressors so Linacs are good candidates for FEL. On the other hand, prop-

erties of the electron beams in storage rings are equilibrium values which involve

radiation damping, quantum excitation, intra beam scattering and many other fac-

tors. The equilibrium rms energy spread is usually high when comparing with Linacs

due to the quantum excitation. Also the bunch current is kept low so that many of


the instabilities (intensity dependent) don’t appear to destroy the storage of electron

beam.

A FEL technique called oscillator FEL (shown in Fig. 1.3) was invented to over-

come these weaknesses of storage rings. It requires the implementation of a relatively

shorter and weaker undulator together with optical cavities to store the radiation

waves. During every revolution, the electron beam interacts with optical wave in

undulator and losses a little fraction of its energy to the radiation. The radiation

keeps growing and reaches saturation in many interactions with the electron beam.

The growth of the power stored in the cavities needs to be larger than the losses at

the optical mirrors so that the growing process can be continued. The gain of FEL

usually does not have to be high as long as a mirror with high reflectivity is chosen

so that it does not require such a high current electron beam. Because the oscillator

FEL reaches saturation rather slowly comparing with SASE FEL and requires many

interactions between electron beam and optical laser, it requires very fine alignment

of the optical mirrors for the least degradation of FEL performance. If one is tar-

geting hard X-ray which can be used for biology material science experiments and

crystallography experiments, one would have difficulties in finding proper materials

for reflection mirrors. Recently, some mirrors made of diamonds have been proposed

but none experimental data has proven its validity. Some innovative ideas have come

out in the effort of solving this problem.

1.3.1 Angular-Modulated Harmonic Generation (AMHG) and

Echo Enable Harmonic Generation (EEHG)

In the standard High-Gain Harmonic Generation (HGHG) scheme [3], two stages of

undulators are implemented. A seed laser with wavelength λ is first used to generate

energy modulation in the electron beam in the first undulator–modulator. After

12 1. Introduction

Figure 1.3: Oscillator FEL involves a shorter undulator and two

reflective mirrors to confine the optical wave. Laser

reaches saturation in N turns.

passing through a chicane satisfying constraint R56δ = λ/4, the energy modulation is

converted to density modulation with the information of high harmonics of wavelength

λ. Then the density modulated beam is sent into the second undulator–radiator to

generate coherent radiation at wavelength λ/n. The phase space evolution of this

process is shown in Fig. 1.4. Typically generating the nth harmonic of the seed laser

requires the energy modulation amplitude to be approximately n times larger than

the beam energy spread. Because of the inherent large energy spread of the beam in

storage rings, the harmonic number is limited to about 3 to 5, good for ultraviolet but

still one order of magnitude lower than soft X-ray. One may think about making use

of the tiny vertical emittance of the beam in storage rings and propose an angular

modulation instead of energy modulation to achieve higher harmonics. A vertical

wiggling motion of the electron beam in modulator will introduce angular modulation


Figure 1.4: Initial beam longitudinal phase space (a) with verti-

cal axis the particle momentum. Different colors are

used to depict different regions in initial particle dis-

tribution. Energy modulation is observed after modu-

lator(b). Microbunching forms after chicane with con-

dition R56δ = λ/4(c).

to the beam. A chicane between modulator and radiator with nonzero R54 with

proper focusing magnets will transfer the angular modulation to phase modulation.

The electron beam will further generate coherent radiation at nth harmonic. The

bunching factor is crucial during this process. For the nth harmonic, it scales with

bn ∝ e−1

2n2r2, (1.27)

where n is the harmonic number and r = kσ′

yR54 with k the wave number of seeded

laser and σ′

y the rms angular width for the electron beam. In order to make the

bunching factor for the nth harmonic large enough the modulated angular or energy

amplitude should be n times larger than beam divergence or energy spread respec-

tively. For HGHG, due to the large energy spread in storage rings, laser power needed

for generating such a large energy modulation is high. Also the energy spread after

14 1. Introduction

lasing is high so may affect beam stability. If one uses angular modulation, with very

tiny vertical emittance, laser power needed to generate the angular modulation is

significantly lowered and the energy spread growth during lasing is very low given by

∆γ =√

2Bγεy/λ, (1.28)

where B is the angular modulation amplitude and εy is the vertical emittance. Typi-

cally, the harmonic number using angular modulation can be up to around 50 which

reaches soft X-ray regime.

Going back to HGHG, when two stages of modulators and chicanes are imple-

mented, so called EEHG [4] can also further extend to higher harmonic number. The

electron beam in the first modulator develops energy modulation before entering a

strong dispersive chicane with large R56. As shown in Fig. 1.5, the longitudinal phase

space will be over rotated so a fine strip pattern forms in energy. After it experiences

further energy modulation in the second modulator and a bunch compressor(2nd

chicane), a very fine microbunching exists in the electron beam, which can radiate

ultrashort X-ray in radiator. The longitudinal phase space evolution during the whole

process is shown in Fig. 1.6. The bunching factor of EEHG decays much slower over

harmonics comparing with HGHG or AMHG

bn ∝ n−1/3. (1.29)

For EEHG, harmonic number can go up to a few hundred. But the phase space

control is very constringent considering multiple stages are being implemented. Also

the coherent synchrotron radiation (CSR) and incoherent synchrotron radiation (ISR)

introduces a large energy spread which will smear out the fine energy bands required

for EEHG. Thus the beam current cannot be high as CSR scales with the beam

current. The beam energy also cannot be very high as ISR scales with the beam

energy to the 7th power. Many other practical issues such as rf power jittering will

severely affect the performance of EEHG and experimental proof is yet to come.


Figure 1.5: Particles over microbunch when passing through a

large dispersive chicane with R56δ = λ(a) and R56δ =

2λ(b).

1.3.2 Optics Free FEL Oscillator (OFFELO)

Recently, people have invented a new method–Optics Free FEL Oscillator (OFFELO)

to overcome the shortage of proper mirrors working in X-ray regime. OFFELO re-

quires two circulating beams (storage rings or Energy Recoverty Linacs) with one

being low energy and the other being high energy. Schematic drawing is shown in

Fig. 1.7. At first, a fresh low energy beam goes into modulator and develops a little

bit of the microbunching structure. Then it circulates one turn and enters radiator

with microbunching structure preserved. In radiator, the low energy beam gener-

ates a light at the wavelength of its microbunching period and goes to dump after

the radiator. This generated light acts as a seed and interacts with the high energy

circulating beam in a long undulator and enters high gain regime thus power of the

light grows up fast. After the interaction with high energy beam, the light with high

power goes into modulator as a seed and interacts with another fresh beam with low

energy in the modulator. Due to the high power the light carries, this low energy

beam starts out in its 2nd run with stronger microbunching structure thus radiates

16 1. Introduction

Figure 1.6: Particles form very fine energy strips after the large

dispersive chicane(a).Particles further experience en-

ergy modulation in second modulator(b). At the end

of 2nd chicane, a density modulation with ultrashort

period is formed(c). Microbunching is observed in cur-

rent distribution at the end of 2nd chicane.

a light with higher power in the radiator. After a few iterations, the power of the

light radiated will reach saturation and FEL is done without using any of the optical

mirrors. One important and most difficult problem for this process to realize is to

preserve the longitudinal phase space structure for low energy beam when it circulates

one turn. As we have discussed before, particles are doing oscillation (rotation) in

longitudinal phase space and their time structure will be completely destroyed after

14

of the synchrotron period. However, if phase slip factor is zero(up to a few orders of


Figure 1.7: Optics Free FEL Oscillator requires two circulating

beams. Low energy beam is the information carrier

from modulator to radiator. Radiation reaches satura-

tion in a few turns.

δ), the synchrotron motion is very slow. Thus the particle distribution in longitudinal

direction freezes during the transportation from modulator to radiator so that any of

the microbunching structure (starts out to be small) developed in modulator can be

preserved. Zeroth order of η can be made zero with stronger focusing quadrupoles

so that particles with different energies are strongly focused to have the same pass

length. Higher orders of η are also important in this sense and they can be tuned

to zero with the help of higher order magnets such as sextupoles, octupoles, etc. A

systematic study of how magnets affect phase slip factor or momentum compaction

factor will be discussed in the appendix of this thesis.

Another important issue is the repetition rate. If the high energy beam’s current

is very high, although the power of the light grows up fast and can reach saturation in

fewer iterations, the energy spread growth in high energy beam induced by this FEL

interaction is very large. If high energy beam is a storage ring beam, it usually take

18 1. Introduction

a few thousand turns of revolutions to damp the energy spread back to a small value.

This would greatly reduce the reptition rate of the lasing. If both beams are from

ERL and the energy spread blown up by this process is still within the acceptance of

energy recovery to work, then it requires superconducting cryomodules and does not

have advantage in rep rate comparing with what is now achieved in ERL based FEL.

It only has advantage in extending the radiation spectrum to X-ray or even hard

X-ray which cannot be possible with the current ERL based FEL where reflective

cavities are used. This is being achieved at a very high cost (construction of two

ERLs). A technical breakthrough in mirror design is probably more economical and

affordable.

Linear Lattice for 10 pm Storage Ring 19

Chapter 2

Linear Lattice for 10 pm Storage

Ring

Most electron storage rings in the world are designed to make use of synchrotron

radiation. The brightness of a storage ring is given by B =Fphoton

(2πεx)(2πεy)where εx, εy

are the transverse electron beam emittances. The photon flux Fphoton is the number

of photons per unit time in a given bandwidth ∆ω/ω which by convention is usually

chosen to be 0.1%. For a typical 3rd generation light source, to achieve high beam

brightness, the transverse emittances are usually very small (≈ 1 nm for horizontal

emittance and less than 1% for vertical emittance) and beam current is high. The

beam brightness usually ranges from 1020 to 1023 photons/(s mm2-mrad2 0.1% of

bandwidth).

When a storage ring has a small emittance that reaches the diffractive limit of

the radiation, the brightness will greatly increase not only because the transverse

beam size is small but also because the photons add up coherently. This condition is

satisfied when εy ≤ λ/4π, where y stands for both horizontal and vertical directions.

Due to the lack of vertical bending magnets, the vertical emittance is usually induced

20 2. Linear Lattice for 10 pm Storage Ring

by field errors or misalignments in magnets. Skew quadrupoles can be implemented

to couple the horizontal emittance to vertical direction. The vertical emittance for a

typical 3rd generation light source can reach diffractive limit by a very small coupling

coefficient (≤ 1%). To have both planes reaching diffractive limit, we need to design

a storage ring with ultimate low horizontal emittance.

In this chapter, we will describe the linear lattice design of an ultimate storage

ring using n-BA structure.

2.1 10 pm storage ring and n-BA structure

In order to maximize the number of straight sections, achromat structure with n bend-

ing magnets (n-BA) is used. Dispersion function outside of the achromat structure

is zero so insertion devices or user beamlines can be implemented. Due to the break

of symmetry, its theoretical minimum emittance (TME) is larger than nonachromat

lattice which can be purely symmetric and have nonzero dispersion everywhere.

Typically, the natural emittance for a storage ring is given by

εx = FlatticeCqγ2θ3, (2.1)

where Cq = 3.83 × 10−13m is a radiation constant, and θ is the total bending angle

in each dipole. The scaling factor Flattice is

Flattice =〈H〉dipJxρdipθ3

, (2.2)

where 〈H〉dip is the average H-function over all the dipoles and Jx is the horizontal

damping partition number. Flattice is a quantity that depends on the design of the

storage ring lattice. For a Theoretical Minimum Emittance (TME) lattice with non

zero disperion, it is 1/(12√

15Jx).

From Eq.(2.1), we know that the most efficient way of making natural emittance

small is to reduce θ. Thus we need to increase the total number of dipoles. For 10

2.1 10 pm storage ring and n-BA structure 21

pm storage ring, we use 11BA structure with total of 440 dipoles separated into 40

superperiods. Each superperiod has 11 dipoles with non zero dispersion inside the

superperiod. Dispersion is closed between superperiods so a 10 meter long straight

section can be used for insertion devices and user beamlines.

