Wakefield Simulations and Measurements for the CLIC ... - inspire-hep

Wakefield Simulations and Measurements for the CLIC RF Accelerating Structure

THÈSE NO 6185 (2014) PRÉSENTÉE LE 5 MAI 2014 À LA FACULTÉ DES SCIENCES DE BASE

LABORATOIRE DE PHYSIQUE DES ACCÉLÉRATEURS DE PARTICULES PROGRAMME DOCTORAL EN PHYSIQUE

ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE POUR L'OBTENTION DU GRADE DE DOCTEUR ÈS SCIENCES

PAR

Giovanni DE MICHELE

acceptée sur proposition du jury: Prof. C. Hébert, présidente du jury Prof. L. Rivkin, Dr A. Grudiev, directeurs de thèse Dr A. Faus Golfe, rapporteur Dr T. Garvey, rapporteur Prof. V. G. Vaccaro, rapporteur

Suisse 2014

To Martina, Sabrina, Mattia, Angela, Andrea, Emily, Alessandra, Sofia: looking to the future

Contents Acknowledgements

v

Abstract

vii

Sintesi

ix

Introduction

1

1 Linear Colliders

5

1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.2

The Compact Linear Collider: overview . . . . . . . . . . . . . . . . . .

5

1.2.1

7

Two Beam Concept . . . . . . . . . . . . . . . . . . . . . . . . .

2 CLIC Accelerating Structures

11

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.2

RF Design and Parameters . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.3

Wakefields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

2.3.1

Long-Range Wakefields . . . . . . . . . . . . . . . . . . . . . . .

17

2.3.2

Wakefields from a Bunch of Particles . . . . . . . . . . . . . . . .

19

2.3.3

Multipole Expansion of the Wakepotential . . . . . . . . . . . . .

22

2.3.4

Longitudinal and Transverse Impedances . . . . . . . . . . . . .

24

Wakefield Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

2.4.1

Heavy Damped Waveguide . . . . . . . . . . . . . . . . . . . . .

25

3D EM Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

2.5.1

Direct Method . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

2.5.2

Indirect Method . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

2.5.3

Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

2.4 2.5

i

CONTENTS

2.6

2.5.3.1

Material Models . . . . . . . . . . . . . . . . . . . . . .

28

2.5.3.2

GdfidL Simulations . . . . . . . . . . . . . . . . . . . .

29

2.5.3.3

CST Particle Studio Simulations . . . . . . . . . . . . .

32

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

3 EM characterization of damping materials

35

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

3.2

The Coaxial Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

3.2.1

Study of Feasibility of the Method . . . . . . . . . . . . . . . . .

38

3.2.2

Measurements of Dielectric Materials: Air-Gap Correction . . . .

41

The Reflection Method . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

3.3.1

Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

3.3.2

Check of the Routine for the Extraction of the EM Properties . .

57

3.3.3

Measurements with the Reflection Method . . . . . . . . . . . . .

58

The Transmission Method . . . . . . . . . . . . . . . . . . . . . . . . . .

61

3.4.1

Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

3.3

3.4 3.5

4 Measurements of Transverse Wakefields

73

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

4.2

Direct Measurements of Wakefields . . . . . . . . . . . . . . . . . . . . .

73

4.2.1

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

4.2.2

The FACET Facility . . . . . . . . . . . . . . . . . . . . . . . . .

76

4.2.3

Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . .

79

4.2.3.1

Alternative Locations for the CLASSE Experiment . .

83

4.2.3.2

Simulation of the Experiment . . . . . . . . . . . . . . .

83

Indirect Measurements of Wakefields . . . . . . . . . . . . . . . . . . . .

94

4.3.1

Structure Alignment using Beam-Induced Signals . . . . . . . . .

94

4.3.2

SwissFEL Injector X-band Structure . . . . . . . . . . . . . . . .

94

4.3.3

The RF Design . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

4.3.4

Wakefields Monitor Measurements at PSI . . . . . . . . . . . . .

98

4.3.5

Higher bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.3.6

Wakefields Monitor Measurements at Sincrotrone Trieste . . . . 103

4.3

4.4

Mechanical Prototype for CLIC Accelerating Structures . . . . . . . . . 107

ii

CONTENTS

5 Optimization of the HOM Damping Load

111

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.2

The Baseline Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.3

RF Characterization of the Damping Load . . . . . . . . . . . . . . . . . 112

5.4

Microwave Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.5

Cavity Attached to the Waveguide . . . . . . . . . . . . . . . . . . . . . 115

5.6

Towards the Final Load Design . . . . . . . . . . . . . . . . . . . . . . . 118

5.7

The Proposed Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.8

Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Conclusions 5.9

131

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.10 Future of the CLASSE Experiment . . . . . . . . . . . . . . . . . . . . . 132 A Fundamental Theorem of Beam Loading

