Analysis of Surface Roughness Wake Fieldsand Longitudinal Phase Space

in a Linear Electron Accelerator

Dissertationzur Erlangung des Doktorgrades

des Fachbereichs Physikder Universitat Hamburg

vorgelegt von

Markus Huningaus Borken in Westfalen

Hamburg2002

Gutachter der Dissertation Prof. Dr. P. SchmuserProf. Dr. M. Tonutti

Gutachter der Disputation Prof. Dr. P. SchmuserDr. R. Brinkmann

Datum der Disputation 5. August 2002

Vorsitzender des Prufungsausschusses Dr. D. Grundler

Vorsitzender des Promotionsausschusses Prof. Dr. G. Huber

Dekan des Fachbereichs Physik Prof. Dr. F.-W. Bußer

Zusammenfassung

Ein Experiment ist in der TESLA Test Anlage (TTF) durchgefuhrt worden, umStorwellenfelder zu untersuchen, die durch Elektronenpulse mit PicosekundenLange in engen Strahlrohren mit einer kunstlich aufgerauhten Innenoberflacheangeregt werden. In einem magnetischen Spektrometer wurde die durch dieStorwellen erzeugte Energiestruktur der Elektronenpakete analysiert. Starke har-monische Energiemodulationen wurden beobachtet. Mit Hilfe einer longitudina-len Phasenraumtomographie wurden die Wakepotentiale direkt vermessen. Dazuwar die Implementierung eines neuen Rekonstruktionsalgorithmus basierend aufder Maximum-Entropie-Methode notwendig. Mit einem mm-Wellen Interferome-ter konnte die zugehorige THz-Strahlung beobachtet werden. Die beobachtetenEffekte werden mit Modellrechnungen verglichen.

Abstract

An experiment has been carried out at the TESLA Test Facility (TTF) linacto investigate the wake fields generated by picosecond electron bunches in nar-row beam pipes with artificially roughened inner surface. The energy structureimposed on the bunches by the wake fields has been analyzed with a magneticspectrometer. Strong harmonic wake field effects were observed. By means oflongitudinal phase space tomography the wake potentials were studied directly.This required the implementation of a new reconstruction algorithm making useof the maximum entropy method. With a mm-wave interferometer the corre-sponding THz radiation was observed. The observed effects are compared withmodel calculations.

To Tina and Jonas

Contents

1 Introduction 1

2 Wake Fields 32.1 Introduction to wake fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Waveguide coated with a dielectric layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Surface Roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.1 Surface structure with a rectangular shape . . . . . . . . . . . . . . . . . . . . . . . . 92.3.2 Smooth and shallow corrugations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3.3 Time Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.4 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Stepchange in the Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Introduction to Free Electron Lasers 163.1 Pendulum Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Dimensionless FEL Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 The TTF FEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Longitudinal Phase Space Tomography 224.1 Algebraic Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2 Maximum Entropy Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2.1 The Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2.2 Limits of Applicability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2.3 Combining Independent Sources of Information . . . . . . . . . . . . . . . . . . . . . . 27

5 The TESLA Test Facility Linac 295.1 Simulation of the Longitudinal Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6 Experimental Tomography 346.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.2 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

7 The Wake Field Experiment 387.1 Energy profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387.2 Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417.3 Microwave Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

8 Conclusion and Outlook 46

A Derivatives 47

List of Figures

2.1 Phase Velocity in Dielectric Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Impedance of Dielectric Beam Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Pulse Length of Dielectric Wake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Wake Potential for various Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 Normalized Peak Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6 Surface Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.7 Rectangular Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.8 Dohlus Model: Beam Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.9 Stupakov Model: Beam Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.10 Mean Correlation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.11 Surface Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.12 Spoiler Roughness Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.13 Undulator Roughness Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.14 Step in the Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.15 Boundary Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.16 Impedance Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1 FEL Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 FEL Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3 FEL with Energy Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.4 Permitted Energy Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.1 Kaczmarz’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2 ART:Full Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3 ART:Reduced Set of Projection Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.4 ART Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.5 MENT:Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.6 MENT:Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.7 MENT:Bunch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.8 MENT:Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.9 MENT:Independent Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.10 MENT:Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.11 MENT:Combined Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.12 MENT:Projection Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.1 TTF: Schematic Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.2 BC2: Schematic and Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.3 Impedance of Collimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.4 Cut Collimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.5 Bin Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.6 Simulation: BC2 Entrance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.7 Simulation: BC2 Exit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.8 Simulation: BC2 and CSR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.9 Simulation:Collimator and Wake Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6.1 Sensitivity os Screen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

i

6.2 Tomography Maximum Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366.3 Comparison of Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366.4 CTR Autocorrelation and Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.5 Streak Camera Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.6 Wake Experiment: Longitudinal Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.7 Spectrometer Beam Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

7.1 Difference Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.2 Phase Space Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.3 Frequency vs Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407.4 Energy Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407.5 Phase Space after Reference Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417.6 Phase Space after Sandblasted Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427.7 Tomography and Simulation: Reference Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . 427.8 Tomography and Simulation: Sandblasted Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . 427.9 Radiated Wake 4 mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427.10 Atmospheric Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437.11 Radiated Wake, Sandblasted Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437.12 Interferometer: Grooved Test Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437.13 Length of Wake Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447.14 Inductance vs Structure Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447.15 Wake Time Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

ii

Chapter 1

Introduction

One of the fundamental principles of modern sci-ence is the experiment. By designing sophisticatedexperimental setups and analyzing the measureddata the scientist gains understanding of the prin-ciples of nature. The 20th century has experiencedan overwhelming progress in terms of extending themeasurement techniques to step further and furtherinto fundamental processes.

A very successful tool is the scattering measure-ment. It is applicable to problems in all fields ofphysics and many other disciplines. By scatter-ing elementary particles at a target it is possibleto uncover hidden properties and reactions of thetarget material and sometimes the elementary par-ticle itself. Depending on the material under studythe projectiles may be electrons, positrons, protons,neutrons, or photons. Owing to the particle-waveduality of quantum theory scattering is basicallyequivalent to diffraction. The spacial resolution ofthe scatter experiment is given by the wavelengthrespectively energy of the projectile. If the energyof the projectile is large enough the nature of inter-action with the target may change drastically andnew particles be created. In this way the focus ofthe analysis changes from the initial state of thetarget to the final state of the reaction, funding anew field of research, the particle physics. Mod-ern elementary particle physics is mainly concernedwith the investigation of the basic constituents ofmatter and their forces and interactions, and theartificially produced second and third generation ofleptons and quarks is an essential part of experi-mental and theoretical particle physics. Almost allexperimental data are described with high preci-sion by the standard model of the unified electro-magnetic and weak interactions, and the quantumchromodynamics. One essential cornerstone is miss-ing, however, the Higgs particle, which is thoughtto be responsible for the short range of the weakforce and the lepton and quark masses.