2.1.1 Theoretical Minimum Emittance (TME)

To match to the TME lattice, we need to minimize the 〈H〉dip. Starting from transport

matrix theory, the dispersion function and its first derivative in dipoles are given by

D = ρ(1 − cosφ) +D0 cosφ+ ρD′0 sinφ, (2.3)

and

D′ = (1 − D0

ρ) sinφ+D′

0 cos φ, (2.4)

where D0 and D′0 are the dispersion function and its first derivative at the entrance

of the dipole and φ is the phase advance along the dipole. Thus H-function which is

defined as

H = γD2 + 2αDD′ + βD′2, (2.5)

can be expressed as

H(φ) = H0 + 2(α0D0 + β0D′0) sinφ− 2(γ0D0 + α0D

′0)ρ(1 − cos φ) + β0 sin2 φ

+ γ0ρ2(1 − cosφ)2 − 2α0ρ sinφ(1 − cosφ).

After averaging the phase advance in the dipole, we arrive at

〈H〉 = H0 + (α0D0 + β0D′0)θ

2E(θ) − 1

3(γ0D0 + α0D

′0)ρθ

2F (θ)

+β0

3θ2A(θ) − α0

4ρθ3B(θ) +

γ0

20ρ2θ4C(θ),

where

E(θ) = 2(1 − cos θ)/θ2, F (θ) = 6(θ − sin θ)/θ3, A(θ) = (6θ − 3 sin 2θ)/(4θ3),

B(θ) = (6 − 8 cos θ + 2 cos 2θ)/θ4, C(θ) = (30θ − 40 sin θ + 5 sin 2θ)/θ5.


Under small angle approximation, we have A = B = C = D = E = F = 1. We note

that θ here is the bending angle in a dipole.

To find out the minimum H-function without the achromat condition, we can

simply take derivatives with respect to initial D0 and D′0

∂ 〈H〉∂D0

=∂ 〈H〉∂D′

0

= 0,

then we finally arrive at the matching conditions

D∗0 =

1

12Lθ, (2.6)

and

β∗0 =

L√60, (2.7)

where the L is the total length of the dipole. Both dispersion and beta function

have waists at the center of dipoles. In the 10 pm lattice, we use quadrupole triplets

in between two dipoles to match the optics to the TME conditions listed above.

In order to make beta function and dispersion minimum at the center of dipoles,

we choose a layout of QD-QF-QD for the quadrupole triplets. Quadrupoles with

lengths of 25 cm are used and a drift space of 40 cm between quadrupoles is kept

to accommodate sextupoles and avoid collision of magnets. The quadrupole field

gradients are −1.5561/m2, 2.1631/m2 and −1.5561/m2 respectively.

In 10 pm storage ring, we have both achromatic region and non-achromatic region.

We need a good transition between these two regions. In other words, we need to

match the H-functions in both region. For an isomagnetic storage ring, the lengths

of the center dipole and outer dipole should satisfy the condition

L2 = 31/3L1 (2.8)

with L2 the length of center dipoles and L1 the length of the edge ones. For 10 pm

storage ring, we choose middle dipoles to be 1.95 m and edge ones 1.3 m to satisfy this


Table 2.1: Parameters for 10pm storage ring.

Parameter Value

Beam energy 5 GeV

Ring circumference 2663 m

Equilibrium energy spread ∆E/E (rms) 0.0378%

Natural emittance (rms) 9.1 nm-mrad

Natural horizontal chromaticity -595.339

Natural vertical chromaticity -148.741

Horizontal betatron tune 202.89

Vertical betatron tune 33.88

Momentum compaction factor 1.223e-5

requirement. In a storage ring design, another important property is the fractional

energy spread δ = dEE

. Derived from the equilibrium longitudinal emittance, the

fractional energy spread is given by

(σEE

)2 = Cqγ2

JEρ, (2.9)

with JE the longitudinal damping partition number. And we have relations

Jx = 1 − D, Jz = 1, JE = 2 + D. (2.10)

For isomagnetic storage rings with separated function magnet, D = αcRρ

is a small

number. For 10 pm storage ring, we have Jx = 0.99996 and JE = 2.00004. We

choose the bending radius of the dipoles to be 78 m and the length of the edge

dipoles to be 1.3 m so the rms energy spread is 0.0378%. The total circumference

of the ring is 2663m. Table. 2.1 shows the main parameters for this design. Since


0.0 10. 20. 30. 40. 50. 60. 70. 80.s (m)

δ E/ p 0c = 0 .

Table name = TWISS

MINIMUM EMITTANCE FBA1 LATTICELinux version 8.23/08 03/05/11 05.54.14

0.0

2.5

5.0

7.5

10.0

12.5

15.0

17.5

20.0

22.5

25.0

β(m

),DX

100

β x β y DX100

Figure 2.1: Plot of TWISS parameters for 11BA structure. Hori-

zontal dispersion is magnified by 100 times.

in vacuum undulator with period of 1 cm is available and we are aiming at hard

X-ray lasing so we choose nominal beam energy to be 5 GeV and the storage ring

can accommodate energy from 4 GeV to 7 GeV. The optics for one superperiod is

shown in Fig. 2.1. The beta-function at the center of middle dipoles is matched

to Ldip/√

60 and dispersion matched to Ldipθdipole/12 and natural emittance is 9.1

pico-meters for this lattice. We note that this is still 3.3 times larger than the TME

predicted emittance – 2.77 pm because of the breaking of symmetry. The betatron

tunes are chosen to be νx = 202.89 and νy = 33.88 respectively so the zeroth order

tunes stay relatively far away from lower order resonances. In order to move tunes


Figure 2.2: Plot of tune space with up to 8th order resonance lines.

Red square is the location for 10 pm storage ring’s

tunes.

to a safe location without changing optics and lattice properties much, we vary the

quadrupole triplet in the non-dispersive region. The beta functions and dispersions

in the central dipole regions are not changed thus the achieved minimum emittance is

not affected. Fig. 2.2 shows the tune space with up to the 8th order resonance lines.

2.1.2 Effort in shortening the circumference

As we can see, 10 pm storage ring design has a large circumference due to the large

number of dipoles. This is common for ultimate storage ring designs because tiny


0.0 20. 40. 60. 80. 100. 120. 140. 160.s (m)

δ E/ p 0c = 0 .

Table name = TWISS

MINIMUM EMITTANCE FBA1 LATTICELinux version 8.23/08 03/05/11 06.16.57

0.0

2.5

5.0

7.5

10.0

12.5

15.0

17.5

20.0

22.5

25.0

β(m

),DX

100

β x β y DX100

Figure 2.3: Plot of TWISS parameters for 25BA structure. Hori-

zontal dispersion is magnified by 100 times.

dispersion is required to reach low emittance. Large circumference is very costly

especially for the construction of beam tunnel and vacuum system. An alternative

design with shorter circumference, on the other hand, is preferable if the natural

emittance could be maintained at the same level. As we have discussed above, the

emittance of a lattice is determined by the bending angle in the dipole

εx = FlatticeCqγ2θ3. (2.11)

In order to get the same emittance, we need to keep bending angle in each dipole the

same. While keeping the n-BA structure, we change the number of dipoles in each

superperiod and number of superperiods to get the same bending angle. To match the


Table 2.2: Parameters for 10pm storage ring with 25BA.

Parameter Value

Beam energy 5 GeV

Ring circumference 2334 m

Equilibrium energy spread ∆E/E (rms) 0.0378%

Natural emittance (rms) 9.5 nm-mrad

Natural horizontal chromaticity -585.427

Natural vertical chromaticity -141.382

Horizontal betatron tune 189.9

Vertical betatron tune 22.88

Momentum compaction factor 1.4e-5

H-function between dispersive region and non-dispersive region, a ratio of 1/3√

3 should

be maintained between the length of the central dipoles and edge dipoles. A simple

calculation of two cases, 11BA with 40 superperiods and 25BA with 17 superperiods

is shown to have same bending angles

2π

17 ∗ (1.5 ∗ (25 − 2) + 2)∗ 1.5 = 0.015189 ≈ 0.01512 =

2π

40 ∗ (1.5 ∗ (11 − 2) + 2)∗ 1.5.

(2.12)

The optics of this new layout with 25BA structure is shown in Fig. 2.3. Beta function

and dispersion remain the same as previous 11BA structure thus the emittance is

matched to theoretical minimum. Main parameters for 25BA structure is shown in

Table. 2.2. The natural chromaticities remain huge because the total number of mag-

nets and minimum beta functions are similar to the 11BA design. The circumference

of the 25BA lattice is reduced to 2334 m. This seems to be the limit of what we


can achieve unless we implement combined function magnets. The momentum com-

paction factor, αc ∝ Ldipθdip

R∝ Dmin

Rwith R the average radius of the circumference,

is slightly larger than 11BA structure due to the reduction of ring circumference.

2.2 Combined function magnet lattics

0.0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50s (m)

δ E/ p 0c = 0 .

Table name = TWISS

MINIMUM EMITTANCE CFM LATTICELinux version 8.23/08 25/08/10 10.27.37

0.0

1.

2.

3.

4.

5.

6.

7.

8.

9.

β(m

),DX

10 β x β y DX10

Figure 2.4: TWISS parameters for a superperiod of the combined

function magnet lattice.

A straightforward way of thinking to make a storage ring with shorter circum-

ference is to use combined function magnets. By making dipoles with defocusing

gradient, we can absorb two quadrupoles from the quadrupole triplet into the central

dipole and shorten the circumference. The new lattice will be composed of dipoles

with defocusing gradient and single quadrupole in between two dipoles to match the

2.2 Combined function magnet lattics 29

optics to TME. A simple lattice is made in the effort of achieving this purpose. Fig-

ure. 2.4 shows the optics for a central dipole cell in the combined function magnet

lattice. The entire ring is composed of identical structures as previous 11BA lattice

except the dipoles have gradients. The whole lattice resembles FODO lattice. We use

0.8 1 1.2 1.4 1.6 1.8 2 2.210

−3

10−2

10−1

100

101

102

103

Kq(1/m2)

Dx(m

)

Figure 2.5: Dispersion vs matching quadrupole strength for the

combined function magnet lattice. The different colors

represent different drift space lengths. Longer drift

space requires weaker matching quadrupole strength.

Boundary reaches stability limits.

a sort of analytical way to search for all possible solutions. The parametric space for

this structure is relatively simple – the central dipole’s bending angle and bend radius

are determined by the emittance and rms energy spread that we want to achieve with


this lattice. We have only 3 parameters to change: central dipole’s gradient – Kc; drift

space length between dipole and matching quadrupole – L and quadrupole’s gradient

– Kq. By choosing one value for Kc, we can scan through all the L’s and Kq’s that can

possibly provide a periodic solution. We then can obtain a set of beta functions and

dispersion functions and compare all of them with the theoretical minimum values.

If they don’t match, we change the value of Kc and iterate until we find the solution

that is matched to the TME lattice.

0.8 1 1.2 1.4 1.6 1.8 2 2.210

−2

10−1

100

101

102

103

Kq(1/m2

βx(m

)

Figure 2.6: Beta funtion vs matching quadrupole strength for the

combined function magnet lattice. The different colors

represent different drift space lengths. Solution found

from Fig.2.5 does match to theoretical value. Kc =

0.5(1/m2) is too small for this case. Plot’s boundary

reaches stability limits.

2.2 Combined function magnet lattics 31

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

νx

ν y

L=0.1

L=0.5

L=0.9

L=1.3

L=1.7

L=2.1

L=2.5

L=2.85

Figure 2.7: Beta tune space plot for the combined function mag-

net lattice. The different colors represent different drift

space lengths. Longer drift space results in a larger ra-

tio between βx and βy thus a change in the quadrupole

strength is not sensitive in changing βy. Thus vertical

betatron tune is not changed much. Plot’s boundary

reaches stability limits.