133

B Breakdown Limit

137

C Technical Aspects and Organization of the Experiment

141

Bibliography

147

Curriculum Vitae

153

iii

CONTENTS

iv

Acknowledgements

First of all, I would like to thank three people for supporting my work, my education and profusely devoted their time to me when necessary. In particular Prof. Leonid Rivkin and Dr. Terence Garvey for giving to me the possibility to join, as PhD student, the LPAP laboratory at EPFL and PSI. I am grateful to them for their willingness to support my attendances to the USPAS schools in Michigan and North Carolina among other courses and conferences. I also would like to thank them for their advices and for putting me in contact with researchers both at EPFL and PSI. I am in debt with my supervisor at CERN, Dr. Alexej Grudiev for the amount of time, weekly and sometime daily based, of our discussions and exchange of ideas. It has been a privilege and a great honour for me working with you and I really enjoyed it. I will never forget how, in your advices, you were never wrong! A special thanks goes to the people that I met and made possible the starting of my PhD journey (in alphabetical order): T. Pieloni, R. Wegner, W. Wuensch, C. Zannini. I would like to thank people that directly or indirectly have given contributions to this work. From CERN : A. D’Elia, M. Filippova, J. M. Giguet, A. Grudiev, G. Hagmann, A. Latina, V. Khan, I. Kononenko, S. Lebet, T. Pieloni, S. Ramberger, G. Riddone, T. Schneider, D. Schulte, A. Solodko, D. Valuch, M. Van Stenis, W. Wuensch, C. Zannini. From PSI : S. Bettoni, P. Craievich, M. Dehler, T. Garvey, L. Rivkin, R. Zennaro. From EPFL: L. Rivkin, J-F. Zuercher. From Sincrotrone Trieste S.C.p.A.: G. D’Auria, F. Gelmetti, M. Milloch, C. Serpico. From SLAC : E. Adli, C. Adolphsen, E. Bong, C. Clarke, S. DeBarger, G.L. Gassner, U. Wienands. From Fermilab: P. Berrutti, G. Gallo, D. Passerelli, P. Stabile, L. Ristori, M. Merio for sharing their experiences and working time while at the USPAS schools.

v

ACKNOWLEDGEMENTS

I thank Dr. W. Bruns for the GdfidL support and useful hints for the material implementation. A special thanks to C. Serpico and G. D’Auria for arranging in short time the wakefields monitor measurements in Trieste. Finally I would like to thank the Swiss National Science Foundation for the financial support of this work and the USPAS office and the US Department of Energy for the school grants. Infine un ringraziamento speciale ai miei genitori che mi hanno sostenuto moralmente durante questi anni all’estero: sono sicuro che porrete questa copia vicino alla tesi di laurea di cui siete tanto fieri.

vi

Abstract

Physics at the terascale is considered to hold the answers to many of the key open questions in particle physics, ranging from the electroweak symmetry breaking mechanism to searches and possible verification of new symmetries and Dark Matter, and a wealth of other possible phenomena beyond the Standard Model. A Linear Collider at the terascale such as the Compact Linear Collider (CLIC) will be able to perform detailed and precise studies of the physics at this scale, perfectly complementing the LHC discovery potential with precision measurements and in several cases also extending the discovery range. In order to reach multi-TeV energies high accelerating gradients are paramount, and the CLIC two-beam concept as proposed in 1986 provides a unique opportunity to reach multi-TeV energies with an e+ e− machine [1]. One of the most important objectives of the CLIC study is to demonstrate the design accelerating gradient of 100 MV/m in a fully featured accelerating structure under nominal operating conditions including pulse length and breakdown rate. Among different requirements, the structure also requires strong long-range wakefield suppression in order to transport the long bunch trains needed for RF power to beam power efficiency. In this work the suppression of the wakefields in the CLIC RF accelerating structure is investigated. Different mechanisms can be used in order to suppress the higher order modes (HOMs) in an accelerating structure. This thesis focuses on the heavy damping mechanism by means of rectangular waveguide attached to the main accelerating cells. The CLIC tapered and damped (TD) RF accelerating structure has been simulated by means of 3D EM simulators in order to map the transverse wakefields and impedances. An extensive study of the damping material has been carried out by measuring its EM properties. Those properties have been implemented in the EM simulators. The damping material

vii

ABSTRACT

to be used has been defined and simulations show that the results fulfill the beam dynamics requirements in terms of wakefield damping. A prototype of the CLIC TD26 RF structure has been built to directly measure long-range wakefields. The experiment has been simulated and we show that is possible in an existing facility to perform such an experiment within our specifications. Furthermore even indirect measurements of wakefields have been investigated on existent RF structures. Measurement and simulations of wakefields monitors have shown both the reliability of our EM simulation tools and the possibility to align the RF structure to micron level precision. Keywords: accelerator, wake functions, impedances, EM simulators, CST MWS, CS PS, Ansoft HFSS, GdfidL, EM material properties, permittivity, EkasicF, EkasicP, CerasicB1, RF structure, CLASSE experiment, wakefield monitors.

viii

Sintesi

La fisica nella scala dei tera elettronvolt è considerata possedere le risposte a molte questioni aperte nella fisica delle particelle, dalla rottura della simmetria elettrodebole alla ricerca di nuove simmetrie e della Materia Oscura ad altri fenomeni al di l` a del Modello Standard. Un collisore di particelle come il Compact Linear Collider (CLIC) permetter` a di performare dettagliati e precisi studi fisici in questa scala di energia, perfettamente complementari al potenziale di scoperta di LHC e in alcuni casi potr` a anche estendere la scala energetica di scoperta. Per raggiungere energie dei multiTeV sono essenziali alti gradienti acceleranti e il concetto di due fasci di CLIC come proposto nel 1986 fornisce un’unica opportunità per raggiungere tali energie con una macchina a collisioni e+ e− . Uno dei pi` u importanti obiettivi di CLIC è quello di dimostrare un gradiente accelerante di 100 MV/m in una completa struttura accelerante operante in condizioni nominali di lunghezza dell’impulso e frequenza di scarica. Tra differenti specifiche, la struttura richiede anche lo smorzamento dei forti campi scia a lungo raggio in modo da poter accelerare lunghi treni di pacchetti di particelle necessari per avere una buona efficienza in termini di rapporto tra potenza fornita alla struttura rispetto alla potenza effettivamente somministrata al fascio. In questo lavoro viene studiata la soppressione dei campi scia nella struttura RF accelerante di CLIC. Diversi meccanismi possono essere impiegati per sopprimere i modi di ordine superiore in una struttura accelerante. In questa tesi si focalizza l’attenzione su un meccanismo di ampio smorzamento utilizzando guide d’onda rettangolari annesse alle celle acceleranti. La struttura RF accelerante di CLIC con il suo profilo rastremato e smorzato è stata simulata con codici elettromagnetici 3D in modo da mappare i campi scia e