To increase the sensitivity and to decrease thestatistical errors of the measurements a large par-ticle flux is required. This is especially true in par-ticle physics where interaction cross section usually

scales with the inverse square of the center of massenergy. Closely related to the development of scat-tering techniques is the development of particle ac-celerators. The development of particle acceleratorsled to the availability of particle beams of very highparticle flux (brightness) and particle energy. Theformer leads to ever increased sensitivity, whereasthe latter leads to improved spatial resolution. Inparallel the development also leads to smaller tem-poral resolution by shortening the beam pulses.

The TESLA (Tera Electronvolt SuperconductingLinear Accelerator) collaboration proposes to builda machine that will be at the forefront in two verydistinct areas of research. First, it will deliver elec-tron and positron beams for e−e+ collision exper-iments at a center of mass energy of 0.5 TeV upto 0.8 TeV at luminosity of 1034cm−2s−1. Sec-ondly, included will be a X-ray free electron laser(FEL) delivering wavelengths around 1 A with apeak brilliance around 1033 photons/(s mrad2 mm2

0.1% bandwidth) and pulse length of 100 fs. In thisway TESLA will carry on a long-lasting traditionat DESY to use the accelerators built for particlephysics for entirely different research fields as well.

At DESY a test accelerator (TESLA Test Facil-ity – TTF) has been built and operated to performR&D for the superconducting acceleration struc-tures and to do an integrated system test. Atthe TTF linac also an UV FEL has been oper-ated to prove the viability of the Self AmplifiedSpontaneous Emission (SASE) mode of free elec-tron lasers. The requirements on the electron beamquality of such a SASE-FEL are very demandingin terms of small transverse and longitudinal emit-tances. Especially the requirements for the longitu-dinal phase space with 1 nC of charge concentratedin less than 1 ps and with small energy spread arethe subject of intense theoretical and experimen-tal studies. The very short but intense currentpulses give rise to strong wake fields and coher-ent sychrotron radiation effects in the accelerator.These radiation effects in turn influence the energydistribution of the electron bunch.

One main goal of this thesis is the experimental

1

investigation of the wake fields excited by the resid-ual surface roughness inside the vacuum chamberof the FEL undulator. The gap of this undulatoris very small (12 mm) leaving space for a vacuumchamber of 9 mm inner diameter only. Calcula-tions show that the rough surface in the order of1 µm present in standard 10 mm pipes will lead tostrong wake fields by the ultrashort bunches. To-gether with the resistive wall wake fields this dom-inates the effects of wake fields on the performanceof the FEL. Because of the importance of the effectand some uncertainty in the model predictions anexperiment has been conducted at the TTF linac tostudy the surface roughness wake field effect.

Considering the stochastic nature of the surfacestructure the model predictions may appear surpris-ing: A resonant excitation at a certain frequency ispredicted, with the frequency determined by the ra-dius of the beam pipe and the depth of the surfacestructures. This harmonic wave could be verifiedexperimentally both from the energy distributionin the bunches as well as by spectral analysis of theemitted radiation. For measuring the energy mod-

ulation of the bunches tomographic methods havebeen applied to reconstruct the longitudinal phasespace. The longitudinal phase space tomographyin a linear accelerator suffers from the limitation ofthe angular range which excludes the use of con-ventional reconstruction algorithms. A maximumentropy method has been utilized to overcome thislimitation.

The second chapter of this thesis will give a in-troduction to wake fields with the focus on differentmodels to describe surface roughness wake fields.In chapter three the fundamentals of FEL physicsare described in the one-dimensional theory rele-vant for the longitudinal phase space. The theoryof phase space tomography is explained in chapterfour. As an introduction to the discussion of exper-imental findings the chapter five gives an overviewof the TTF linac and the evolution of the longitu-dinal phase space. The sixth chapter deals with theresults of tomographic measurements of the longitu-dinal phase space done independently of the wakefield experiment. The seventh chapter is reservedfor a detailed analysis of the data taken in the wakefield experiment.

2

Chapter 2

Wake Fields

A bunch of charged particles moving in an accel-erator carries the Coulomb field of its constituents.In the limit of ultrarelativistic motion, i.e. the par-ticle energies are much higher than the rest ener-gies, the field is concentrated in a disk perpendicu-lar to the trajectory of motion. Depending on thestructure of the environment the bunch self field isperturbed and can be reflected onto the beam axisand interact with the particles in the bunch itselfor with following bunches. The perturbed fields arecalled wake fields. The radiated energy might im-pair the proper functioning of the accelerator sys-tems. Wake fields in accelerating structures can beexpanded in terms of eigenmodes of the cavities andare then referred to as higher order modes. Depend-ing on whether the wakes act on the bunch itself oron the following bunches they are called short rangeor long range. The former ones may degrade thelongitudinal and transversal emittances of individ-ual bunches, the latter can cause collective instabil-ities in the accelerator.

The short range wake fields contain wavelengthsin the order of the bunch length. In the case oflinear colliders (LC) or free electron lasers (FEL)the bunch length is in the range of millimeters andeven below. At the same time the bunches containa large numbers of electrons, e.g. ∼ 1010 in theTESLA Test Facility linac. Strong peak fields haveto be expected.

Devices to analyze the emitted wake field radia-tion have to use the quasi-optical techniques devel-oped for far infrared radiation.

To create long range wake fields time constantsin the order of the bunch spacing are needed. Thebunch separation in the TTF linac is 0.4 to 1 µs.Such long fill and decay times may occur in thesuperconducting cavities. With respect to the longrange wake fields for TESLA the major concern aretherefore the higher order modes of the accelerationcavities.

In general longitudinal wake fields scale inverselywith the distance of the structures from the beam,the transverse wake fields scale with the inversecube. The synchronous mode wake fields discussed

below scale with the inverse square of the radius.Therefore the wakes generated inside the narrowvacuum chambers of the undulator magnets of theFEL are particularly harmful.