Figure. 2.5 shows the plot of calculated dispersion function versus Kq’s over dif-

ferent drift space lengths indicated by different colors. The horizontal dashed line

indicates the theoretical value and the vertical lines are the solutions of Kq we get

from matching the dispersion. When the drift between dipole and quadrupole gets

longer, weaker matching quadrupole is needed. The boundary of plot reaches the


stability limit where the periodic solution no longer exists and either x or y motion

will be unstable. Phase Necktie diagram Fig. 2.7 shows the transverse betatron tunes

encounter half integers when L’s and Kq’s go to their limits. When L is big, βy at

quadrupole location is small thus the change in the quadrupole gradient would not

change βy dramatically thus there would be no big change in vertical tune. When

L is small, βy at the quadrupole location is large so the change in the quadrupole

gradient would change βy dramatically and result in a significant vertical tune change.

Because the matching quadrupole is horizontal focusing, the increase in its strength

would always result in a decrease in βx and increase in βy thus a increase in νx and

a decrease in νy are shown in the Fig. 2.7.

After we find out the required Kq’s for different L’s to match the dispersion

function to TME, we go to the beta function plot Fig. 2.6. The intersections are

higher than the horizontal dashed line – TME beta function solution thus there is no

solution for this case – Kc = 0.5 (1/m2) is too small to match the lattice to TME. We

note that the intersections on Fig. 2.6 have same beta function values which indicates

the drift space length does not affect the matching to TME – only the gradient of

central dipole matters.

The first solution that both dispersion and beta function are matched occurs

when Kc = 1 (1/m2). This lattice has a circumference of 1004 meters which is much

shorter than separated function magnet lattice we mention previously and its natural

emittance is 6.8 pm. But unfortunately, the magnetic gradient B1

B= Kcρdip = 78

(1/m) is too big – comparable with the gradient sectors used in nonscaling FFAGs [10]

so that the implementation of such dipoles in storage ring is unrealistic. We abandon

the idea of combined function magnets and the simulation in the rest of this thesis

will be based on the 11BA lattice presented earlier.

Nonlinear Lattice and Dynamic Aperture (DA) optimization 33

Chapter 3

Nonlinear Lattice and Dynamic

Aperture (DA) optimization

The parameters listed in previous chapter are equilibrium properties. In storage rings,

damping and fluctuation act together and develop the equilibrium state over at least

a few thousand turns of beam revolutions. For 10 pm storage ring, the transverse

damping time is about 200 ms or 23000 turns of beam revolution. Modern storage

rings also require accumulation to achieve high beam current and intensity. It usually

takes a long time which is many times of the damping time to charge the ring up to

the desired current. Thus a beam life time of at least a few hundreds of thousands of

turns is required for 10 pm storage ring.

When a particle is off the design orbit with an offset, it sees a different betatron

tune and may cross some strong resonances that cause particle loss. Strong sextupoles

may also produce nonlinear resonance driving terms that drive certain low order

strong resonances. Due to all these concerns, we need to calculate and optimize the

dynamic aperture and understand the mechanism of nonlinear driving terms and tune

shift with amplitude – how they affect dynamic aperture.

34 3. Nonlinear Lattice and Dynamic Aperture (DA) optimization

3.1 Positive chromaticities and sextupole correc-

tion.

As we will discuss in the next chapter, in real accelerators, discontinuities and im-

perfections of the chamber wall will generate wakefield [8]. This time dependent

short range wakefield which is also called broadband impedance will induce a time-

dependent transverse force on beam passing by thus introduce an forced oscillation.

When a bunch of particles passes by and encounters this broadband impedance, at

mode n, the particles execute collective motion with a coherent frequency ωn,w given

by

ωn,w = (n+Q)ω0, (3.1)

whereQ is the betatron tune and ω0 is the angular revolution frequency. The collective

amplitude at time t can be expressed as

yk = Ykej(ωt−nθ). (3.2)

When n is a positive number, it represents a transverse oscillation faster than nominal

betatron oscillation – fast wave. When n is negative but n + Q is still positive, it

represents an oscillation slower than betatron oscillation – backward wave. When n

is negative and n + Q is also negative, the coherent frequency becomes negative –

slow wave. A further analysis shows the collective wave frequency has a frequency

spread due to the off momentum variable δ

ωn,w = ωn,w0 + [Cy − nη]ω0δ, (3.3)

where Cy = dQ/dδ is the chromaticity and η is phase slip factor. The frequency

spread vanishes at mode number

n0 =C

η. (3.4)

3.1 Positive chromaticities and sextupole correction. 35

When frequency spread is zero, any collective motion will not be damped by Landau

damping. The collective frequency ω in Eq.(3.2) can be written as

ω = ωn,w0 + jeβIZ⊥

4πRγmQω0

, (3.5)

with Zbot the transverse impedance. For a fast wave or backward wave, the real

part of ω is positive thus the real part of the transverse impedance becomes positive.

Collective motion given in Eq.(3.2) will be damped. On the other hand, for a slow

wave, the real part of ω is negative thus the real part of the transverse impedance

becomes negative. Collective motion given in Eq.(3.2) will grow over time. A beam

with zero frequency spread will suffer slow wave collective instability. Because η is

a positive number (above transition) and usually order of 10−3 for a 3rd generation

light source and 100 times smaller for 10 pm storage ring, when we have a negative

chromaticity, it will result in having a large negative n0 thus result in a slow wave.

This will cause instability and we need to change chromaticity to be zero or positive.

Natural chromaticities, arising solely from quardrupoles and given by

Cy,nat = − 1

4π

∮

βyKyds, (3.6)

are very large negative numbers for 10 pm storage rings as shown in previous chapter

due to the large number of strong focusing quadrupoles we use for optics matching.

To correct these large natural chromaticities to positive numbers, we need to use

many families of strong sextupoles. In the presence of sextupoles, the chromaticity is

shifted by

Cx = − 1

4π

∮

βx[Kx(s) − S(s)D(s)]ds, (3.7)

Cz = − 1

4π

∮

βz[Kz(s) + S(s)D(s)]ds, (3.8)

where S(s) = −B2/Bρ is the effective sextupole strength with the convention of pos-

itive meaning horizontal focusing. D(s) is the dispersion function at location s. To


change chromaticities most efficiently, we need to place chromatic sextupoles close to

quadrupoles where βxD(s) and βzD(s) are maximum. We also need to put the focus-

ing sextupoles at a place where a large ratio of βx/βz is satisfied and vice versa for

the defocusing sextupoles so that we can change the transverse chromaticities inde-

pendently with different sextupole families. For 10 pm storage ring, we use 8 families

of sextupoles. Four of them are located in dispersive region close to quadrupoles to

move chromaticities to positive. Four of them are located in non-dispersive region to

change driving terms and tune shift with amplitude which we will discuss later in this

chapter. Due to the large natural chromaticities, the required sextupole strengths are

large – maximum at about 350 1/m2. Therefore, nonlinear resonances induced by

sextupoles are strong and dynamic aperture is very small.

3.2 Dynamic aperture and tune shift with ampli-

tude

Using simulation code ELEGANT [7], we were able to calculate and optimize dynamic

aperture. In the input “ele” file, we optimize the DA by varying sextupoles’ strengths.

We match both of the chromaticities to 1 and set a constraint that second order

chromaticities less than 200. We also set a constraint that the nonlinear driving

terms should be less than 150. A single particle tracking for 4000 turns shows an

aperture of about 1.5 mm by 1.5 mm, as is shown in Fig. 3.1. In order to understand

how the tune shift with amplitude affects DA size, we track particles with different

initial offsets in the DA using MAD8 [6]. We calculate betatron tunes from the

turn by turn tracking data and find a large tune shift with amplitude as is shown in

Fig. 3.2. The dashed line shows the nominal tunes as we discuss in the last chapter

and the dots are the tracking results. Betatron tunes cross half integers and even

3.2 Dynamic aperture and tune shift with amplitude 37

integers quadratically. In order to better understand the mechanism of resonances,

Figure 3.1: 4000 turn dynamic aperture for 10 pm storage ring.

1.5 mm aperture is obtained after correcting the large

tune shift with amplitude.

we use Independent Component Analysis (ICA) [11, 9] to analyze the turn by turn

tracking data from MAD8. A typical ICA mode for particle with small initial offset

(Xoffset = 0.01 mm) is shown in Fig. 3.3. We observe oscillatory spatial and temporal

wave functions (up-left and up-right plots), which indicates this mode is dominated

by a betatron oscillation. Fast Fourier transform of the temporal wave function shows

a single spike at 3 times of the betatron tune. When particle’s initial offset is large

(Xoffset = 1.3 mm), the ICA result for the same mode is shown in Fig.3.4. We

observe chaotic phenomenon due to the crossing of many different resonances. In

this case, instabilities (correspondent to large amplitude modulation) are observed in


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−0.5

0

0.5

1

x0(mm)

tune

νx

νy

Figure 3.2: Quadratic tune amplitude dependence.

both spatial and temporal wave functions and correlation function plot has noise like

pattern. It fully correlates with itself and gradually decays with longer time delay τ

to zero.

The many different resonances mainly come from two factors: one is the driving

terms which drives certain order of resonances and the other one is the large tune shift

with amplitude. These two effects can be changed and balanced by tuning sextupole

families. Where we are at with the sextupole settings seems not to be a bad choice for

DA. More systematic way of calculation will be carried out using Genetic Algorithm

(GA) for a brute force search in parametric space.

3.3 Injection issues 39

3.3 Injection issues

Modern storage ring requires off-axis accumulation to reach high beam current. This

requires large DA (≥ 10mm) and low coupling. For 10 pm storage ring, the size

of DA is too small, although sufficient for storing the beam due to the tiny size of

the beam (rms beam size less than 30 µm), to have off-axis accumulation. Also, the

coupling needs to be big to alleviate IBS effect which will be discussed in the next

chapter. Thus instead of accumulation, used beam should be transported to dump

in one shot and a fresh bunch comes in using on-axix injection. This requires high

performance injector and a possible booster with considerably low emittance and full

energy as the main storage ring.

If we inject bunch trains, the fractional variation D in train intensity is given by

D = TinjNtrains/τ , with Tinj the interval bewteen trains, Ntrains the number of trains,

and τ the beam lifetime. The required injector current is Iinj = IringC(/cτD), with

Iring the beam current in the ring and C the ring circumference. An simple example

of D = 0.05 with Iring = 100 mA and τ = 2 hrs results in Iinj = 2.46 nA. Assuming

we have 200 bunch trains with 20 bunches in each train, the time interval between

trains should be Tinj = Dτ/Ntrains = 1.8 s. Thus the charge per train is 4.44 nC with

0.22 nC per bunch. These are all reasonable numbers within the specs of a modern

linac.


0 1000 2000 3000−4

−2

0

2

4x 10

−7

distance(m)

A11

0 20 40 60−1

−0.5

0

0.5

1

tao

sv

sv = 0.000000

0 200 400 600−0.1

−0.05

0

0.05

0.1

turns1

1

0 0.2 0.40

0.1

0.2

0.3

0.4

tune

pow

er

Figure 3.3: ICA analysis for a particle in DA with a small initial

offset. It experiences mostly betatron oscillation al-

though some of the instability induced by nonlinear

effect can be observed in temporal wave function(up-

right plot).

3.3 Injection issues 41

0 1000 2000 3000−2

−1

0

1

2x 10

−3

distance(m)

A11

0 20 40 60−0.5

0

0.5

1

tao

sv

sv = 0.004379

0 200 400 600−0.2

−0.1

0

0.1

0.2

turn

s11

0 0.1 0.2 0.3 0.40

0.01

0.02

0.03

tune

pow

er

Figure 3.4: ICA analysis for a particle in DA with a large initial off-

set. This mode shows frequency spectrum with noisy

peaks which indicates the particle experiences many

different resonances at the boundary of DA.