ix

SINTESI

le impedenze trasverse. Uno studio estensivo dei materiali di smorzamento è stato effettuato misurando le proprietà EM dei materiali stessi. Tali proprietà sono state poi implementate nei simulatori EM. Il materiale di smorzamento è stato cos`ı definito e le simulazioni hanno mostrato che le sue caratteristiche EM soddisfano le specifiche della dinamica del fascio per quanto riguarda lo smorzamento dei campi scia. Un prototipo della struttura RF di CLIC, TD26 (dall’inglese, tapered and damped ), è stato costruito per misurare in modo diretto i campi scia a lungo raggio. L’esperimento è stato simulato e in questo lavoro si mostra che è possibile, negli esistenti laboratori, performare tale esperimento con le nostre specifiche. Inoltre sono state effettuate anche misure indirette di campi scia su strutture a radiofrequenza. Le misure e le simulazioni di dispositivi di monitoraggio per campi scia hanno mostrato sia l’affidabilità dei nostri simulatori EM e la possibilità di allineare le strutture RF ad un livello di precisione dell’ordine dei micrometri.

Keywords: acceleratori, campi scia, impedenze, simulatori EM, CST MWS, CS PS, Ansoft HFSS, GdfidL, proprietà EM dei materiali, permittività, EkasicF, EkasicP, CerasicB1, strutture RF, esperimento CLASSE, dispositivi di monitoraggio per campi scia.

x

Introduction

The Large Hadron Collider (LHC) at CERN is probing the terascale energy region with a rich program of physics at a new high energy frontier. The latest collisions in 2012 brought the discovery of the Higgs boson. Nevertheless the LHC experiments need to be complemented by experiments at a lepton collider in the teraelectron volt (TeV) energy range. The required beam collision energy range will be better defined following physics requirements based on LHC results when the luminosity will almost double after the restart of the machine in 2015. The highest energy lepton collisions, 209 GeV, were reached with electrons and positrons colliding in the Large Electron-Positron Collider (LEP) at CERN. In this case the limiting factor was the synchrotron radiation. In order to overcome this limitation, there was a proposal to realize electron-positron colliders where two opposing linear accelerators accelerate the particles to their final energy in one pass before focusing and colliding in a central interaction point inside a detector. Global collaborations are currently developing two alternative technologies, each with a different final energy: the International Linear Collider (ILC) [2] with colliding beam energy of 500 GeV (upgradeable to 1 TeV) based on beam acceleration by superconducting radiofrequency (RF) structures; and the normal conducting Compact Linear Collider (CLIC) [3] which should reach a final energy of 3 TeV [4]. CLIC could be built in different stages, starting at the lowest energy required by physics, with successive luminosity and energy upgrades. The CLIC study is focused on the design of a linear collider with a colliding beam energy of 3 TeV and a luminosity of 2 × 1034 cm−2 s−1 . In order to be efficient and cost effective, CLIC aims at an acceleration of 100 MV/m. At the moment only room temperature travelling wave structures at high frequency (12 GHz) are likely to achieve this gradient. In order to reduce the power

1

INTRODUCTION

consumption, the accelerating structures have to be designed with high RF-to-beam efficiency but at CLIC the RF-to-beam efficiency is limited at higher gradient because acceleration is proportional to field level while ohmic wall losses scale as the square of the fields. The basic mechanism to maintain RF-to-beam efficiency in the range of 20−30% in high-gradient normal-conducting linear collider designs is to accelerate bunch trains. With trains of sufficiently high current, power is transferred to the beam before it is lost to the cavity walls [1]. The bunch trains are also susceptible to a bunch-to-bunch instability driven by long-range transverse wakefields. Relative offsets, between the beam and the structure, of the leading bunches in a train excite higher-order modes which then give transverse kicks to the following bunches. In order to counteract this effect the accelerating structures have two characteristics which suppress the effect of the long-range transverse wakefields: detuning and damping. The solution which has been adopted for CLIC are heavily, waveguide-damped structures with a moderate level of detuning. Alternative designs are also being pursued including an adaptation of the NLC/JLC DDS design [5] to 100 MV/m and choke-mode damping [6]. In this PhD thesis the study of wakefield damping for CLIC accelerating structure has been investigated. The work has been divided into five chapters as follows: • Chapter 1 introduces the CLIC complex and its major features. • Chapter 2 describes the concepts of wakefields and impedances. Furthermore it introduces the wakefield damping mechanism for CLIC accelerating structures and the simulation methods used to simulate them. • Chapter 3 addresses the EM characterization of damping materials by introducing the coaxial method. For the first time, damping materials for the CLIC RF accelerating structure have been measured over a wide range of frequencies. With such measured properties, simulations of long-range transverse wakefields have been performed on the baseline design of the CLIC RF accelerating structures and on the prototype for direct wakefield measurements. • In Chapter 4 two different approaches to measure wakefields in RF structures are presented. The direct method makes use of a drive and witness bunch in order to sample the excited wakefield. A location for this experiment has been found and the experiment itself has been simulated. It turned out that at the

2

Introduction

time of writing the only location is the SLAC linac in CA, USA. Simulations of the experiment with the tracking code PLACET have been performed and show that the location fulfills our constraints and requirements. A proposal has been submitted on October 2011 and accepted by the FACET user facility committee but to date no funding has been provided by the US Department of Energy for the commissioning of positrons needed for such measurements. A prototype structure has been built and all details and ancillaries and integration in the SLAC tunnel and experiment location have been cross-checked with SLAC colleagues and with the support of the CLIC mechanical engineering team at CERN. In addition, the results of indirect measurements of wakefields on X-band structures are presented. • Chapter 5 describes the geometry optimization process for the damping loads of the CLIC accelerating structure. A new geometry is proposed which is more compact with respect to the nominal design.