2.1 Introduction to wake fields

Consider a point charge q moving in free space at aconstant velocity v. The electric field in the restframe of the charged particle is spherically sym-metric and drops as ∼ 1/r2, with r being the dis-tance from the point charge in cylindrical coordi-nates (r, ϕ, z). In an electron accelerator the beamsare highly relativistic so that the field has to beLorentz transformed to the laboratory frame, yield-ing [21]

E‖(r, t) =q

4πε0

γvt

[r2 + (γvt)2]3/2e‖,

E⊥(r, t) =q

4πε0

γr

[r2 + (γvt)2]3/2e⊥, (2.1)

B⊥(r, t) =1c2

v ×E⊥(r, t), (2.2)

where the relativistic factor γ is defined by

γ =1√

1− β2withβ =

v

c. (2.3)

The unit vector e‖ is chosen to be parallel to thevelocity v of the charge and e⊥ perpendicular toit. Unless specified otherwise, it is assumed in thefollowing that the vacuum chamber is cylinder sym-metric and that the beam is moving on its symme-try axis. The peak value of E⊥ is reached at a timet = 0 when the particle passes the point of mini-mum distance to the observer,

E⊥(r, 0) =q

4πε0

γ

r2e⊥. (2.4)

The time interval in which the amplitude of thetransverse electric field at the radius b exceeds halfthe peak value is approximately given by

4t ≈√

2b

γv. (2.5)

3

For γ 1 the electric field is concentrated in asmall disk with opening angle ∼ 1/γ. In the ultra-relativistic limit γ → ∞ the field reduces to a δ-distribution in the z-direction

Er(r, z, t) =qZ0c

2πrδ(z − ct) Bϕ =

1cEr, (2.6)

with E⊥ = Erer, B⊥ = Bϕeϕ, and Z0 =√

µ0/ε0

the impedance of the vacuum. Because of its ap-pearance the electromagnetic field eq. 2.6 is some-times referred to as “pan-cake” term.

Here I consider only wake fields with azimuthalsymmetry. Then the Helmholtz equations for theelectric field in vacuum and non conducting mate-rials can be written as

µε

c2∂2

t Ez =1r∂r(r∂rEz) + ∂2

zEz (2.7)

µε

c2∂2

t Er = ∂r

(1r∂r(rEr)

)+ ∂2

zEr, (2.8)

with ∂x = ∂∂x and µ, ε the permeability and the

permittivity respectively of the material. For mostbeam pipe materials it is justified to assume µ = 1,hence this factor will be omitted. The permittivityhas to be accounted for in a beam pipe covered witha thin dielectric layer.

Often it is advantageous to use the Fourier trans-form of the fields

Ez/r(r, z, t) (2.9)

=12π

∞∫−∞

∞∫−∞

Ez/r(r, kz, ω)ei(kzz−ωt)dωdkz.

Then the wave equation simplifies to

1r∂r(r∂rEz) +

(k2ε− k2

z

)Ez = 0(2.10)

∂r

(1r∂r(rEr)

)+(k2ε− k2

z

)Er = 0,

with k = ω/c. The pan-cake term transforms into

Eδr =

qZ0c

2πr. (2.11)

When there is no charge the electric field fulfills theMaxwell equation

∇ ·E =1r∂r(rEr) + ikzEz = 0. (2.12)

This equation relates Ez and Er. The magneticfield can be derived from

− µ0∂H

∂t= rotE

ikZ0Hϕ = ikzEr − ∂rEz. (2.13)

For the analysis of the influence of the wake fields onthe beam the wake potential is calculated. It repre-sents the effective voltage seen by a particle movingwithin the bunch. The coordinate ζ is the relativelongitudinal coordinate moving with the bunch

W‖(ζ) =1q

∫dzEz(ζ, z), ζ = z − ct, (2.14)

The wake potential can be calculated in three steps.First the impedance of the accelerator is calculatedfrom the electric field in the Fourier space. In theultra relativistic limit it is

Z‖(k) =1qc

∫dzEz(r = 0, kz, k)ei(kz−k)z. (2.15)

From the impedance it is possible to calculate thewake function. It is the Green’s function, i.e. thewake potential induced by a δ-like charge distribu-tion.

W δ‖ (ζ) =

c

2π

∞∫−∞

dkZ‖(k)eikζ (2.16)

The wake potential then is the convolution of thewake function with the line charge density ρ(ξ) ofthe bunch

W‖(ζ) =

∞∫−∞

dξW δ‖ (ζ − ξ)ρ(ξ). (2.17)

Certain wake fields are generated continuouslyalong the beam pipe. The wake arising from adielectric layer discussed in the next section is anexample. For quasi infinite beam pipes and con-tinuous wake generation the impedance is usuallyquoted per unit length and only calculated for thefields synchronous with the bunch

Z‖(k) =Ez

qc. (2.18)

In this case it may be advantageous to use the sur-face impedance to derive the longitudinal field

Z =Ez

Hϕ. (2.19)

The contribution of a finite piece of the beam pipeto the total impedance can be calculated via

Z‖(k, L) =1qc

L∫0

dzEz(r = 0, kz, k)ei(kz−k)z,(2.20)

with L the length of the considered piece of beampipe.

4

100 200 300 400 500 6000.85

0.9

0.95

1

1.05

1.1

1.15

frequency / GHz

velo

city

v/c

Dispersion Relation for circular waveguide with dielectric layer

phase velocity

group velocity

Figure 2.1: Phase velocity of a waveguide mode in a beampipe with a dielectric surface layer. The phase velocity ω/kz

and the group velocity dω/dkz are shown. The pipe has aradius of 5 mm and the layer has a thickness of 30 µm.

2.2 Waveguide coated with adielectric layer

A metallic beam pipe acts as a waveguide. The elec-tromagnetic fields inside this waveguide can be ex-panded into modes, each individually fulfilling thewave equation 2.10 with appropriate boundary con-ditions at the metallic walls. The solutions for thisequation can be written using Bessel functions ofthe first kind [8]

Ez = EJ0(krr), (2.21)

Er = − ikz

krEJ1(krr), (2.22)

Hϕ =k

kzZ0Er, (2.23)

with kr =√

k2 − k2z . Note that this solution is even

valid for k2r < 0. The metallic boundary is assumed

to be perfectly conducting. Hence the longitudi-nal electric field vanishes at the surface. This canonly be accomplished if kr · b equals a root of J0,b being the pipe radius. This yields k2

r > 0 andhence kz < k. The phase velocity vph = c · k/kz

is larger than the speed of light for all modes in aperfectly conducting waveguide. The particles arealways moving with a speed v < c. Even in theultra-relativistic limit assuming v = c the velocityof any frequency component of the pan-cake term issmaller than the phase velocity of the correspond-ing waveguide mode. Thus a coupling to the modeis not possible in the time average.