Intra-beam Scattering (IBS) and Microwave Instability (MI) 43

Chapter 4

Intra-beam Scattering (IBS) and

Microwave Instability (MI)

In reality, when an electron bunch passes through the vacuum chamber, the elec-

trons in the bunch will interact with the discontinuity or imperfection of chamber

wall or cavities and generate wake field. This wake field can affect the bunch itself

or the coming bunches and cause collective motion. When the collective motion’s

eigenfrequency satisfies a certain condition, the perturbation amplitude will grow ex-

ponentially and cause emittance growth and bunch lengthening thus beam loss. The

beam encounters collective microwave instability under such condition.

Longitudinal impedance of an accelerator is the Fourier transform of the wakefield

generated by the electron beam.

Z||(ω) =

∫ ∞

−∞W||(t)e

−jωtdt, (4.1)

where W||(t) is the wake function. Similarly, we can obtain wake function by using

inverse Fourier transformation

W||(t) =1

2π

∫ ∞

−∞Z||(ω)ejωtdω. (4.2)

44 4. Intra-beam Scattering (IBS) and Microwave Instability (MI)

Longitudinal impedance can be divided into two kinds: one is induced by the space

charge and one is induced by the chamber wall. The space charge induced impedance

can be described byZ||,scn

= −j g0Z0

2βγ2. (4.3)

It is independent of beam frequency and only determined by geometric factor g0 which

is given by

g0 = 1 + 2ln(b

a), (4.4)

where a is the outer radius of the beam and b is the inner radius of the vacuum

chamber.

Impedance caused by the chamber wall comes from resistive wall and also the

inductance between the beam and the chamber wall. Resistive wall impedance is

inversely proportional to the skin depth δskin =√

2µσcω

and it reduces to zero when

beam frequency ω drops to zero. Depending on the bandwidth, there are two types of

wakes or impedances: one is a long range wakefield or narrowband impedance and one

is a short range wakefield or broadband impedance. Narrowband impedance comes

from parasitic modes in rf cavities and cavity-like structures in accelerators. Higher

order modes (HOM) of the cavity-like structure is usually the source of narrowband

impedance. parameters for narrowband impedance are largely dependent on the

material and geometry of cavity-like structures. Broadband impedance arises from

vacuum chamber breaks, bellows, and other discontinuities in accelerator components.

It has very fast response in time domain thus can affect single bunch properties. We

will analyze this broadband impedance induced single bunch microwave instability in

this chapter.

The longitudinal narrowband and broadband impedances can conveniently be rep-

resented by an equivalent RLC circuit

Z(ω) =Rsh

1 + jQ( ωωr

− ωr

ω), (4.5)

Intra-beam Scattering (IBS) and Microwave Instability (MI) 45

where ωr is the resonance frequency and Rsh is the shunt impedance. For broadband

impedance, usually the quality factor Q = 1. The cut off frequency is given by

ωr,bb = ω0R/b = βc/b. (4.6)

A schematic drawing of a longitudinal impedance is shown in Fig.4.1. The longitu-

Figure 4.1: Schematic drawing of a longitudinal impedance.

dinal impedance is symmetric and the real part of the impedance is an odd function

and the imaginary part of the impedance is an even function.

Longitudinal impedance induces collective motion in the beam. Similar to what

has been discussed in last chapter on transverse collective motion, a quantity Ω is

defined as the eigenfrequency of the longitudinal collective motion. Beam encounters


the collective microwave instability only when the imaginary part of Ω is negative,

i.e. ImΩ < 0, the perturbation amplitude will grow exponentially.

When the momentum spread is negligible, the eigenfrequency is given by relation

(Ω

nω0

)2 = −j eI0Z||/n

2πβ2Eη. (4.7)

So the solution of stable collective motion requires a real Ω which results in −jZ||/nη >

0 otherwise one of the two solutions (with a sign off) with negative imaginary part

will induce microwave instability. For a capacitive impedance which could be in-

duced by space charge effect, η needs to be negative which means below transition.

In other words, for an inductive impedance requires above transition to have a real

Ω. Resistive impedance will always be unstable.

4.1 Single bunch collective instability

In reality, the perturbations arising from wakefields, rf phase error, dipole field error,

ground vibration, etc., consist of a spectrum of frequency distributions. The mean

field of the perturbations will change the potential well seen by particles and induce

incoherent synchrotron oscillation thus cause a synchrotron tune shift and modify

the unperturbed tune of the system. This is called potential well distortion and it

will change the bunch length of the beam. On the other hand, it will not affect the

energy spread which is determined by the quantum fluctuation due to the synchrotron

radiation. Under the potential well distortion, a single bunch length is given by

(σlσl0

)3 − (σlσl0

) − αIbIm

Z||/n

√

2π(E/e)ν2s0

(R

σl0)3 = 0, (4.1)

where σl0 and σl are the unperturbed and perturbed bunch length due to the potential

well distortion, α is the momentum compaction factor, Ib is the average beam current,

Im

Z||/n

is the imaginary part of the longitudinal impedance and R is the average

4.1 Single bunch collective instability 47

radius of the storage ring. When the Im

Z||/n

is negative (capacitive), the potential

well distortion will shorten the bunch length and vice versa. For 10 pm storage ring,

due to the small momentum compaction factor and large beam energy, the potential

well distortion effect can be neglected.

The single bunch microwave instability starts to be significant when the beam

current reaches a threshold defined by Keil-Schnell equation

I =2πβ2(E/e)σ2

δ |η|F∣

∣

∣

Z||

n

∣

∣

∣

, (4.2)

where I = FBI0 is the peak current of the electron beam, FB =√

2πσθ

is the bunching

factor and η is the phase slip factor. The form factor F = 1 for a gaussian beam

which is being assumed in the simulation in this chapter.

Above the threshold current, the bunch length and energy spread follow the Keil-

Schnell formula. We have relations

σθ =|η|νsσδ = ω0σt, (4.3)

and

νs =

√

heV |cos(φs)η|2πβ2E

. (4.4)

We can derive that

σδ =

Ib

∣

∣

∣

Z||

n

∣

∣

∣νs

√2πβ2(E/e)η2

1

3

, (4.5)

so we have scaling law σδ ∝ I1

3

b . If we apply similar derivation on the bunch length,

we have

σt =|η|σδνsω0

=

Ib

∣

∣

∣

Z||

n

∣

∣

∣η

√2πβ2(E/e)ν2

sω30

1

3

, (4.6)

and

σl = cσt =

Ib

∣

∣

∣

Z||

n

∣

∣

∣c2ηR

√2πβ2(E/e)ν2

sω20

1

3

, (4.7)


thus we have scaling law σl ∝ I1

3

b . Since both bunch lengthen and rms energy spread

grow up when beam current increases, there are limits on the highest beam current

one can achieve before severe beam loss takes place.

Single bunch microwave instability has very fast growth rate. It can deterio-

rate beam quality before damping process can react. Usually in 3rd generation light

sources, the beam properties are largely limited by the single bunch microwave in-

stability as can be seen in ALS at LBNL (Fig. 4.2, Fig. 4.3) and SPEAR3 at SLAC

(Fig. 4.4, Fig. 4.5). For 10 pm storage ring, the encountering of the microwave

0 10 20 30 40 50 600

50

100

150

200

250

300

als single bunch current(mA)

bunch length(ps)peak current(A)

Figure 4.2: Bunch length vs beam current for ALS. Bunch length-

ening is observed due to single bunch microwave insta-

bility.

instability is earlier due to the tiny momentum compaction factor – 100 times smaller

4.1 Single bunch collective instability 49

0 10 20 30 40 50 600.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4x 10

−3

als single bunch current(mA)

rms energy spreadρ

FEL

Figure 4.3: Rms energy spread vs beam current for ALS. Energy

spread can be blown up by a few times under single

bunch microwave instability.

than a typical 3rd generation light source. As is shown in Fig. 4.6 and Fig. 4.7, the

peak current of the beam can reach a few kA when the average beam current is high.

However, the microwave instability induces a very large rms energy spread growth

which is much greater than the calculated ρFEL. SASE FEL lasing is not possible.

Due to the tiny momentum compaction factor, the threshold current for 10 pm stor-

age ring calculated by Eq.(4.2) is 0.7 µA. Under most of the circumstances, 10 pm

storage ring will be operated above threshold thus the vacuum chamber should be

carefully designed to be very smooth.


0 5 10 15 20 250

50

100

150

200

250

spear3 single bunch current(mA)


Figure 4.4: Bunch length vs beam current for SPEAR3. It has

similar performance as ALS due to the similar param-

eters of the storage ring.

4.2 Intra-beam scattering (IBS) and its effect

For modern storage rings especially high intensity rings with high peak current, in-

trabeam scattering (IBS) is very important. IBS effect is a small angle multiple

Coulomb scattering within the beam. During this scattering process, emittance and

energy spread are diluted. An important scaling factor for the IBS growth rate is

given as IBS parameter Aibs:

Aibs =Nbcr

20

64π2σsσδεxεzβ3γ4, (4.1)

4.2 Intra-beam scattering (IBS) and its effect 51

0 5 10 15 20 250.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10

−3

spear3 single bunch current(mA)

rms energy spreadρ

FEL

Figure 4.5: Rms energy spread vs beam current for SPEAR3. The

calculated FEL parameter is much lower than rms en-

ergy spread so SASE FEL is not possible.

where Nb is number of charged particles per bunch, σs is the rms bunch length, σδ is

the rms momentum spread, r0 is the classical radius of the electron and β and γ are

relativistic factors. IBS grow rate is a very complicated function of Aibs but higher

IBS parameter results in a faster IBS grow rate thus bigger IBS effect.

We can see from Eq.(4.1) that when the beam current is high and beam energy is

low, IBS effect is stronger. Also it gets magnified by small transverse and longitudinal

emittances. These are all what 10 pm storage ring has and thus we need to calculate

its effect on beam properties especially emittances and energy spread. As we have

noticed, a larger coupling between horizontal and vertical results in a smaller IBS


0 20 40 60 80 1000

200

400

600

800

1000

1200

1400

1600

1800

2000

10pm single bunch current(mA)


Figure 4.6: Bunch length vs beam current for 10 pm storage ring.

Bunching factor is very big that peak current can reach

few kA when beam current is high.

parameter thus alleviates IBS effect. Since IBS usually has a growth rate comparable

to radiation damping, quantum excitation and other scattering mechanisms such as

gas scattering etc, a final equilibrium will be developed over a few thousand turns.

This whole process has been implemented in a subroutine of ELEGANT called ib-

sEmittance. We use the tracking results from the linear lattice calculation and input

into ibsEmittance to evaluate the final transverse and longitudinal emittances. We

assume 100% beam coupling to relief the IBS effect. Figure. 4.8 shows the scaled

emittance (with respect to initial emittance) growth under different bean energies

and peak currents. IBS can blow up transverse emittance 4 times when beam energy


0 20 40 60 80 1000

0.005

0.01

0.015

0.02

0.025

0.03

0.035

10pm single bunch current(mA)

rms energy spreadρ

FEL

Figure 4.7: Rms energy spread vs beam current for 10 pm storage

ring. FEL parameter is closer to rms energy spread

when beam current is low.

is 5 GeV and peak current is 200 Amps. A slightly higher beam energy will signifi-

cantly reduce emittance growth induced by IBS effect. This can gain when the beam

energy increases from 5 GeV to 6 GeV and 7 GeV. If the beam energy keeps growing,

the IBS effect is already negligible and the initial emittance scales with γ2 thus this

will result in a total emittance increase as is shown in Fig. 4.9. IBS effect can double

the rms energy spread which we want to keep at a low value for beam storage and

also for FEL lasing. As is shown in Fig. 4.10, when peak current is relatively low,

the IBS grow is very steep and a saturation type of pattern happens when the peak

current is very high. There are two ways of operating the machine, either at very low


0 50 100 150 2001

1.5

2

2.5

3

3.5

4

4.5

Ipeak

(A)

Em

ittan

ce g

row

th fa

ctor

5GeV6GeV7GeV8GeV9GeV10GeV




current where the rms energy spread is very tiny or at very high current where ρFEL

could be larger than the energy spread. Both cases are possible for FEL process to

be implemented. We will show that high current operation is impossible due to mi-

crowave instability. We also observe bunch lengthening as Fig. 4.11 shows. This will

significantly reduce peak current thus ruin FEL process. To put everything together,

we have Fig. 4.12 and Fig. 4.13 showing the equilibrium emittance and rms energy

spread under different operation modes. Using the equilibrium beam properties, we

can plot out the IBS parameter as is shown in Fig. 4.14. Lower peak current has

smaller Aibs. When peak current is high, the IBS effect is so strong that equilibrium


0 50 100 150 2000.5

1

1.5

2

2.5

3

3.5x 10

−11

Ipeak

(A)

ε equi

libriu

m(m

−ra

d)

5GeV6GeV7GeV8GeV9GeV10GeV




emittance has been blown up so much that Aibs becomes smaller.