3

INTRODUCTION

4

Chapter

1

Linear Colliders 1.1

Introduction

The Large Hadron Collider at CERN is exploring the TeV energy range but nonetheless an electron-positron collider could be a complementary machine due to its unique combination of high energy and experimental precision and could also extend the discovery range of the LHC. In linear colliders the charged particles are accelerated linearly through thousands of accelerating cavities. This means that the RF parameters and the technologies involved have to be carefully selected in order to have a cost effective design in terms of power consumption and accelerator length.

1.2

The Compact Linear Collider: overview

The research and development for linear colliders in the TeV range is being carried out within the framework of a world-wide collaboration. The Compact Linear Collider study aims at a center-of-mass energy range for electron-positron collisions of 0.5 to 5 TeV, optimised for a nominal center-of-mass energy of 3 TeV. In order to reach this energy in a realistic and cost efficient scenario, the accelerating gradient has to be very high. CLIC aims at a loaded accelerator gradient of 100 MV/m that can be reached by room temperature travelling wave structures at high frequency (12 GHz). Superconducting technology is fundamentally limited to lower gradients [7]. Figure 1.1 shows the layout of the CLIC accelerator complex at the energy of 3 TeV and Figure 1.2 a schematic of the underground installation.

5

1. LINEAR COLLIDERS

Figure 1.1: Layout of CLIC at 3 TeV. Image credit: CERN

Figure 1.2: Schematic of the underground CLIC installation. Image credit: CERN

6

1.2 The Compact Linear Collider: overview

The main electron and positron beams are generated and pre-accelerated in the injector linacs and then enter the pre-damping rings (PDR) and later into the damping rings (DR) in order to reduce the beam emittance [1]. The small emittance beams are further accelerated in a common linac before being transported through the main tunnel to the turnarounds (TA). After the turnarounds the acceleration of the main beam begins with an accelerating gradient of 100 MV/m. The power generation of the main linac is not done through classical RF amplifiers such as, e.g., klystrons because the CLIC target energy would require a huge number of these devices. In the CLIC acceleration scheme the klystron powering is replaced by the generation of a drive beam at an energy of about 2.4 GeV with a working frequency of 1 GHz. Delay loops (DL) and combiner rings (CR1 and CR2) are used for the time compression of the drive beam pulses. The time-compressed drive beam reaches a current of about 100 A at a beam energy of about 2.4 GeV. This compressed drive beam is transported through the main linac tunnel to 24 individual turnarounds [1]. At this stage the beam enters the power extraction and transfer structure (PETS) where the power is extracted from a high-current, low-energy drive beam in order to accelerate the low-current main beam to high energy (about 1 A and 9 GeV-1.5 TeV). The beams collide after a long beam delivery section (BDS) (collimation, final focus) in one interaction point (IP) in the centre of the complex [1] . Figure 1.3 shows a possible implementation of the CLIC complex in the Geneva area. The generation of the main beams, the drive beams, and the central collision point would fall into existing CERN territory, whereas the two 24 km long acceleration tunnels would extend into the local area as underground installations. The blue dots show the tunnel length needed for a collision energy of 3 TeV, whereas the pink dots indicate the size of the installation for 500 GeV [1].

1.2.1

Two Beam Concept

The concept of the drive beam and main beam comes from the necessity to power thousands of accelerating structures (AS) with short RF pulses. The peak RF power per unit length required to establish an accelerating gradient of 100 MV/m is about 270 MW per meter of active structure length. A total of about 9.2 TW of RF peak power is required for both linacs. Such power cannot be maintained for very long, the bunch train to be accelerated is only 156 ns long with a repetition frequency of 50 Hz. If one takes into account the filling time of the accelerating structures, the RF pulse

7

1. LINEAR COLLIDERS

Figure 1.3: Potential location for the CLIC accelerator complex. Image credit: CERN

is 244 ns long. The extremely demanding peak power requirement is met in CLIC by adopting the two beam acceleration scheme [1]. The two beam concept consists of two RF structures: a low impedance PETS, with large aperture and high group velocity, which decelerates the drive beam and a high impedance accelerating structure, with small aperture and low group velocity, which accelerates the main beam (Table 1.1).

Parameter

PETS

AC

Aperture Radius [mm] R0 [kΩ/m] Q vg [%] c Gradient [M V /m]

11.5

3.15 ÷ 2.35

2.2

15 ÷ 18

49

1.65 ÷ 0.83

-6.3

100

Table 1.1: Two beam structure parameters.

The available klystrons on the market can only deliver power into pulses which are about one order of magnitude longer with respect to the CLIC RF pulse. A klystron powered linear collider with 100 MV/m accelerating cavities would need about 35000

8

1.2 The Compact Linear Collider: overview

high power klystrons (about 50 MW each) with each klystron having a factor of five pulse compression [1]. Such a number of klystrons is not feasible in terms of cost and maintenance, for this reason the two beam scheme of CLIC, at the present, turns out to be the most efficient solution. In this case the power is transported to the accelerating structures by a second electron beam, the drive beam, which runs parallel to the main beam at a distance of about 60 cm. Figure 1.4 shows the layout of the PETS feeding the accelerating structures. One PETS provides RF power for two accelerating structures. This scheme also provide a simplification of the tunnel layout, without any RF amplifiers (klystrons), which houses the main linac, the drive beam and the transfer lines.