In a waveguide with dielectric coating the situ-ation is fundamentally different. Assume a beampipe with a thin dielectric layer on its inner sur-face, with a dielectric constant ε and a thicknessδ b. Because the thickness of the dielectric layeris much smaller than the radius of the beam pipe,the variation of r inside the layer is negligible and

equation 2.10 simplifies to(∂2

y + ∂2z + εk2

)Ez/r = 0, (2.24)

y ≡ b + δ − r.

This approximation is equivalent to assuming thepipe surface being locally flat. The new variable yhas the origin at the metallic wall and is pointinginwards. Inside the dielectric layer the solution isfound to be

Ediez = Edie sin(κry), (2.25)

Edier = − ikz

κrEdie cos(κry), (2.26)

with κr =√

k2ε− k2z . The fields have to match at

the boundary between the dielectric and the vac-uum

Er(b) = εEdier (δ) Ez(b) = Edie

z (δ). (2.27)

Inserting 2.23 and 2.25 into 2.27 and dividing thetwo equations yields the surface impedance

Z(k, kz) =Ez

Hϕ

(2.28)

= Z0kzEz

kEr

= iZ0κr

εktan(κrδ).

Without a source current inside the pipe the surfaceimpedance has to match the ratio Ez/Hφ calculatedfrom (2.21) and (2.23)

ZZ0

= ikrJ0(krb)kJ1(krb)

. (2.29)

From this equation it is possible to derive the dis-persion relation for the dielectrically coated waveg-uide.

Trying to calculate kz = kz(k) one finds that(2.29) is a transcendental equation. The dispersioncan be calculated in closed form when kr is taken asthe independent variable. Additionally one has toassume tanκrδ ≈ κrδ which is justified due to thesmall layer thickness, κrδ 1. Then the equations2.28 and 2.29 read

κ2rδ

εk=

krJ0(krb)kJ1(krb)

(2.30)

=2J0(krb)

bk(J0(krb) + J2(krb)).

Here the identity J1(x)/x = 1/2 (J0(x) + J2(x))is used to continue the function steadily aroundk2

r = 0. The factor k on both sides cancels andκ2

r = k2(ε− 1) + k2r . Then the mode frequency ωm

is given as followsωm

c= km(kr) (2.31)

=

√2ε

(ε− 1)bδ

(J0(krb)

J2(krb) + J0(krb)− bδ

2εk2

r

),

5

kz(km, kr) =√

k2m − k2

r .

vph =km

kz,

An example for the phase velocity is shown in figure2.1. The phase velocity equals the speed of lightwhen kz = k ⇔ kr = 0. A higher phase veloc-ity is found for k2

r > 0, a smaller phase velocity isfound for k2

r < 0. Obviously the variable kr switchesfrom real to imaginary values. For the calculationof derivatives it is therefore necessary to use thevariable ξ = k2

r instead.The group velocity can be derived from

vgr

c=

dkm

dkz. (2.32)

The differentials can be calculated by

dkm =dkm

dkr

dkr

dξdξ (2.33)

dkz =(

∂kz

∂kr+

∂kz

∂km

dkm

dkr

)dkr

dξdξ

Dividing the differentials yields

vgr

c=

dkm

dkr

∂kz

∂kr+ ∂kz

∂km

dkm

dkr

, (2.34)

where the different terms are given as follows (seeappendix A).

dkm

dkr=

εkrb

2km(ε− 1)δ

(J2(krb)J0(krb)

J21(krb)

− 2δ

bε− 1)

∂kz

∂kr= −kr

kz(2.35)

∂kz

∂km=

km

kz

The terms dξdkr

cancel and have been dropped. Atk2

r = 0 there is a solvable discontinuity in this for-mula. The equation 2.47 shows the series expansionaround this point.

With a dielectric surface layer the electromag-netic waves are slowed down so that there is acertain frequency at which the phase velocity ofthe wave equals the speed of light. At this fre-quency there is the possibility for a continuous en-ergy transfer from a ultra-relativistic beam to thewaveguide mode. In the following this will be calledthe synchronous mode wake field.

The frequency of the synchronous mode can befound by calculating k for kr = 0. The right handside of equation 2.29 becomes b/2 in the limit kr →0. Using again the approximation tan(x) ≈ x, onefinds

kres =

√2ε

(ε− 1)bδ. (2.36)

This the wavenumber of the synchronous mode.To obtain the excitation strength and pulse formof the mode one has to calculate the longitudinalimpedance Z‖(kz,4k).

100 200 300 400 500 6000.95

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

δ=30µmfres

=261 GHz

frequency [GHz]v ph

/c

Figure 2.2: Absolute value of the impedance of a dielectri-cally coated beam pipe in the frequency-phase velocity plane(eq 2.40). The value of the impedance is coded into the grayscale of the picture, the coding is logarithmic to accountfor the diverging behaviour when matching the waveguidemodes. The bright lines in the map correspond to the diver-gence of the impedance at a certain waveguide mode. Thephase velocity of the lowest mode is plotted as black curveaccording to eq. 2.31. Note that the impedance was calcu-lated exactly whereas eq. 2.31 is based on an approximationof the tangent. The two calculations deviate slightly at veryhigh frequencies.

The longitudinal impedance is found by evaluat-ing the quotient Ez(0, kz, k)/q. To do so the surfaceimpedance can be utilized

Ez(b, kz, k) = ZHϕ,total (2.37)

= Z ·( qc

2πb+ Hϕ(b, kz, k)

).

Inserting the general solution of the wave equa-tion in vacuum, eq. 2.21 the longitudinal beamimpedance (per unit length) is found

qcZ0

2πb= E

(ik

krJ1(krb) +

Z0

ZJ0(krb)

)(2.38)

⇔ Z‖ =Ez

qc

=Z0

2πb· 1

ikkr

J1(krb) + Z0Z J0(krb)

.(2.39)

Inserting the surface impedance from equation 2.28yields

Z‖ =Z0

2πb· 1

ikkr

J1(krb)− ikεκr

J0(krb) cot(κrδ).(2.40)

Figure 2.2 shows the impedance map for a beampipe of 4 mm radius and 30 µm dielectric surface

6

layer. In the case of an infinitely long beam pipeonly the impedance along the line vph = c has tobe taken into account. This is equivalent to takingkr = 0

Z‖ = − 12πb

· Z1 + ikbZ

2Z0

(2.41)

⇔ Z‖ = iZ0

πb2· k

k2res − k2

. (2.42)

The longitudinal wake function can be calculatedvia Fourier transform

wδ‖(ζ) =

c

2π

∞∫−∞

dkZ‖(k)e−ikζ

= −Z0c

πb2cos(kresζ)Θ(ζ), (2.43)

with ζ = z − ct the relative coordinate behind thesource charge and Θ the Heaviside step function.Note that this wake function contains a cosine termeven though the impedance is purely imaginary.