4.2.1 Comparison between microwave instability and IBS ef-

fect

As we have discussed before, both MI and IBS will blow up beam emittances and

energy spread thus cause beam dilution or even beam loss. We want to compare them

and as is shown in Fig. 4.15, under the same condition, rms energy spread induced by

microwave instability is at least 3 times as large as IBS effect. Microwave instability


0 50 100 150 2001

1.2

1.4

1.6

1.8

2

2.2

2.4

Ipeak

(A)

σ Ef/σ

Ei




is more destructive to beam qualities than IBS effect.

4.3 SASE FEL performance study under microwave

instability

As we have shown before, the microwave instability can cause severe energy spread

growth. If we want to apply SASE FEL process, the relation σE ≤ ρFEL has to be

4.3 SASE FEL performance study under microwave instability 57

0 50 100 150 2001

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

Ipeak

(A)

σ sf/σ

si




satisfied, which can be further expressed in the way

ρFEL =1

2

(

I

IA

λ2wK

2w

2πγ3

1

4πσxσy

)1/3

, (4.1)

where the IA = 17kA is the Alfven current. Thus we have

σE =

Ibω0R∣

∣

∣

Z||

n

∣

∣

∣νs

√2πβ2(E/e)cη2

1

3

≤ ρFEL, (4.2)

andνsω0R

∣

∣

∣

Z||

n

∣

∣

∣

√

(2π)Ecη2Ib ≤

1

8

I

IA

λ2wK

2w

2πγ3

1

4πσxσy, (4.3)


5 6 7 8 9 105

10

15

20

25

30

Ebeam

(GeV)

ε equi

l(pm

)

Ipeak

=10A

Ipeak

=40A

Ipeak

=100A

Ipeak

=200A




then

32πσxσyνsω20R

∣

∣

∣

∣

Z||n

∣

∣

∣

∣

σtIAγ3 ≤ Ecη2λ2

wK2w, (4.4)

and finally we reach

16(2π)5/6σxσyν1/3s

∣

∣

∣

∣

Z||n

∣

∣

∣

∣

4/3

I1/3b IAγ

5/3 ≤ E4/30 η5/3λ2

wK2w. (4.5)

In order to make Eq.(4.5) to be satisfied, the key is to increase η. Increasing λw and

Kw can also help but they would increase the wavelength of the laser. We cannot

reduce the energy which will also increase the laser wavelength. Another option is

to operate the machine under low current condition. The idea of using oscillators


0 50 100 150 2001

1.2

1.4

1.6

1.8

2

2.2

2.4

Ipeak

(A)

σ Ef/σ

Ei




may be killed by the lack of operable reflection mirrors in the hard X-ray working

regime. A new idea of OFFELO (optics free FEL oscillator) using two energy beams

has hereby risen. The difficulty lies in the preservation of longitudinal phase space

which can be overcome by using higher order magnets to control phase slip factor.

This will be covered in the Appendix.


5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5x 10

−18 Aibs current energy

Ebeam

(GeV)

Aib

s

Ipeak

=10A

Ipeak

=40A

Ipeak

=100A

Ipeak

=200A





3 4 5 6 7 8 9 100

0.002

0.004

0.006

0.008

0.01

0.012

0.014Microwave instability vs IBS effect

Ebeam

(GeV)

σ δ

IBS I

peak=10A

IBS Ipeak

=40A

IBS Ipeak

=100A

IBS Ipeak

=200A

MI Ipeak

=10A

MI Ipeak

=40A

MI Ipeak

=100A

MI Ipeak

=200A




Conclusions 63

Chapter 5

Conclusions

In this thesis, a design of ultimate storage ring with natural emittance less than

10 pico-meter has been reported. As a storage ring, it has the advantage of stable

operation and high average power and flux. As an ultimate storage ring, its trans-

verse emittances reach the diffractive limit for hard X-rays thus peak brightness is

significantly enhanced due to coherent radiation. This dissertation is dedicated to

designing the linear lattice using n-BA and Theoretical Minimum Emittance(TME)

structure and understanding how the dynamic aperture is affected by nonlinear ele-

ments and a selfconsistent analysis of effects of instabilities which may degrade the

ring performance.

We presented the linear lattice design in Chapter 2. The 10 pm storage ring

is composed of 11-BA structure with TME structure. Total circumference is long

to bring the emittance down. This lattice has a very small momentum compaction

factor and large natural chromaticities. An effort of shortening the ring circumference

using combined function magnets has been studied. Although it can be matched to

TME, the large gradient of the combined function dipole makes it impossible to build

and install in storage rings.

64 5. Conclusions

Following the design of linear lattice using 11-BA structure, we correct the natu-

ral chromaticities in Chapter 3. Eight families of sextupoles have been used and the

large sextupole strengths induce large nonlinear driving terms and tune shift with

amplitude which makes the dynamic aperture of the design very small. An imple-

mentation of ICA method shows the tracking particle at DA boundary experiences

many different resonances and a chaotic motion is discovered. A further optimization

of DA using Genetic Algorithm (GA) is undergoing.

Due to the tiny momentum compaction factor of 10 pm storage ring, we en-

counter the microwave instability threshold much earlier comparing with a typical

3rd generation light source. We evaluate the single bunch microwave instability us-

ing Keil-Schnell formula in Chapter 4. The transverse and longitudinal emittances

and rms energy spread grow up dramatically when beam current is high. We also

observe a large Intra-Beam Scattering effect on emittances and energy spread when

beam energy is low and beam current is high. Comparing these two effect, microwave

instability is more severe thus a good design of vacuum chamber is key in maintaining

a good beam quality for ring operation and furthermore FEL process.

Undulator theory and laser study 65

Appendix A

Undulator theory and laser study

A.1 Background

Undulator radiation theory gains a great success in FEL and many light sources use

insertion devices like wigglers and undulators to get the coherent radiation. The peri-

odic structure of undulator adds up the radiation with certain wavelength coherently

and thus amplifies a single frequency light from noise background. Also the extra

radiation from beam wiggling motion in wigglers changes the damping partition thus

equilibrium emittance. In this section, we will discuss Klein-Nishina formula at first,

then we will prove that the laser beam interaction is equivalent to treating laser as

undulator. At last, we will discuss the damping effect induced by beam wiggling

motion in the laser undulator.

A.2 Elastic photon-electron collision

We consider a collision between an electron (Ee, Pe) and an incoming photon (EL, PL)

at angle θ1. The scattered electron (E′

e, P′

e) and photon (Eγ, Pγ), at angle φ and θ2

66 A. Undulator theory and laser study

relative to the initial electron coming direction, can be obtained from the conservation

of momentum and energy:

Pe + PL cos θ1 = Pγ · cos θ2 + P′

e · cosφ, (A.1)

PL sin θ1 = Pγ · sin θ2 + P′

e · sin φ, (A.2)

Ee + EL = Eγ + E′

e (A.3)

with

Pγ =Eγc, PL =

ELc, (A.4)

and

P 2e c

2 +m2ec

4 = E2e , P

′2e c

2 +m2ec

4 = E′2e . (A.5)

Taking the square of first two equations and summing them up, we have

P′2e = P

′2e · cos2 φ+P

′2e · sin2 φ = (Pe +PL cos θ1 −Pγ cos θ2)

2 + (PL sin θ1 −Pγ sin θ2)2,

(A.6)

while according to Eq.(A.5), we have

E′2e = P

′2e c

2 +m2ec

4 = (Ee + EL −Eγ)2, (A.7)

Substituting Eq.(A.6) into Eq.(A.5) so we obtain

(P 2e + P 2

L + P 2γ + 2PePL cos θ1 − 2PePγ cos θ2 − 2PLPγ cos(θ2 − θ1))c

2 +m2ec

4

= E2e + E2

L + E2γ + 2EeEL − 2EeEγ − 2ELEγ .

By using Eq.(A.4) and Eq.(A.5), we finally come up to

Eγ =EL(1 − β cos θ1)

(1 − β cos θ2) + EL

Ee(1 − cos(θ2 − θ1))

, (A.8)

where θ1 is the initial angle between the photon and electron while θ2 is scattered

angle of electron. If we choose the back scattering case with θ1 = π and θ2 = θ then

we arrive at

Eγ =EL(1 + β)

(1 − β cos θ) + EL

Ee(1 + cos θ)

. (A.9)

A.2 Elastic photon-electron collision 67

It is worthwhile to point out that EL is measured in the lab frame thus there is a

Doppler shift effect in the frame transformation. At ultra relativistic, we have

ν ′ = ν

√

1 + β

1 − β= ν

√

21

2γ2

= 2γν, (A.10)

where ν and ν ′ are the photon frequencies measured in lab frame and electron rest

frame respectively. We define a unitless quantity α = hν′

mec2measured in the electron

rest frame. By reorganizing Eq.(A.9), we have

Eγ =EL(1 + β)

(1 − β cos θ) + α2γ2 (1 + β cos θ)

=EL(1 + β)

(1 + α2γ2 ) − β · cos θ(1 − α

2γ2 ). (A.11)

With the help of

ELEe

=hν

γmec2=

2γhν

2γ2mec2=

hν ′

2γ2mec2=

α

2γ2, (A.12)

when θ increases from 0 to π, Eγ decreases monatonically, so

θ = 0, (Eγ)max =EL(1 + β)

(1 − β) + α2γ2 (1 + β)

=EL(1 + β)

1 − β(1 − α

2γ2(1 + β

1 − β) +

α2

4γ4(1 + β

1 − β)2 +O(α3))

= 4γ2EL(1 − 2α + 4α2 − 8α3 + · · · ),

θ = π, (Eγ)min =EL(1 + β)

(1 + β) + α2γ2 (1 − β)

=EL

1 + α2γ2

1−β1+β

=EL

1 + α8γ4

= EL(1 − α

8γ4+

α2

64γ8+ · · · ).

So we have

(Eγ)min=EL, (Eγ)max=4γ2EL(1 − 2α)=4γ2EL. (A.13)

Using incident laser with 1µm wavelength colliding with 50MeV electron beam would

give hν = 1.24eV , γ = 100, hν ′ = 2γhν = 248eV and α = 4.8 × 10−4 is a very small

quantity.


A.3 Klein-Nishina formula

Klein-Nishina formula gives a precise prediction of the differential cross section for

photon-electron collision based on quantum electrodynamics. Comparing with the

Thomson scattering or the Compton scattering, Klein-Nishina formula gives a second

order correction which agrees very well with experimental data. It has the form of

dσ

dΩ=r20

2(P (Eγ, θ) − P (Eγ, θ)

2 sin2 θ + P (Eγ, θ)3) (A.14)

with

P (Eγ, θ) =1

1 + Eγ

mec2(1 − cos θ)

(A.15)

the ratio of photon energy after and before the electron-photon collision.