Figure 1.4: CLIC Module Layout. Image credit: CERN

CLIC is divided into sectors, each sector is on average 878 m long and consists of 2 984 accelerating structures with a length of 23 cm. Each linac consists of 24 such sectors, each accelerating the main beam by about 62 GeV. The total ratio of active length to linac length is almost 80%. Each pulse train of the drive beam carries a peak power of about 240 GW but the average beam power transported to each linac sector is only 2.9 MW given the pulse length of 244 ns and the repetition rate of 50 Hz.

9

1. LINEAR COLLIDERS

10

Chapter

2

CLIC Accelerating Structures 2.1

Introduction

This chapter describes the RF design, the RF parameters and the geometry of the CLIC RF accelerating structure. In addition, it introduces the concepts of wakefields and impedances that will be applied in Chapter 4 and Chapter 5.

2.2

RF Design and Parameters

The CLIC accelerating structure parameters (see Table 2.1) [1] follow the baseline design reported in [8]. Figure 4.1 shows the vacuum part of the 3D geometry of one CLIC RF accelerating structure, which is composed of a tapered chain of 26 damped cells with double-feed couplers for the input and the output power. The cell geometry varies along the length of the accelerating structure such that the synchronous phase velocity at the operating frequency of 11.994 GHz is always c. The cell irises vary in radius (3.15 to 2.35 mm) and thickness (1.67 to 1.00 mm). The total length of one accelerating structure is about 25 cm and the transverse dimensions fit in a circle of 10 cm radius. Each cell is equipped with four rectangular waveguides in order to heavily damp the HOMs. The three most important parameters are shunt impedance per unit length r, Q factor and the group velocity vg i.e. the rate at which RF energy flows along the

11

2. CLIC ACCELERATING STRUCTURES

Parameter

Symbol

Value

Main Linac RF Frequency [GHz] Linac repetition rate [Hz] No. of particles/bunch No. of bunches/pulse Bunch separation [ns] Bunch train length [ns] Total pulse length [ns] Unloaded/loaded gradient [MV/m] Acceleration structure length (active) [m] Average a/λ Input/Output iris radius [mm] Input/Output iris thickness [mm] Input/Output group velocity [%] No. of regular cells/acc. structure Filling time/rise time [ns] First/Last cell Q-unloaded First/Last cell shunt impedance [MΩ/m] Accelerator structure input power [MW] RF to beam efficiency [%]

fRF frep N Nb ∆tb τtrain τpulse Gunl/l lstruct ha/λi a t vg /c Nc τf , τr Q rs Pacc ηb,RF

11.994 50 3.72×109 312 0.5 (6 RF periods) 156 244 120/100 0.23 0.11 3.15 - 2.35 1.67 - 1.00 1.65 - 0.83 26 67/21 5536 - 5738 81 - 103 61.3 28.5

Table 2.1: CLIC main linac and accelerating structure parameters.

12

2.2 RF Design and Parameters

Figure 2.1: CLIC travelling wave accelerating structure.

structure [9]: r=

2 Eacc −dP/dz

(2.1)

Q=

ωU/L −dP/dz

(2.2)

vg = dω/dβ

(2.3)

where Eacc is the accelerating gradient over the cell, L the cell length and U the stored energy in the cavity. The power flow is equal to the energy per unit length times the group velocity: P = vg U/L

(2.4)

and the attenuation characteristics are: dP = −2αP dz dEacc = −αEacc dz

where

13

(2.5)

α=

ω 2vg Q

(2.6)


The term α is the attenuation parameter in unit of [1/m]. It depends on the group velocity and the Q value of the structure. The most critical parameters for the design of a TW disk loaded structure are the frequency, the phase advance per cell and the attenuation per section. Regarding the attenuation, the RF design can have a constant attenuation i.e. the so called constant impedance structure. Alternatively, the group velocity can be reduced along the length by tapering the dimension of the iris radii in order to keep stored energy and thus the accelerating gradient constant compensating the power losses along the structure. In this latter case the field is kept constant and the structure is called constant gradient [10]. For a constant impedance structure, the field and the power flow, by integrating the equations 2.5 and 2.6, is attenuated as: P = P0 e−2αz ; Eacc = E0 e−αz

(2.7)

where P0 is the input power. For a structure of length L, the exponential becomes αL = τ , the so called attenuation parameter of the structure. For a constant gradient structure, by assuming that r and Q are constant (this is done by modifying the structure dimensions to adjust α(z)) we can redefine the attenuation and the power as: Z τ=

L

α(z) dz;

PL = P0 e−2τ

(2.8)

0

Because r and Eacc are constant, from the 2.1 also dP/dz has to be constant i.e. the power decreases linearly along an RF structure of length L: P (z) − P0 z = (2.9) PL − P0 L By substituting the 2.8 in 2.9 we obtain the relation between the power and distance in a constant gradient RF structure: h i z P (z) = P0 1 − (1 − e−2τ ) (2.10) L The choice of the operating mode is determined by the dispersion characteristics of the structure and the operating frequency. Figure 2.2 shows a dispersion plot for a single cell of the CLIC main linac. Figure 2.3 shows the dispersion for the first dipolar band as well.

14

2.2 RF Design and Parameters

Figure 2.2: Dispersion curve for the accelerating mode of the middle cell. The red line corresponds to the speed of light line.

Figure 2.3: Dispersion curve for the accelerating mode and first dipolar mode of the middle cell. The red line corresponds to the speed of light line.