To evaluate the wake function for a finite beampipe one has to take into account the impedance inthe whole (kz, k)-plane. To ease the calculation ofthe residua one can try to write the impedance asa product of polynomial quotients

Z‖ =Z0

πb2· ik

krεJ0(krb)J1(krb)

− (k2ε− k2z)δ

· krb

2J1(krb)· (k2ε− k2

z)δ

k2

=Z0

πb2· ik

krεJ0(krb)J1(krb)

− k2rδ − k2(ε− 1)δ

· krb

2J1(krb)· (k2ε− k2

z)δ

k2

=Z0

πb2· ik

ε(ε−1)bδ

[krb

J0(krb)J1(krb)

− bδε

k2r

]− k2

· krb

2J1(krb)· k2ε− k2

z

k2ε− k2

Using equation 2.31

Z‖ = iZ0

πb2· k

k2m − k2

· 1

J0(krb) + J2(krb)· k2ε− k2

z

k2ε− k2.

(2.44)

To evaluate the Fourier integral it is preferable towrite k as function of kz, which is achieved by us-ing the Taylor series expansion up to the 3rd order.First one finds the series expansion of k and kz fromequation 2.31 with ξ = k2

r (See appendix A)

km

kres= 1− 1

2amξ −

(a2

m

8+

b4

384

)ξ2 (2.45)

−(

a3m

16+

amb4

768+

b6

9216

)ξ3 + · · ·

kz

kres= 1− 1

2arξ −

(a2

r

8+

b4

384

)ξ2

−(

a3r

16+

arb4

768+

b6

9216

)ξ3 + · · ·

with

am =b2

8+

bδ

2εar =

b2

8+

bδ

2(2.46)

The Taylor series then reads

km = kres +am

ar(kz − kres) (2.47)

+b4 − 48 amar

48 a3rk2

res

· (kz − kres)2

2kres

+

(amarb4 − 48a2

ma2r

16 a5rk2

res

−48a2

rb4 − 4arb6 + b8

768 a5rk2

res

)(kz − kres)3

6k2res

For small values of kz the influence of the dielec-tric vanishes. Then a better approximation is givenby the well known formula km =

√k2

z + α21/b2, with

α1 being the first zero of the Bessel function J0. Thecoefficient of the first order is the group velocityclose to the synchronous mode frequency

vg = am/ar. (2.48)

Given these functions it is possible to construct theFourier transform of equation 2.44 in two dimen-sions

wδ‖(z, t) =

Z0c

πb2

∞∫−∞

∞∫−∞

dk√2π

dkz√2π

ik

k2m − k2

(2.49)

· k2ε− k2z

(ε− 1)kkz· krb

2J1(krb)· ei(kzz−kct)

The integration along k can be performed analyti-cally using the residue theorem, if the zeros of theBessel function in the third factor are neglected.They belong to distinct modes of higher order (seethe bright lines in figure 2.2) which can be analyzedseparately. Due to their high phase velocity, a cou-pling to the beam is not to be expected. Then,

wδ‖(ζ, l) = (2.50)

Z0c

πb2

∞∫0

dkzk2

mε− k2z

(ε− 1)kmkz· krb

2J1(krb)· sin(∆kl)

∆k· cos(kzζ),

where l is the position of the bunch in the beampipe (0 ≤ l ≤ L, ∆k = kz−km). The sine term orig-inates from the finite duration of excitation. The in-tegrand gives the wavelength spectrum of the radia-tion pulse depending on the position l of the bunch.The pulse can then be calculated numerically from

7

0 0.5 1 1.5 2 2.5 3 3.5−20

−10

0

10

20

position ζ [mm]

ampl

itude

[arb

. uni

ts]

0 0.5 1 1.5 2 2.5 3 3.5 4−0.2

−0.1

0

0.1

wavenumber 1/λ [mm−1]

ampl

itude

[arb

. uni

ts]

0 5 10 15

-15

-10

-5

0

5

10

15

position ζ [mm]

ampl

itude

[arb

. uni

ts]

Figure 2.3: Spectrum and wake field pulse of a dielectric layer wake. The spectrum has been calculated with eq. 2.50, thepulse via fast Fourier transform (FFT) from the spectrum. The left hand picture shows the spectrum in the upper part and thepulse in the lower part. W δ

‖ and wδ‖ are shown. The momentary pulse wδ

‖ is approximately rectangular. The integrated wake

function W δ‖ then drops linearly from the source charge to a distance vgL behind the source. The right hand picture shows the

wake calculated after 50 cm, 1 m, and 2 m of beam pipe

the spectrum. The overall effect is obtained by in-tegration along the path of the beam,

W δ‖ (ζ, L) =

Z0c

πb2

L∫0

dl wδ‖(ζ, l). (2.51)

The only quantity in the equation 2.50 that dependson l is the term sin(∆kl)/∆k which is easily inte-grated. Hence the overall wake function is

W δ‖ (ζ, L) =

Z0c

πb2

∞∫0

dkzk2

mε− k2z

(ε− 1)kmkz· krb

2J1(krb)

·1− cos(∆kL)

∆2k

· cos(kzζ). (2.52)

The momentary wake field pulse described by equa-tion 2.50 has a roughly rectangular shape. The in-tegrated wake function then drops linearly from thesource charge to a distance vgL behind the source(see figure 2.3 left).

For a given longitudinal bunch profile the wakepotential is found by convolution of the wake func-tion and the charge distribution. In figure 2.4 thisis shown for a gaussian charge distribution and sev-eral different ratios of the wavelength and the σ ofthe Gauss function. The resulting corrections to theaverage and peak energy loss of the electrons in thebunch are shown in figure 2.5.

2.3 Surface Roughness

In the previous section it has been shown that wakessynchronous to the beam are excited if the phase ve-locity of the waveguide modes is slowed down to thespeed of the beam. Such a wave can exist if there is

−4 −2 0 2 4 6 8 10−200

−150

−100

−50

0

50

100

150

200

z/σ

W|| [a

rb. u

nits

]

λ/σ=2.6

λ/σ=4.5

λ/σ=7.7

λ/σ=13.3

λ/σ=21.5 λ/σ=37.5λ/σ=64.0

−4 −2 0 2 4−15

−10

−5

0

5

10

15

20

z/σ

charge

λ/σ=2.6

1.0

Figure 2.4: Wake potentials for different wavelengths λ ofthe synchronous mode. The inset shows a zoom into thearea around the bunch and smaller amplitudes. The dot-ted plot gives the charge distribution, which is a gaussian∼ exp(−z2/(2σ2)).