A.3.1 Total cross section and differential cross section

If we integrate Eq.(A.14) over all angles and then we get the total cross section

σ =2πNe4

m2ec

4

1 + α

α[2(1 + α)

1 + 2α− 1

αlog(1 + 2α)] +

1

2αlog(1 + 2α) − 1 + 3α

(1 + 2α)2, (A.16)

where α which is defined in last section is usually a small term which we can Taylor

expand the total cross section around. This results in

σ =8π

3r20(1 − 2α + 5.2α2 − 13.3α3 + · · · ). (A.17)

The unit of the total cross section is cm2, where the 0th order is simply Thomson

cross section. The Klein-Nishina modification happens at the 2nd order term which

is quite small. As discussed before, typical α is very small of about 10−4 order so

calculation using Thomson cross section should be enough for our consideration. If

A.4 Laser-Beam Interaction 69

we take derivative with respect of Eγ then we have

dσ

dEγ=

πr20

2

m2ec

4

ELE2e

[m4ec

8

4E2LE

2e

(Eγ

Ee − Eγ)2 − m2

ec4

ELEe

EγEe − Eγ

+Ee −EγEe

+Ee

Ee − Eγ]

=πr2

0

2

4

Eγ[E2

0

4γ2E2L

E2γ

E2e

− E20Eγ

ELE2e

+ 2]

= 2πr20

1

Eγ[E2γ

4γ4E2L

− Eγγ2EL

+ 2]

= 2πr20

4

Eγ[(Eγ

Eγ− 1

2)2 +

1

4],

with E0 = mec2 and Ee = γE0 and Eγ = 4γ2EL. The differential cross section has a

parabolic relation with scattered photon energy and it reaches half of the peak value

when Eγ = 12Eγ as is shown in Fig. A.1. A straightforward 3D plot with θ and Eγ as

variables can be seen in Fig.A.2. From Eq.(A.9), we can see that the photon energy

is so concentrated in the forward direction that we can apply small angle expansion,

Eγ =4γ2EL

1 + γ2θ2. (A.18)

For an electron beam with 50 MeV, γ = 100 and energy is concentrated in a cone with

angle 1γ

= 0.01 rad, when the angle goes to 3 times of this, which is still a very small

angle, the radiated photon energy drops an order. So from Fig. A.1, most contribution

of the cross section lies in a very small angle cone in the forward direction.

A.4 Laser-Beam Interaction

In this Section, we prove that the laser-beam inverse Compton scattering is equivalent

to treating the laser as an undulator. The laser-electron interaction can be considered

as inverse Compton scattering or can be considered as electron passing through an

undulator of electromagnetic field.


0 50 100 150 2000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Eγ[keV]

dσ/d

Eγ/σ

Tom

pson

[keV

−1 ]

γ=50γ=100γ=150γ=200

Figure A.1: A 2D plot of differential cross section vs photon energy

according to Klein-Nishina formula. At half of the

peak energy, the differential cross section is half of

the value of its peak cross section.


01

23

4

0

0.5

1

1.5

2

x 105

0

0.2

0.4

0.6

0.8

1

x 10−28

θ[Radian]Eγ[eV]

dσ/d

Eγ[c

m2 /e

V]

γ=50γ=100γ=150γ=200

Figure A.2: A 3D plot of differential cross section vs photon en-

ergy and emission angle θ according to Klein-Nishina

formula.


We first consider the laser-beam interaction as inverse Compton scattering. The

rate of back-scattered photon production in head-on collision is

dNγ

dt= f

NeNL

4πσxσzσT , (A.19)

where Ne andNL are, respectively, the numbers of electrons and photon in the electron

bunch and laser pulse, σx,z are the rms transverse beam sizes, f is the frequency of

collision, and

σT =8π

3r2e (A.20)

is the total Thomson cross section and re = e2/(4πε0mc2) is the electron classical

radius. Note that 4πσxσy is the effective cross section of the electron beam.

The number of photons emitted in a solid angle per electron per collision in the

electron rest frame is

dNx

dΩ=

1

fNe

dNx

dtdΩ=

NL

4πσxσy

r2e

2

(

1 + cos2 θe)

. (A.21)

Now the energy in the laser pulse of duration τL is EL = NLEL = NLhfL = PLτL,

where PL is the power of the laser beam and EL = hfL is energy of laser photon. In

terms of these new variables, the number of photons per solid angle is

dNx

dΩe=

PLτL4πσxσzhfL

re2

e2

4πε0mc2(

1 + cos2 θe)

=PLτL

4πσxσzfL

αre4πmc

(

1 + cos2 θe)

, (A.22)

The strength of the magnetic component B in the laser beam is related to the

power byPL

4πσxσy=cB2

µ0. (A.23)

Integrating over all outgoing directions, the total number of photons produced by

each electron for a single passage is

Nx =Neffλ

2L

c

cB2

µ0

α

4πmc2e2

4πε0mc2

∫

(

1 + cos2 θe)

dΩe

=4π

3αNeffK

2u, (A.24)


where the undulator parameter is

Ku =eBλL2πmc

. (A.25)

To compute the photon energy, we transform the electron rest frame to the lab

frame:

cos θe =cos θ − β

1 − β cos θ≈ 1 − γ2θ2

1 + γ2θ2,

dΩe

dΩ=

1

γ2(1 − β cos θ)2≈ 4γ2

(1 + γ2θ2)2(A.26)

The back scattered photon energy is

Eγ(θ) =EL(1 + β)

1 − β cos θ≈ E

1 + γ2θ2(A.27)

where E = 4γ2EL is the maximum energy of the back-scattered photons. Thus the

number of photons and the total energy emitted are

Nx =

∫

αNeffK2u

4

(

1 +

(

1 − γ2θ2

1 + γ2θ2

)2)

4γ2

(1 + γ2θ2)2dΩ (A.28)

U =

∫

αNeffK2u

4

(

1 +

(

1 − γ2θ2

1 + γ2θ2

)2)

E

1 + γ2θ2

4γ2

(1 + γ2θ2)2dΩ

=2π

3αNeffK

2uE. (A.29)

Equation (A.28) reproduces the result of Eq. (A.24). The average energy of photon

emitted is

〈Eγ〉 =U

Nx

=1

2E. (A.30)

The differential cross-section of the inverse Compton scattering (Klein-Nishina for-

mula) is

dσ

dEγ=

8πr2e

E

[

(

Eγ

E− 1

2

)2

+1

4

]

. (A.31)


The average energy of Eq. (A.30) agrees with that obtained from the Klein-Nishina

formula:

〈Eγ〉 =1

σT

∫

Eγdσ

dEγdEγ (A.32)

Now, we consider the laser as a set of undulators with varying magnetic field. The

energy radiated from a wiggler is

Uw =CγE

4

2π

∮

ds

ρ2w

=CγE

4

2π

e2

p2· B2

w · Lw =2π

3αK2NwE. (A.33)

where Cγ = 4π3

r0(mc2)3

= 8.846×10−5 m/(GeV)3 and ρw the bending radius of curvature

of wiggler, α = e2/(2hcε0) is the fine structure constant and we have taken into

account both the electric and magnetic fields of the laser-beam. The result agrees

with that of Eq. (A.29), and thus we have proved the theorem that the laser-electron

inverse Compton interaction is equivalent to interaction of electron traversing through

the set of undulators made of electromagnetic fields. We proved that the undulator

theory is self consistent in the sense that the total energy of radiation is the sum of

every photon emitted.

A.5 Laser induced damping

When electrons accelerate, they radiate photons and lose energy. With rf cavity im-

plemented, longitudinal momentum will be compensated and transverse ones damped

thus transverse emittances are improved. We also know the fact that higher energy

particles lose more energy than lower energy particles thru this radiation process (pro-

portional to particle energy to 4th power). The compensated energy from rf cavity

does not differ very much (depending on whether it is below or above transition en-

ergy), thus the energy spread is also suppressed by this process. Overall beam quality

is improved with synchrotron radiation. In this section, we are going to calculate and

study the decay time induced by laser-beam interaction.

A.5 Laser induced damping 75

Damping due to radiation could come from many factors: dipole, damping wiggler,

laser interaction etc. They have the same properties that electrons change velocities

and radiate when they pass through these elements. Thus the total damping time is

determined by:

τtotal =τ0

1 +Uwig

U0+ Ulaser

U0

, (A.34)

where τ0, τwig and τlaser are the decay times from radiation in dipole, wiggler and

laser respectively. U0, Uwig and Ulaser are the energy loss in dipole, wiggler and laser

respectively. If we take the inverse of this equation and reorganize it, we can find out

that:1

τtotal=

1

τ0+

1

τwig+

1

τlaser(A.35)

with

τwig =τ0Uwig

U0

, τlaser =τ0

Ulaser

U0

. (A.36)

We will calculate this ratio of damping time and absolute value respectively.

First of all, let’s take a look at what is U0,

U0 =CγE

4

2π

∮

dipole

ds

ρ2=CγE

4

ρ, (A.37)

where Cγ = 4π3

r0(mc2)3

= 8.846 × 10−5m/(GeV )3 is the radiation constant and ρ the

radius of curvature for the main dipoles. For the case of ALPHA project at Indiana

University [5], beam energy is 50MeV and ρ = 1.273m, we can get U0 = 0.434505

GeV , a very small number.

For energy loss due to laser interaction, according to Eq.( A.33) we have relation

Ulaser =CγE

4

2π

e2

p2

∮

Lw

ds ·B2L =

CγE4

2π

e2

p2LwB

2L (A.38)

by using

BLρL =p

e. (A.39)


If we assume the electron beam has a geometry with (σx, σz, σt) as its horizontal,

vertical and longitudinal dimensions, we can arrive at the relation between the laser

field and laser power1

µ0B2L = ρlaser =

Plaserπσxσzc

. (A.40)

Plug this back into Eq.(A.38) and we have for the energy loss in laser interaction

Ulaser =CγE

4

2π

e2

p2Lwµ0

Plaserπσxσzc

, Lw = cσt. (A.41)

Thus the ratio between τlaser and τ0 is

τlaserτ0

=U0

Ulaser=

2π

µ0ρ

p2

e2πσxσzPlaserσt

. (A.42)

This ratio is proportional to the transverse beam size while inversely proportional to

longitudinal beam size and inversely proportional to the laser power and independent

of directions. The synchrotron radiation damping time τ0 is given by

τ0 =2ET0

U0

1

J, (A.43)

with T0 the revolution period and J damping partition. For the damping partition,

we have relations

Jx = 1 − I4I2, Jz = 1, JE = 2 +

I4I2

(A.44)

where I4 =∫

dipoleDρ3ds and I2 =

∫

dipole1ρ2ds are the radiation integrals. Here we

already assume the dipole is separated function magnet. For 10 pm storage ring, due

to the tiny dispersion and large bending radius, I4I2

≈ 0. MAD8 simulation gives us

the values of damping partitions Jx = 0.99996, Jz = 1 and JE = 2.00004.

For ALPHA project, electron beam size is at maximum 250 µm horizontally and

vertically without mini-beta insertion. With the beta function minimized, the beam

size can go down to about 50 µm so we study various beam size from 50 µm to

250 µm with 50 µm step size and we assume our bunch length is 10 ps. We can


102

104

106

108

1010

1012

1014

10−8

10−6

10−4

10−2

100

102

104

106

108

Laser Power(W)

τ L/τ0

250µm200µm150µm100µm50µm

Figure A.3: Ratio between damping times induced by laser and

dipoles under different laser cross section and laser

power.

calculate the τlaser and τlaser

τ0respectively. The results are shown as below. From

the Figs.A.3–A.6, we observe that for the same laser power, larger the beam size (or

laser cross section area) is, smaller damping effect laser induces due to the smaller

electromagnetic field the laser beam carries. On the other hand, for a fixed beam

size, the larger laser power, the stronger damping effect it has. When the laser power

reaches 0.1 TW for a beam with 50 µm rms beam size, the damping time induced by

laser is comparable with the one induced by main dipoles.


102

104

106

108

1010

1012

1014

10−6

10−4

10−2

100

102

104

106

108

Laser Power(W)

τ Lx


Figure A.4: Horizontal damping time induced by laser under dif-

ferent laser cross section and laser power.


102

104

106

108

1010

1012

1014

10−6

10−4

10−2

100

102

104

106

108

Laser Power(W)

τ Ly


Figure A.5: Vertical damping time induced by laser under different

laser cross section and laser power.