15


phase advance [degree]

Q-values

0 30 60 90 120 150 180

7.974 8.150 8.516 8.785 8.899 8.919 8.917

Table 2.2: Phase advance vs Qs for the first dipolar band of the middle cell.

Table 2.2 shows the Q values versus the phase advance for the first dipolar band of the middle cell. As said before, the TW linacs can be designed as constant impedance all along the accelerating cells or quasi-constant gradient as in the case of CLIC. The baseline CLIC RF accelerating structure is a disk loaded structure with waveguides attached to the accelerating cell to allow damping of the HOMs. The RF structure has been designed by optimizing the geometric parameters a (iris radius), b (cavity radius), t (iris thickness) of the single cells [11]. The parameter b is determined from tuning of the frequency. The Q value increases as thickness decreases. The group velocity (the ratio of the propagation power to the stored energy) increases as the aperture increases. The shunt impedance and r/Q increase as thickness and/or aperture decrease. Small thickness of the disk increases the ratio Es /Eacc , as does larger a because the field concentration near the beam axis is less for larger apertures [9].

2.3

Wakefields

A bunch traveling along a device carries EM (electromagnetic) fields that induce surface charges and currents in the walls. Such charges and currents become sources of fields that act back on the same or following bunches. In the case of relativistic charged particles, the electric field distribution is Lorentz contracted to a disk perpendicular to the motion with a small angular spread on the order of 1/γ, where γ is the relativistic factor. In this case the longitudinal component of the electric field is almost zero.

16

2.3 Wakefields

Even in a smooth perfectly conducting pipe the electric field contraction is present. When the bunch encounters geometric variations such as RF cavities, steps, bellows, diagnostic components, scattered EM radiation is produced and can act on trailing particles within the same bunch or on following bunches. This scattered radiation is called the wakefield and such a definition is strictly true for relativistic electron beams. For particles with velocity less than the speed of the light, the scattered radiation does not always travel behind the source particles. In the case of a cavity, such wakefields resonate inside and can be described as a sum of all resonant modes excited within the structure. The frequencies higher than the cutoff frequency of the beam pipe will propagate away while the lower frequencies, below the cutoff frequency of the beam pipe, remain inside the structure in which they were excited and decay according to their quality factors. These last modes can affect the motion of the particles in the trailing bunches [12].

2.3.1

Long-Range Wakefields

The wakefields are classified either as short-range or long-range wakefields. For ultrarelativistic bunches, short-range wakefields generated by the particles at the head of the bunch affect the motion of particles in the tail of the bunch causing energy loss and a transverse deflection of the particles in the tail. The effect of the long-range wakefields is driven by high Q transverse dipolar modes that induce time-varying transverse deflections in trailing bunches. These deflecting modes can cause beam breakup (BBU) instabilities and lead to emittance growth or beam losses on the walls. In this thesis we are above all interested in long-range wakefields because they can affect the performances of the CLIC RF accelerating structures. In particular we are interested in the integrated effect of the scattered fields on following bunches as they pass through the same region that has been perturbed by the wakefields. One defines the wakepotential that characterizes the net impulse delivered to the trailing bunches that are moving along the same or parallel paths with respect to the sources. Let’s assume that the source point charge q0 is moving parallel to the central axis of a generic geometry and is displaced by an amount r0 (see Figure 2.4) according to z0 = c · t and that the witness charge is displaced by an amount r and travels with the same velocity at a distance s according to z = c · t − s.

17


Figure 2.4: Generic geometry with cross section variation.

One defines the δ-function longitudinal and transverse wakepotentials [12, 13] per unit source charge as: 1 wz (r, r0 , s) = − q0

L

Z

Ez (r, z, t)t=(z+s)/c dz

(2.11)

0

and

w⊥ (r, r0 , s) =

1 q0

Z

L

[E⊥ + c(ˆ z × B)]t=(z+s)/c dz

(2.12)

0

where L is the interval in which the witness charge experiences the wakefields. The minus sign in the longitudinal wakepotential means that the witness charge loses energy when the wake is positive. Positive transverse wake means that the transverse force is defocusing. These functions are characteristic properties of the geometric perturbation and they can be seen as a transfer function of the system. The equations 2.11 and 2.12 are useful as Green’s functions i.e. as impulse response from which the wakepotentials of an arbitrary charge distribution can be calculated via the convolution of the Green’s function with the arbitrary charge distribution itself. Because of the wakefields the witness charge receives a longitudinal force which changes its energy and a transverse force which deflects its trajectory. For the longitudinal one we have that the momentum

18

2.3 Wakefields

change ∆pz or energy change ∆U is:

∆pz (r, r0 , s) =

∆U (r, r0 , s) −q q0 wz (r, r0 , s) = c c

(2.13)

and for the transverse momentum change:

∆p⊥ (r, r0 , s) =

2.3.2

q q0 w⊥ (r, r0 , s) c

(2.14)

Wakefields from a Bunch of Particles

The δ-function wakepotentials can be used as Green functions weighted by the charge distribution. Let us consider ultra-relativistic particles characterized by a line charge density Λ(z). An element of charge dq = Λ(z0 )dz0 while passing through a scattering geometry will produce a potential at z, either longitudinal or transverse, given by [12]:

dV (z) = w(s)Λ(z0 )dz0

(2.15)

The bunch wakepotential for the linear distribution at z is the integral overall the source charges ahead of z divided by the total charge in the bunch.