100

101

102

10−4

10−3

10−2

10−1

100

λ/σ

ener

gy lo

ss [n

orm

.]

peak

average

Figure 2.5: Maximum and average energy loss as function ofthe ratio of the wavelength λ of the synchronous mode andthe rms spread σ of the bunch charge distribution. At λ ≈ σthere is the transition to a differentiating wake potential.The final value is obtained by multiplying with qZ0c/(πb2).

8

a non-vanishing surface impedance, i.e. a longitu-dinal electric field at the surface is present. Surfaceroughness at the boundary surface will also producea longitudinal electric field. The consequences arediscussed in the following.

0 1 2 3 4 5 6 7 8−30

−25

−20

−15

−10

−5

0

5

10

15

20

longitudinal position [mm]

surf

ace

prof

ile [µ

m]

0 0.5 1 1.5 2

x 105

0

1

2

3

4

5

6

7x 10

−16

wavenumber 2π/λ [m−1]

pow

er s

pect

rum

[m3 ]

δrms

=8.7µm

Figure 2.6: Surface profile of a sandblasted beam pipe. Theright picture shows the corresponding spectrum S(k). Thenormalization is such that

∫Sdk = δ2

rms, the rms height ofthe surface structures.

The figure 2.6 shows the surface profile of a beampipe used in the wake field experiment.

2.3.1 Surface structure with a rect-angular shape

A first attempt to model the surface roughness maybe by means of periodic rectangular grooves. Sincethe height of the surface structures is much smallerthan the radius of the beam pipe the surface ismodelled as a plane. The electric and magneticfield components, however, are still labelled withthe subscripts r, z, ϕ. The fields of the mode at thesurface are written as

Ez = ZH Hϕ = H Er = Z0H. (2.53)

Let the periodicity of the gaps be d, g the widthof the gaps, and δ the depth (see figure (2.7). Thefields inside the gaps can be expanded into eigen-modes

E(n)z (y, z) = an sin(

√k2 − α2

ny) cos(αnz) (2.54)

E(n)r (y, z) = − αnan√

k2 − α2n

cos(√

k2 − α2ny) sin(αnz)

H(n)ϕ (y, z) =

ikan

Z0

√k2 − α2

n

cos(√

k2 − α2ny) cos(αnz)

αn =nπ

g.

-g-d

6?δ

Symmetry Axis

6

b

Figure 2.7: Sketch of the rectangular surface structure. Thestructures are axially symmetric.

Assuming that the wavelength of the mode insidethe beam pipe is much larger than the gap width,one finds that the longitudinal electrical field is theaverage of the field in the gaps and that it is suffi-cient to calculate the lowest order mode inside thegap

E = ZH =g

da0 sin(kδ). (2.55)

Under this assumption it is clear that no periodicityis required any more. The coefficient a0 can becalculated from the boundary conditions at the gapentrance

H(0)ϕ = Hϕ, (2.56)

E(0)z = iZ0H

(0)ϕ (2.57)

⇔ Zrec = iZ0g

dtan(kδ) ≈ iZ0

g

dkδ. (2.58)

Comparison with eq. 2.29 reveals some similaritywith the surface impedance due to the dielectriclayer when setting kz ≈ k. The effective dielectricconstant is

εeff =d

d− g. (2.59)

Hence wakes induced by this kind of surface rough-ness are equivalent to the wakes due to a thin di-electric layer. On page 76 of ref [52] the result ofthe above calculation has been compared to numer-ical calculations. Good agreement has been foundfor symmetric gaps g ≈ d/2. Deviations are foundwhen moving away from this situation, higher ordermodes have to be taken into account in this case.

Detailed numerical calculations have been per-formed at the Technical University Darmstadt tostudy the surface roughness wake fields for differentgeometrical structures [33, 52, 34]. The layer thick-ness δ is calculated from the rms height of the sur-face structure. In many cases a permittivity ε ≈ 2was found.

2.3.2 Smooth and shallow corruga-tions

In [12] Dohlus derives the surface impedance dueto random corrugations. An analytical approachis developed using different approximations for theboundary conditions. Axial symmetry and longitu-dinal periodicity is assumed but no restrictions onthe period length Λ. The boundary condition at thesurface of the beam pipe is given by

∇Hϕ · ~n + (iωε0Zb + (~er · ~n)/b) Hϕ = 0, (2.60)

with ~n the normal to the surface and

Zb(ω) =

√iωZ0

cσ=

1 + i

σδs, (2.61)

9

σ the conductivity and δs the skin depth. The elec-tromagnetic field is expanded into eigenmodes upto order N

Hϕ = Hδϕ +

∑n

CnHϕn (2.62)

Er = Eδr +

∑n

CnErn (2.63)

Ez =∑

n

CnEzn, (2.64)

with n = −N . . .N , kzn = k + nk1, k1 = 2π/Λ,

Hδϕ = Hδ

ϕeikz Hϕn = − ik

Z0krnJ1(krnr)eikznz (2.65)

Eδr = Eδ

reikz Ern = − ikzn

krnJ1(krnr)eikznz (2.66)

Ezn = J0(krnr)eikznz (2.67)

and krn =√

k2 − k2zn. In this notation the beam

impedance is

Zbeam =C0

qc. (2.68)

The left hand side of equation 2.60 can be evaluatedfor each eigenmode

hδ(z) =

∇Hδ

ϕ · ~n +

(ik

Zb

Z0+

~er · ~nR

)Hδ

ϕ

R=b(z)

,(2.69)

hn(z) =

∇Hϕn · ~n +

(ik

Zb

Z0+

~er · ~nR

)Hϕn

R=b(z)

.(2.70)

The expansion coefficients Cn have to be chosensuch that the boundary condition is fulfilled. Ingeneral there will be a deviation d which has to beminimized

d(z) := hδ(z) +∑

n

Cnhn(z) → 0, (2.71)

There are several ways to approximate this prob-lem which is continuous in z. The method chosenhere is to fulfill the condition (2.71) exactly for theFourier coefficients Fmd(z) with m = −N . . .N ,and

Fmf =1Λ

∫ Λ

0

f(z)e−imk1zdz. (2.72)