102

104

106

108

1010

1012

1014

10−6

10−4

10−2

100

102

104

106

108

Laser Power(W)

τ Lz


Figure A.6: Longitudinal damping time induced by laser under dif-

ferent laser cross section and laser power.

Multipole effect on higher order momentum compaction factor 81

Appendix B

Multipole effect on higher order

momentum compaction factor

B.1 Multipoles’ effect

As we have discussed in the last in first chapter, phase slip factor or momentum

compaction factor including their higher order terms (in term of fractional energy

spread δ) can be changed by multipole magnets. Multipole’s effect on momentum

compaction factor depends on the lattice. In the first two sections, we are going to

use a simple FODO lattice to study how sextupoles and octupoles change higher order

dispersion and momentum compaction factor. The simplified lattice is composed of

pure dipoles and alternating thin-focusing quadrupole and thin-defocusing quadrupole

in between two dipoles. We will discuss explicit expression of higher order magnets

in the last section using Hamiltonian dynamics.

82 B. Multipole effect on higher order momentum compaction factor

B.1.1 Momentum compaction factor

In FODO cells, the dispersion at focusing quad and defocusing quad are respectively

D0 ≈l0θ0(1 + S/2)

S2, (B.1)

and

D0 ≈l0θ0(1 − S/2)

S2(B.2)

with l0 the half-cell length and

S = l0|∫

dsB

′

B0ρ0|. (B.3)

The term in integral is the integrated strength of a half-quadrupole. If the centrifugal

focusing or the diplole term ( 1ρ2

) is neglected,

S = sinΦ

2(B.4)

with the Φ to be the phase advance of a FODO cell.

Consider a half cell shown in Figure. B.1. The half-focusing quadrupole is located

at FF’ and the half-defocusing quadrupole is located at DD’. The dipole in between

has a bending angle of θ0 = l0/ρ0. FD is designed orbit while F’D’ is off-momentum

orbit with a bending angle θ = l/ρ. Particle passing through the quadrupole is

experiencing the Lorentz force while it is bending along a circular orbit. We have

relation at FF’

dp⊥dt

= ev||BF = ev||(B′

Fx+1

2B

′′

Fx2 +

1

6B

′′′

F x3 +

1

24B

′′′′

F x4), (B.5)

and since dt = dldv||

so we have

p⊥ =

∫

dtev||BF =

∫

dle(B′

Fx+1

2B

′′

Fx2 +

1

6B

′′′

F x3 +

1

24B

′′′′

F x4). (B.6)

Thus,

tan(∆φF ) =p⊥p||

=e

p

∫

dl(B′

Fx+1

2B

′′

Fx2 +

1

6B

′′′

F x3 +

1

24B

′′′′

F x4). (B.7)

B.1 Multipoles’ effect 83

Figure B.1:


Because we have relation

ds = dl cos(∆φF ) ⇒ dl =ds

cos(∆φF ), (B.8)

then we have

tan(∆φF ) =p0

p

∫

dl

B0ρ0(B

′

Fx+1

2B

′′

Fx2 +

1

6B

′′′

F x3 +

1

24B

′′′′

F x4)

=p0

p

∫

ds

cos(∆φF )

1

B0ρ0

(B′

Fx+1

2B

′′

Fx2 +

1

6B

′′′

F x3 +

1

24B

′′′′

F x4)

with defining

S

l0=

∫

dsB′F

B0ρ0, focusing quadrupole

SF =

∫

dsB′′F

2B0ρ0

, focusing sextupole

OF =

∫

dsB′′′F

6B0ρ0, focusing octupole

TF =

∫

dsB′′′′F

24B0ρ0

, focusing decapole

so

sin(∆φF ) =1

1 + δ(S

l0Dδ + SF D

2δ2 +OF D3δ3 + TF D

4δ4 +O(δ5)), (B.9)

and

∆φF =1

1 + δ(S

l0Dδ + SF D


4δ4)

+1

6

1

(1 + δ)3(S

l0Dδ + SF D


4δ4)3

=1

1 + δ(S

l0Dδ + SF D


4δ4)

+1

6

1

(1 + δ)3(S3

l30D3δ3 + 3

S2SFl20

D4δ4 +O(δ5).

Same procedure can be carried out at DD’ location with all multipole effects consid-

ered:

tan(∆φD) =p⊥p||

=e

p

∫

dl(B′

Dx+1

2B

′′

Dx2 +

1

6B

′′′

Dx3 +

1

24B

′′′′

D x4), (B.10)


and

ds = dl cos(∆φD) ⇒ dl =ds

cos(∆φD), (B.11)

thus we have

tan(∆φD) =p0

p

∫

dl

B0ρ0

(B′

Dx+1

2B

′′

Dx2 +

1

6B

′′′

Dx3 +

1

24B

′′′′

D x4)

=p0

p

∫

ds

cos(∆φD)

1

B0ρ0(B

′

Dx+1

2B

′′

Dx2 +

1

6B

′′′

Dx3 +

1

24B

′′′′

D x4)

with defining

S

l0= −

∫

dsB′D

B0ρ0, defocusing quadrupole

SD =

∫

dsB′′D

2B0ρ0

, defocusing sextupole

OD =

∫

dsB′′′D

6B0ρ0, defocusing octupole

TD =

∫

dsB′′′′D

24B0ρ0, defocusing decapole.

So very similar to FF’, we finally have

sin(∆φD) =1

1 + δ(−Sl0Dδ + SDD

2δ2 +ODD3δ3 + TDD

4δ4 +O(δ5)), (B.12)

and

∆φD =1


2δ2 +ODD3δ3 + TDD

4δ4)

+1

6

1

(1 + δ)3(−S

3

l30D3δ3 + 3

S2SDl20

D4δ4) +O(δ5).

From the trigonometry relation

θ = θ0 − (∆φF + ∆φD), (B.13)

we have

θ = θ0 −1

1 + δ[S

l0(D − D)δ + (SF D

2 + SDD2)δ2 + (OF D

3 +ODD3)δ3

+1

6

S3

l30(D3 − D3)

δ3

(1 + δ)2+

1

2

S2

l20(SF D

4 + SDD4)

δ4

(1 + δ)2+ (TF D

4

+TDD4)δ4].


Both particles have the same B-field so

ρ = ρ0(1 + δ). (B.14)

Combining these two equations, we have

l = ρθ

= l0[1 + δ(1 − S

θ0

D − D

l0) − 1

θ0(SF D

2 + SDD2)δ2 1

θ0[OF D

3 +ODD3 +

1

6

S3

l30(D3 − D3)]δ3

− 1

θ0[−1

3

S3

l30(D3 − D3) +

1

2

S2

l20(SF D

4 + SDD4) + (TF D

4 + TDD4)]δ4]

= l0[1 + δ(1 − S

θ0

D0 − D0

l0) − 1

θ0(S

l0(D1 − D1) + SF D0

2+ SDD0

2)δ2 − 1

θ0[S

l0(D2 − D2)

+2SF D0D1 + 2SDD0D1 +OF D30 +ODD

30 +

1

6

S3

l30(D3

0 − D30)]δ

3 − 1

θ0[S

l0(D3 − D3)

+SF (2D0D2 + D21) + SD(2D0D2 + D2

1) + 3OF D20D1 + 3ODD

20D1 +

1

2

S3

l30(D2

0D1 − D20D1)

−1

3

S3

l30(D3

0 − D30) +

1

2

S2

l20(SF D

40 + SDD

40) + (TF D

40 + TDD

40)]δ

4].

Here we already expand D and D in power of δ

D = D0 + D1δ+ D2δ2 + D3δ

3 +O(δ4), D = D0 + D1δ+ D2δ2 + D3δ

3 +O(δ4). (B.15)

Compare this with C = C0[1 + α0δ + α1δ2 + · · · ], we get

α0 = 1 − S

θ0

D0 − D0

l0

α1 = − 1

θ0[S

l0(D1 − D1) + SF D

20 + SDD

20]

α2 = − 1

θ0[S

l0(D2 − D2) + 2SF D0D1 + 2SDD0D1 +OF D

30 +ODD

30 +

1

6

S3

l30(D3

0 − D30)

α3 = − 1

θ0[S

l0(D3 − D3) + SF (2D0D2 + D2

1) + SD(2D0D2 + D21) + 3OF D

20D1 + 3ODD

20D1

+1

2

S3

l30(D2

0D1 − D20D1) −

1

3

S3

l30(D3

0 − D30) +

1

2

S2

l20(SF D

40 + SDD

40) + (TF D

40 + TDD

40)].


Consider OF’ in Figure. A.3 as y-axis and O the origin. The x-axis is on the

dipole side of OF’. The point F’ is (0, ρ0 + Dδ) and the arc F’D’ cut OF’ at an angle

∆φF . So the equation of arc F’D’ is

[x+ ρ sin ∆φF ]2 + [y − (ρ0 + Dδ) + ρ cos ∆φF ]2 = ρ2. (B.16)

We rotate the x-axis and y-axis clockwisely by an angle of θ0/2 so that the new y-axis

passes thru the center of the dipole. The equation of F’D’ becomes

[x cos θ0/2+y sin θ0/2+ρ sin ∆φF ]2+[−x sin θ0/2+y cos θ0/2−(ρ0+Dδ)+ρ cos ∆φF ]2 = ρ2.

(B.17)

On the other hand, we can start with OD’ axis. The angle at D’ is −∆φD. So we

have

[x− ρ sin ∆φD]2 + [y − (ρ0 + Dδ) + ρ cos ∆φD]2 = ρ2. (B.18)

The axis is now rotated in counterclockwise direction by theta0/2 also so the equation

of the arc F’D’ becomes

[x cos θ0/2−y sin θ0/2−ρ sin ∆φD]2+[x sin θ0/2+y cos θ0/2−(ρ0+Dδ)+ρ cos ∆φD]2 = ρ2.

(B.19)

These two equations should be the same. By comparing coefficients, we get

x :⇒ ρ sin ∆φF − [ρ cos ∆φF − (ρ0 + Dδ)]t = −ρ sin ∆φD + [ρ cos ∆φD − (ρ0 + Dδ)]t,

(B.20)

y :⇒ tρ sin ∆φF + [ρ cos ∆φD − (ρ0 + Dδ)] = tρ sin ∆φD + [ρ cos ∆φD − (ρ0 + Dδ)],

(B.21)

1 :⇒ −2ρ(ρ0+Dδ) cos ∆φF+(ρ0+Dδ)2 = −2ρ(ρ0+Dδ) cos∆φD+(ρ0+Dδ)

2 (B.22)

with t = tan θ0/2. Any two of them can give us exact solution with the order we


want. Here we expand the first two equations and solve up to 4th order in δ

ρ0(1 + δ) 1

1 + δ[S

l0Dδ + SF D

2δ2 + (OF D3 +

1

6

1

(1 + δ)2

S3

l30D3)δ3 + (TF D

4 +

1

2

1

(1 + δ)2

S2SFl20

D4)δ4] − 1

6

1

(1 + δ)3(S3

l30D3δ3 + 3

S2

l20SF D

4δ4) − [ρ0(1 + δ)1

−1

2

1

(1 + δ)2[S2

l20D2δ2 + 2

SSFl0

D3δ3 + (S2F D

4 + 2S

l0D(OF D

3 +1

6

1

(1 + δ)2

S3

l30D3))δ4]

−(ρ0 + Dδ)]t = −ρ0(1 + δ) 1

1 + δ[−Sl0Dδ + SDD

2δ2 + (ODD3 − 1

6

1

(1 + δ)2

S3

l30D3)δ3

+(TDD4 +

1

2

1

(1 + δ)2

S2SDl20

D4)δ4] − 1

6

1

(1 + δ)3(−S

3

l30D3δ3 + 3

S2

l20SF D

4δ4)

+[ρ0(1 + δ)1 − 1

2

1

(1 + δ)2[S2

l20D2δ2 − 2

SSDl0

D3δ3 + (S2DD

4 − 2S

l0D(ODD

3 − 1

6

1

(1 + δ)2

S3

l30D3))δ4] − (ρ0 + Dδ)]t,

and

tρ0(1 + δ) 1

1 + δ(S

l0Dδ + SF D


4δ4) + [ρ0(1 + δ)1 − 1

2

1

(1 + δ)2[S2

l20D2δ2 + 2

SSFl0

D3δ3 + (S2F D

4 + 2S

l0D(OF D

3 +1

6

1

(1 + δ)2

S3

l30D3))δ4]

−(ρ0 + Dδ)]

= tρ0(1 + δ) 1


2δ2 +ODD3δ3 + TDD

4δ4) + [ρ0(1 + δ)1 − 1

2

1

(1 + δ)2[S2

l20D2δ2 − 2

SSDl0

D3δ3 + (S2DD

4 − 2S

l0D(ODD

3 − 1

6

1

(1 + δ)2

S3

l30D3))δ4]

−(ρ0 + Dδ)].