1 W (z) = Qtot

Z

z

1 w(z − z0 )Λ(z0 ) dz0 = Qtot −∞

Z

∞

w(s)Λ(z − s) ds

(2.16)

0

Figure 2.5 shows the Gaussian longitudinal wakepotential for different bunch lengths of a pillbox cavity. For the case of a short bunch, the wakepotential is positive over the length of bunch i.e. the particles lose energy. For the longest bunch the wakepotential changes sign and the particles in the tail gain energy. The bunch energy loss to wakefields can be expressed as:

d∆U = Λ(z)Qtot Wz (z)dz

(2.17)

and the total energy loss overall of the charge distribution is: Z

∞

∆U = Qtot

Λ(z)Wz (z)dz −∞

19

(2.18)


Figure 2.5: Wakepotentials of a pillbox cavity for different bunch lengths.

20

2.3 Wakefields

The total bunch loss parameter, in units of V/pC, is defined as: ∆U (2.19) Q2tot If the wake integration line has an offset ∆r with respect to the beam line, a ktot =

transverse kick factor, in units of V/pC/m, is defined as: R∞ K⊥ =

−∞ Λ(z)W⊥ (z)dz

(2.20)

∆r Qtot The loss parameter for the high order modes is: kHOM s = ktot −

ωr 4Q0

(2.21) acc.mode

For a Gaussian distribution of the electron beam, the line charge density is: Qtot −z02 /2σ2 (2.22) e Λ (z0 ) = √ 2πσ Following the calculations in [12], the modal loss parameter (energy lost by unit charge) for the Gaussian distribution is: k0n,G = k0n e

2 σ 2 /2c2 −ω0n

= k0n e

−2π 2

σ λ0n

2

(2.23)

where n is the number of the modes. The exponential factor is called the Gaussian form factor and from the expression it can be seen that the bunch loss factor decreases when the RMS bunch length becomes a significant fraction of the wavelength for that particular mode. This is because the wakefields no longer add in phase. The total bunch loss parameter for the Gaussian beam is:

ktot =

∞ X

−2π 2

k0n e

σ λ0n

2

(2.24)

n=0

One can compute the total energy loss of the particles in the bunch to the wakefields. In the case of CLIC, the number of particle per bunch is N = 3.72 × 109 and for the number of RF structures we assume the ratio of the total linac length (21.8 km) to the RF structure length (23 cm). The total loss factor for one RF structure calculated from EM simulations is '114 V/pC. The energy loss per electron for the entire linac is [12]: ∆U = e2 N ktot Nstructures N

21

(2.25)


Substituting the numbers gives

∆U N

= 6.47 GeV i.e. 0.43% of the final energy of

1.5 TeV which can be replaced by additional RF power. For this reason the greater concern associated with wakefields (in the case of transverse wakefields the loss is even less) is not the energy loss but the increased energy spread and the transverse emittance growth.

2.3.3

Multipole Expansion of the Wakepotential

In geometries with cylindrical symmetry the electric and magnetic fields can be written as a multipole expansion [14, 15] of sine and cosine functions of the azimuthal variable φ. From Maxwell equations, it turns out that in the plane φ = 0 both the electric and magnetic fields can be calculated by the radial and longitudinal components of the electrical field. A beam passing through a cavity will interact with the excited modes and this interaction is characterized by the longitudinal modal loss factor kz,mn or by the quantity r/Q. These parameters can be calculated numerically using e.g. the EM R R codes CST MWS [16], CST PS [16], HFSS [17], GdfidL [18] from the longitudinal

voltage and stored energy [12, 15]: kz,mn =

|Vz,mn |2 4Umn

(2.26)

where m = 0, 1, 2.... Let us assume that the source charge is displaced at radius r0 along the x-axis (θ=0) and the witness charge is located at (r, θ). The total m-pole component of the δ-function longitudinal wakepotential can be expressed as [13]: wz,m = 2

w⊥,m = 2m

r m r m 0

a

r m r m−1 0

a

a

a

cos (mθ)

X

kz,mn cos(ωmn s/c)

(2.27)

n

X 1 c ˆ (mθ) rˆcos (mθ) − θsin kz,mn sin(ωmn s/c) a ωmn n (2.28)

The modal loss factor can be calculated from frequency domain calculations (e.g. R HFSS, CST MWS ) while the total loss factor (see Section 2.3.2) and the wakepoR tentials are better determined in time domain simulations (e.g. CST PS , GdfidL).

Furthermore if the radial displacements are small compared to the aperture radius of

22

2.3 Wakefields

the beam tube a, then the dominant terms will be the monopole mode (m = 0) for the δ-function longitudinal wakepotential and the dipole mode (m = 1) for the δ-function transverse wakepotential [14, 15]: wz,m ' wz,0 =

X

2kz,0n cos(ω0n s/c)

for s > 0

(2.29)

n

w⊥,m ' w⊥,1 = r0

X

2

n

1 c kz,1n sin(ω1n s/c) 2 a ω1n

for s > 0

(2.30)

where ωmn are the frequencies of the modes. For dipole modes we can define a transverse kick parameter as: kz,1n c (2.31) ω1n a2 It should be noticed that in this case the δ-function longitudinal wakepotential is k⊥,1n =

independent of the transverse coordinates of both the source and test charges, and the δ-function transverse wakepotential is linearly proportional only to the radial offset of the source charge r0 . From the equation 2.29, for s = 0+ , wz is positive i.e. the force (see equation 2.11) is negative and the test charges immediately behind the source charge are decelerated. The source charge at s = 0 sees half of the wake potential that is induced behind it (see Appendix A). The transverse wakepotential instead is zero at s = 0. As s increases both wakepotentials can have different signs. In terms of r/Q we have [15]: |Vz,mn |2 4kz,mn rmn = = 2m Qmn ωmn U a ωmn wz,m ' wz,0 =