This yields the matrix equation

Mc + v = 0, (2.73)

with

(M)m,n = Fmhn, (c)n = Cn

(v)m = Fmhδ, (2.74)

where n = n + N + 1 so that the numbering of thematrix elements starts at 1. It is advantageous tocalculate the effective surface impedance first

Z(ω) =

⟨Ez(b, z)e−ikz

⟩z

〈Hϕ(b, z)e−ikz〉z. (2.75)

This is accomplished by calculating

(M)m,n =

(M)m,n + ikπb2

qcZ0(v)m if n = 0,

(M)m,n otherwise(2.76)

c = −2πbM−1

v, (2.77)

and then

Zω = − C0

q, Cn = (c)n. (2.78)

The terms hδ(z) and hn(z) in the boundary condi-tion can be linearized with respect to the surfaceprofile δr(z)

hδ(z) =qc

2πb

(−ikz,0δ

′r + (1− δr/b)ik

Zb

Z0

)eikz,0z (2.79)

hn(z) = −[J0(krnb)

(1 + ikδr

Zb

Z0

)(2.80)

+J1(krnb)

(−i

kzn

krnδ′r − krnδr + ik

Zb

Z0

1− δr/b

krn

)]eikznz.

This approximation is valid if δr b, δ′r 1, and| krnδr | 1. A first order approximation for the in-verse matrix M

−1is applied. This then yields a sec-

ond order approximation for the surface impedance

Z(k) = Zb + iZ0k

N∑n=−N

An

(nk1 − i

1

b

Zb

Z0

)|Fnδr|2,

An =

[iJ′1(kr,nb) Zb

Z0+ J1(kr,nb)nk1/kr,n

]J0(kr,nb) + ikJ1(kr,nb)/kr,n

ZbZ0

(2.81)

The influence of the surface resistivity can be fur-ther approximated

Z(k) = Zb + iωZ0

c

N∑n=−N

J1(kr,nb)(nk1)2

kr,nJ0(kr,nb)| Fnδr |2

+O(‖δr‖2Zb), (2.82)

In [12] this last approximation has only been de-rived for a sinusodial surface profile. Due to thesimilarity to the case of the Fourier series this maybe generalized. With a less rigorous treatmentStupakov [48] finds an approximation for the sur-face impedance without using axial symmetry

Z = iωL (2.83)

L =Z0

c

∞∫−∞

∞∫−∞

dκzdκxR(κz, κx)κ2

z

κ, (2.84)

with z parallel to the direction of the beam and xperpendicular. R(κz, κx) is the Fourier transformof the autocorrelation of the surface profile func-tion δr(z, x). The integral L is regarded as surfaceinductance, similar to the second term in equation2.82.

10

400 500 600 700 800 900 1000 1100 1200 1300 1400

0.5

1

1.5

2

2.5

frequency [GHz]

para

llel i

mpe

danc

e |Z

| [ar

b. u

nits

]

400 500 600 700 800 900 1000 1100 1200 1300 1400

100

200

300

400

500

600

frequency [GHz]

inte

gral

of |

Z| [

arb.

uni

ts]

Figure 2.8: Beam impedance due to surface roughness. Thebeam impedance is calculated using the equation 2.82. Theparameters of the pipe are radius b = 5 mm and a surfaceprofile as depicted in figure 2.6. The visual impression ofthe spectrum is dominated by the many narrow resonancesdue to the resonances of the surface impedance. Calculatingthe average contribution to the power integral (right picture)results in a single resonance at 620 GHz.

400 450 500 550 600 650 700 750 8000

5

10

15

20

25

frequency [GHz]

|Z||| [

kΩ/m

]

(a) axisymmetric(b) isotropic

Figure 2.9: Beam impedance due to surface roughness. Thebeam impedance is calculated using the equation 2.83. Theparameters of the pipe are radius b = 5 mm and a surfaceprofile as depicted in figure 2.11. The beam impedance plot-ted with a solid line results if one assumes that the rough-ness is axially symmetric. The dashed-dotted line shows theimpedance assuming that the roughness is isotropic on thesurface.

Setting R = R(κz)δ(κx) resulting in an axisym-metric surface, the two models can be compared.The first and most striking difference is the depen-dence on the pipe radius. In this sense the inter-pretation of the surface impedance as a property ofthe surface only is no longer correct. This differencebecomes important for frequencies ω > 2k1c, whenthere are modes above cutoff, which are no longerlocalized on the surface. In these cases there are res-onances in the surface impedance and due to theiroscillatory behaviour also in the beam impedance.

Note: One of the major advantages of using thesurface impedances for analyzing the wake fields isthe fact that the surface impedance is finite wherethere is a singularity in the beam impedance andvice versa (see eq. 2.39). Therefore the numericalstudy of the resonances is much more precise in thisway.

The figure 2.8 shows the beam impedance due tothe surface roughness measured in a pipe used forthe wake field experiment. The impedance has been

calculated using 2.82. For low frequencies k0 k1

the term J1(krb)/(krJ0(krb)) ≈ 1/kr ≈ 1/k1. Inthis case the equation 2.82 and 2.83 deliver simi-lar results. The additional resonances lead to ad-ditional energy losses. Therefore the resonance infigure 2.8 is approximately 4 times wider than itwould be according to surface resistance and induc-tivity alone.

The figure 2.9 shows the beam impedance cal-culated using equation 2.83. The impedancehas been calculated assuming axial symmetryand for an isotropic roughness distribution onthe surface. For isotropic surface structures theautocorrelation function only depends on ξ =√

(x− x′)2 + (z − z′)2

K(ξ) = < δr(x)δr(x + ξ) > . (2.85)

The Fourier transform of K

R(κz, κx) =1

4π2

∞∫−∞

∞∫−∞

dzdxK(ξ)e−i(κzz+κxx)(2.86)

turns into a Hankel transform

R(κ) =12π

∞∫0

ξdξK(ξ)J0(κξ). (2.87)

The figures 2.10 resp. 2.11 show the functions Kand R for the sandblasted beam pipes. The auto-correlation function (figure 2.10) has been obtainedaveraging 18 sets of 1-dimensional data only. Thisleads to rather poor results for large offsets, wherethe mean values are smaller than the rms. Thesehave been suppressed in the calculation by multi-plying a gaussian function with a suitable σ. Infigure 2.11 the spectra R have been plotted for com-parison. They are given such that one directly canobtain

L =Z0

c

∞∫0

κdκR(κ). (2.88)

In the isotropic case this requires a multiplicationwith κ. The integrals for the two cases differ by afactor 0.7450, for a gaussian distribution one findsa factor π/4 ≈ 0.785. This results in a surface in-ductance according to Stupakov et. al. [48]

Lsymm ≈ 3.6 pH Liso ≈ 2.7 pH. (2.89)

Taking the rms value of the surface profileδrms ≈ 10 µm as the layer thickness, this results ineffective dielectric constants

εsymm ≈ 1.4 εiso ≈ 1.27 . (2.90)

11

0 0.5 1 1.5 2

0

20

40

60

80

100

autocorrelation (raw/filtered)

RMS of 18 samples

offset [mm]

<δ r(x

)δr(x

−x’

)> [µ

m2 ]

Figure 2.10: Autocorrelation function of the surface rough-ness averaged over 18 sample measurements. A Gauss func-tion has been multiplied to the averaged function in orderto suppress random noise at large offsets. The sigma of thegaussian is chosen such that it becomes efficient at the pointwhen the signal is as large as the RMS value of the fluctua-tions.