Thus, by comparing the coefficients, 0th order terms cancel out each other so we have

for 1st order

SD0 − t(l0 − θ0D0) = SD0 + t(l0 − θ0D0), (B.23)

tSD0 − θ0D0 = −tSD0 − θ0D0. (B.24)


For 2nd order we have

SD1 + l0SF D02+ t(

1

2

S2

l0D0

2+ θ0D1) = SD1 − l0SDD0

2 − t(1

2

S2

l0D0

2+ θ0D1), (B.25)

tSD1 + tl0SF D02 − 1

2

S2

l0D0

2 − θ0D1 = −tSD1 + tl0SDD02 − 1

2

S2

l0D0

2 − θ0D1. (B.26)

For 3rd order we have

SD2 + 2l0SF D0D1 + l0OF D03+ t(−1

2

S2

l0D0

2+ θ0D2 + SSF D0

3+S2

l0D0D1) = SD2

−2l0SDD0D1 − l0ODD03+ t(

1

2

S2

l0D0

2 − θ0D2 + SSDD03 − S2

l0D0D1),

tSD2 + 2tl0SF D0D1 + tl0OF D03+

1

2

S2

l0D0

2 − θ0D2 − SSF D03 − S2

l0D0D1 = −tSD2

+2tl0SDD0D1 + tl0ODD03+

1

2

S2

l0D0

2 − θ0D2 + SSDD03 − S2

l0D0D1.

For 4th order we have

ρ0S

l0D3 + D3t+ [ρ0SF (D1

2+ 2D0D2) +

1

2ρ0S2

l20(D1

2+ 2D0D2)t− ρ0

S2

l20D0D1t]

+(3ρ0OF D02D1 + 3ρ0t

SSFl0

D02D1 − ρ0t

SSFl0

D03) + (ρ0TF D0

4+

1

2ρ0t[S

2F D0

4

+2S

l0OF D0

4+

1

3

S4

l40D0

4])

= ρ0S

l0D3 − D3t+ [−ρ0SD(D1

2+ 2D0D2) −

1

2ρ0S2

l20(D1

2+ 2D0D2)t+ ρ0

S2

l20D0D1t]

+(−3ρ0ODD02D1 + 3ρ0t

SSDl0

D02D1 − ρ0t

SSDl0

D03) + (−ρ0TDD0

4 − 1

2ρ0t[S

2DD0

4

−2S

l0ODD0

4+

1

3

S4

l40D0

4]),


(tρ0S

l0D3 − D3) + (tρ0SF (D1

2+ 2D0D2) −

1

2ρ0S2

l20(D1

2+ 2D0D2) + ρ0

S2

l20D0D1)

+(3tρ0OF D02D1 − 3ρ0

SSFl0

D02D1 + ρ0

SSFl0

D03) + (tρ0TF D0

4 − 1

2ρ0[S

2F D0

4

+2S

l0OF D0

4+

1

3

S4

l40D0

4])

= (−tρ0S

l0D3 − D3) + (tρ0SD(D1

2+ 2D0D2) −

1

2ρ0S2

l20(D1

2+ 2D0D2) + ρ0

S2

l20D0D1)

+(3tρ0ODD02D1 + 3ρ0

SSDl0

D02D1 − ρ0

SSDl0

D03) + (tρ0TDD0

4 − 1

2ρ0[S

2DD0

4

−2S

l0ODD0

4+

1

3

S4

l40D0

4]).

Thus, we can get solution from using MATHEMATICA as

α0 = 1 − 1

θ0

S

l0(D0 − D0) = 1 − 2S2t

θ0(S2 + θ20), (B.27)

and

α1 = − 1

θ0[S

l0(D1−D1)+SF D

20+SDD

20] =

S6t3

θ0(S2 + θ20)

3+

3S4tθ0(S2 + θ2

0)3−SF D0

3−SDD03.

(B.28)

For α2 and α3,the expression is too long and not listed here.

In the α1 equation, we find out that the last 2 terms are sextupole’s contribution.

But in higher order α expression, the multipole effects mix up so we have mixing terms.

This is because that in the expansion of the ∆φ, we already have a mixing term S2SF

or S2SD coming out of arcsin expansion (there is a cubic power which gives mixing

terms). So for higher momentum compaction function, the contribution of different

order of multipoles mixs up and can not be seperated. Numerical verification is

required.

B.1.2 Higher order dispersion

In this section, we will derive dispersion dynamics into higher order of δ.


First, we need to define what is higher order and which order do we need to go up

to. The goal for us is to achieve coherent condition during the interaction of beam

and laser until it gets saturated. From the spread sheet calculation, in order to keep

this condition, with 1TW laser seed, we need to keep the phase-slip factor to the order

of 10−9−10−10. If we assume the momentum dispersion is of the order of 10−4−10−5,

then we need to expand the η = η0 + η1δ + η2δ2 + O(δ3) up to 2nd order in δ. The

study of how to make η2 very tiny should be sufficient under our consideration, thus

we need to keep dispersion up to 2nd order which means D2. In fact, the expansion

to higher order is just following the same technique but needs much more labor work.

Let’s look into the Figure. B.2. We have

Figure B.2:

x′

=dl0ds

dx

dl0= (1 +

x

ρ0) tan (φ− φ0). (B.1)


Furthermore we get

x′′

=x

′2

ρ0(1 + x/ρ0)+ (1 +

x

ρ0

)[1 +x

′2

(1 + x/ρ0)2](φ

′ − φ′

0), (B.2)

where the φ0 and φ are angles for on-momentum and off-momentum particles with

respect to a fixed axis OA. And we have

φ′

0 = − 1

ρ0, (B.3)

and we know ρ = peB

= − dldφ

dφ = −eBdlp

(B.4)

with the B-field as

B = B0 +B′

0x+B

′′

0

2x2 + · · · = B0ρ0[

1

ρ0

+Kx], (B.5)

thus we have

φ′

=dφ

ds= −eB

p

dl

ds= −p0

p(1 +

x

ρ0

)(1

ρ0

+Kx) sec(φ− φ0). (B.6)

Substituting Eq.(B.3) and Eq.(B.6) into Eq.(B.2) results in

x′′

=x′2

ρ0(1 + x/ρ0)+ (1 +

x

ρ0)[1 +

x′2

(1 + x/ρ0)2] 1

ρ0− 1

1 + δ[1 +

x′2

(1 + x/ρ0)2]1

2 (1 +x

ρ0)

(1

ρ0+Kx).

With x = Dδ plugged in, we arrive

D′′

=D′2δ

ρ0(1 +Dδ/ρ0)+ (1 +

Dδ

ρ0

)[1 +D

′2δ2

(1 +Dδ/ρ0)2]

1

δρ0

− 1

1 + δ[1 +

D′2δ2

(1 +Dδ/ρ0)2]1

2 (1 +Dδ

ρ0

)(1

δρ0

+KD)

.

After plugging in the relation D = D0 + D1δ +D2δ2 + O(δ3) and D

′= D

′

0 + D′

1δ +

D′

2δ2 + O(δ3) and compare the 0th order, 1st order and 2nd order terms, we get the

relations:

D′′

0 +D0

ρ20

+KD0 =1

ρ0, (B.7)


D′′

1 +D1

ρ20

+KD1 =1

ρ0[1

2D

′20 − (

D0

ρ0− 1)2] +KD0(1 −

2D0

ρ0), (B.8)

D′′

2 +D2

ρ20

+KD2 = D′

0D′

1

ρ0

+2D1

ρ20

− 2D0D1

ρ30

+1

ρ0

+3D

′20

2ρ0

− 2D0

ρ20

− D′20 D0

ρ20

+D2

0

ρ30

−D′20 D0 +K

D1 −4D0D1

ρ0+

2D20

ρ0−D0 −

3D′20 D0

2− D3

0

ρ20

.

B.1.3 Multipole effect using Hamiltonian expansion

To obtain an explicit expression of how higher order magnets affect dispersion function

and momentum compaction factor, we need to refer to Hamiltonian

H = −h[√

(1 + δ)2 − (px −e

p0Ax)2 − (py −

e

p0Ay)2 − e

p0As], (B.9)

where h is geometry factor given by

h = 1 +x

ρ. (B.10)

Assuming we have only longitudinal vector potential Ax = Ay = 0, we can obtain

H = −(1 +x

ρ)√

(1 + δ)2 − p2x − p2

y +1

2(1 +

x

ρ)2 +

1

2k1(x

2 − y2)

+1

6k2(x

3 − 3xy2) +1

24k3(x

4 − 6x2y2 + y4) + · · · ,

where kn = ep0

(∂nBy

∂yn ) is the gradient strength of correspondent order of multipole

magnets.

Thus we can derive equations of motion in horizontal direction(assuming there is

no vertical dispersion)

x′ = (1 +x

ρ)

px√

(1 + δ)2 − p2x

, (B.11)

p′x = −[(1

ρ

2

+ k1)x+1

2k2x

2 +1

6k3x

3] +1

ρ[√

(1 + δ)2 − p2x − 1]. (B.12)


We can expand the x and px in terms of fractional energy spread δ

x = D0δ +D1δ2 +D2δ

3 +O(δ4), (B.13)

px = ξ0δ + ξ1δ2 + ξ2δ

3 +O(δ4). (B.14)

After plugging these expansion into Eqs.B.11 and B.12, we can obtain for the zeroth

order

D′0 = ξ0, (B.15)

ξ′0 = −(1

ρ

2

+ k1)D0 +1

ρ. (B.16)

Thus we have

D′′0 + (

1

ρ

2

+ k1)D0 =1

ρ, (B.17)

which is the well known relation for zeroth order dispersion D0.

For higher order dispersion function, we have relations

D′′n + (

1

ρ

2

+ k1)Dn = fn(D0, · · · , Dn−1). (B.18)

And the nth order dispersion function is given by Greens function

Dn =

√

βx(s)

2 sin πνx

∫ s+C

s

ds1fn(D0, · · · , Dn−1)√

βx(s1) cos(πνx − φ(s1) − φ(s)). (B.19)

The inhomogeneous terms fn(D0, · · · , Dn−1) up to 2nd order (octupole contribu-

tion) are calculated

f0 =1

ρ, (B.20)

f1 = k1D0 −1

2k2D

20 −

1

ρ(1 − 1

2D′2

0 ) +2

ρ2D0 −

1

ρ3D2

0, (B.21)

and

f2 = k1(D1 −D0 −3

2D0D

′20 ) − k2(D1 −

1

2D0)D0 −

1

6k3D

30 +

1

ρ(1 +D′

1D′0 +

3

2D′2

0 )

+2

ρ2(D1 −D0 −D0D

′20 ) − 1

ρ3(2D1 −D0)D0.


Higher order of magnets can be calculated in the same way.

Momentum compaction factor can be expanded in higher orders

αc = α0 + α1 + α2 + · · · , (B.22)

where αn = 1C

∮

Dndsρ

can be calculated after nth order dispersion is calculated from

the Eq.(B.19).

BIBLIOGRAPHY 97

Bibliography

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DESIGN OF AN ULTIMATE STORAGE RING FOR FUTURE LIGHT SOURCE

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