X1 n

w⊥,m ' w⊥,1 = r0

(2.32)

r0n cos(ω0n s/c) Q0n

for s > 0

(2.33)

X 1 r1n c sin(ω1n s/c) 2 Q1n n

for s > 0

(2.34)

2

ω0n

The wakepotential of a Gaussian bunch with charge density as expressed in equation 2.22 is obtained by a convolution integral with the δ-function wakepotential (see equation 2.29). In the hypothesis that σ (z − z0 ) it gives [19]: Wz,m ' Wz,0 =

X

1

2kz,0n cos(ω0n s/c)e− 2 (

n

23

ω0n σ 2 c

)

for s > 0

(2.35)


Notice that once we know the frequencies, the kick factors and the quality factors of a cavity, we know the wakepotential for all s. The reader can find more details about methods and applications for wakefields in several papers [13, 14, 15, 19, 20, 21, 22, 23, 24].

2.3.4

Longitudinal and Transverse Impedances

The long-range wakepotential is often dominated by few modes performing damped oscillations after the source charge has passed. In this case a frequency domain description is useful and e.g. the equations 2.33 and 2.34 can be used by considering only one or few modes. The bunch longitudinal wakepotential in terms of the δ-function wakepotential for the ultra-relativistic case, from the 2.16, can be re-written as [12]: Wz (z) =

1 cQtot

Z

∞

wz (s)I(z − s) ds

(2.36)

−∞

where I = Λc and the integral has been extended to −∞ because wz (s) = 0, for s < 0. If we consider a sinusoidal source current then I(z − s) = I0 ejω(z−s)/c = I(z)e−jωs/c and re-writing 2.36 becomes: Z

Wz (z)Qtot 1 = I(z) c

∞

wz (s)e−jωs/c ds

(2.37)

−∞

The right hand term is defined as beam coupling impedance and is the Fourier transform of the δ-function wakepotential. It is given by the ratio of the voltage response V (ω) = Wz (ω)Qtot and the source current I(ω) = I0 ejωz/c : V (ω) 1 Zz (ω) = = I(ω) c

Z

∞

wz (s)e−jωs/c ds

(2.38)

−∞

In the same way one can define a transverse impedance as: −j Z⊥ (ω) = c

Z

∞

w⊥ (s)e−jωs/c ds

(2.39)

−∞

By the inverse Fourier transformation we get: 1 wz (s) = 2π

Z

j 2π

Z

w⊥ (s) =

∞

Zz (ω)ejωs/c dω

(2.40)

Z⊥ (ω)ejωs/c dω

(2.41)

−∞ ∞

−∞

24

2.4 Wakefield Damping

Furthermore due to causality, the real and the imaginary parts of the impedances are not independent. By causality one means that there is no field ahead of the driving charge and this happens when e.g. a source moves with the velocity of the light therefore its associated EM fields lie perpendicular to the direction of motion [25]. For this reason only the real or the imaginary part of the impedance is needed to write the wakefields [14, 19, 24]: 1 wz (s) = π

Z

∞

Re[Zz (ω)]cos −∞

ωs c

dω

(2.42)

Another important feature of the wakefields is the relationship existing between the longitudinal and transverse impedances. This relation is known as Panofsky-Wenzel theorem [14, 19, 26] and states [12]: Z⊥ (ω) =

c ∇Zz (ω) ω

(2.43)

This expression is often used in EM codes in order to calculate the transverse impedance from the longitudinal impedance of the dipole mode.

2.4

Wakefield Damping

The CLIC operation foresees trains of bunches, therefore transverse wakefields can generate emittance growth and instabilities. For this reason in CLIC accelerating structures, heavy damping and linear detuning of the accelerating cells is foreseen. The CLIC RF structure is equipped with lossy material loads terminating four rectangular waveguides coupled to each accelerating cell in order to damp the high order modes (HOMs) which are the source of long-range transverse wakefields. The lossy material absorbs EM wave energy with little reflection back to the accelerating cells. In the past, computations of the long-range wake of CLIC accelerating modes have been done using perfectly absorbing boundaries to terminate the damping waveguides. In this thesis, 3D EM simulations of CLIC baseline accelerating structure with HOMs damping loads have been performed as well.

2.4.1

Heavy Damped Waveguide

Figure 2.6 shows the current design shape of the loads used to terminate the damping waveguides. The distance between the beam axis and the tip of the SiC load is

25


50 mm. The load has 30 mm long part which is tapered from 1 × 1 mm cross-section to 5.6 × 5.5 mm and a 10 mm long part of the latter cross-section [8]. In this thesis the baseline design for the real geometry has been verified i.e. with damping loads made of realistic lossy material. Different materials have been investigated for their lossy characteristics such as CerasicB1, EkasicF and EkasicP that are different forms of SiC. The EM properties have been measured for some frequency points. The choice of the material for the damping loads is a compromise between EM properties and cost. It turns out that the better material (CerasicB1) is more expensive. More details about measurements of EM properties can be found in Chapter 3 and Chapter 5 describes an optimization of the load shape.

Figure 2.6: Geometry of the baseline damping load into the waveguide.

2.5

3D EM Simulations

All the simulations in this thesis have been performed mainly with three 3D EM codes: R R GdfidL, CST MWS and PS , HFSS. The most important point for every simulation

is the discretization of the structure. An estimation of the maximum relevant frequency is [27]: fmax =

c 2πσz

(2.44)

This frequency should be sampled at least 10 times on the grid. But moving towards short bunches and a longer structure it has turned out that this criterion is not strong

26

2.5 3D EM Simulations

enough [27]. Assuming an equidistant grid, it is necessary to have a grid step width: s ∆z