0 5 10 15

x 104

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

−6

wavenumber k [m−1]

spec

tral

func

tion

R [m

m3 ]

(a) axisymmetric(b) isotropic

Figure 2.11: Spectral functions of the surface roughness.The spectral functions have been calculated (a) by a Fouriertransform (axial symmetric structures) and (b) using theHankel transform (isotropic structures).

Assuming that the same spectrum can be used inthe model 2.82 as well, the curve in figure 2.8 wascalculated. The corresponding inductances accord-ing to the Dohlus model are

Lsymm ≈ 3.6 pH Liso ≈ 3.0 pH, (2.91)

and the corresponding dielectric constants are

εsymm ≈ 1.4 εiso ≈ 1.31 . (2.92)

The Dohlus model (eq. 2.82) fits better with theexperimental data than the Stupakov model (eq.2.83). See chapter 7 for reference. In the fig-ures 2.12 and 2.13 the impedances of the surfaceroughness wake in the collimator and undulator areshown. In the undulator the surface roughness wakeadds little to the wake field due to surface resistiv-ity.

2.3.3 Time Constants

The dielectric layer model can be applied to calcu-late the group velocity of the roughness wake and

0 0.5 1 1.5 2 2.5 3 3.5

x 105

0

1

2

3

4

5

6

7

8

9x 10

−7

wavenumber k [m−1]

spec

tral

func

tion

R [m

m3 ]

(a) axisymmetric(b) isotropic

600 700 800 900 1000 1100 1200 13000

0.5

1

1.5

2

2.5

3

frequency [GHz]

impe

danc

e |Z

| [ar

b. u

nits

]

Figure 2.12: Spectral functions of the surface roughness andthe corresponding impedance of the collimator. The spectralfunctions have been calculated (a) by a Fourier transformand (b) using the Hankel transform. The beam impedanceof the spoiler has been calculated assuming a constant radiusof 3 mm.

0 5 10 15

x 104

0

2

4

6

8

10

12

14

16

18

20x 10

−9

wavenumber k [m−1]

spec

tral

func

tion

R [m

m3 ]

(a) axisymmetric(b) isotropic

0 5 10 15 200

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

frequency [THz]

impe

danc

e |Z

| [ar

b. u

nits

]

(a) Dohlus model(b) inductance and resistance(c) resistance only

Figure 2.13: Spectral functions of the surface roughness andcorresponding impedance of the undulator vacuum cham-ber. The spectral functions have been calculated (a) by aFourier transform and (b) using the Hankel transform. Thebeam impedance of the undulator has been calculated forthe radius of 4.5 mm. The surface roughness wake is a smallcorrection to the resistive wall wake.

12

hence its pulse length. The surface inductance canbe translated into an effective dielectric constant

ε =1

1− Lc/(δZ0). (2.93)

The pulse length Lp = L(c−vg)/c can be calculatedby

Lp = L

(1− am

ar

)= L

(1− b2/8 + bδ/(2ε)

b2/8 + bδ/2

)= L

ε− 1ε

· δ/2b/8 + δ/2

≈ Lε− 1

ε· 4δ

b(2.94)

= 4LLc

Z0b. (2.95)

Losses in the system are accounted for by multipli-cation with the corresponding exponentially decay-ing function with a characteristic decay length ζ0.For the wake function this may result in a changeof the decay constant

W δ‖ =

Z0c

πb2cos(kresζ) · (1− ζ/Lp) · exp(−ζ/ζ0),

≈ Z0c

πb2cos(kresζ) · exp(−ζ/ζ1), (2.96)

ζ1 =

(1

Lp+

1

ζ0

)−1

, 0 ≤ ζ ≤ Lp. (2.97)

Taking into account only the surface resistance be-low 1 THz the damping leads to a small correctionto the lossless pulse length. The large number ofresonances in the Dohlus model causes additionallosses for the harmonic wake. This results in abroadening of the average resonance which can betranslated into a shorter damping constant. Thisshorter decay constant dominates (see figure 7.15).

2.3.4 Further Reading

In recent years the topic of surface roughness wakefields has received considerable attention in many –mainly theoretical – publications and contributionsto conferences. Besides the publications alreadycited in the previous sections [33, 52, 34, 12, 48]other papers are mentioned at this point. The treat-ment in [48] is based on [47], but it should be notedthat in the latter the existence of a synchronousmode is denied. Closely related are the publications[4, 6, 7, 49, 50]. In connection with the dielectriclayer model [32, 5] should be mentioned. In [38]the dielectric layer model is tested for its physicalmeaning. The report [13] applies several models onthe case of the TESLA X-FEL.

6bs

6bl

B -P, R

P+

R=

0

Figure 2.14: Jump in the radius of a beam pipe.

In [1] some doubt is cast on the concept of abeam impedance per unit length. The calculationspresented there result in a slowly oscillating energytransfer between beam and radiation field. The cen-ter frequencies are predicted to be linearly growingwith the beam energy

kn = ±αnγ

b, (2.98)

with αn the zeros of the Bessel function J0 and b theradius of the beam pipe. In the wake field exper-iment with beam energy γ ≈ 460 and pipe radiusb = 5 mm the lowest frequency would have to beexpected at 10 THz.

In [2] the surface roughness is treated as beingregular with a rectangular shape resulting in a res-onance at

kres =αn√bδ

, (2.99)

independent from the roughness structure in con-trast to the treatment in section 2.3.1. In [27] thechange of phase velocity of the waveguide modes iscalculated depending on a statistical surface rough-ness. The conclusion drawn is that no resonantmode exists which is in contradiction to the exper-imental results described in chapter 7.

2.4 Stepchange in the CrossSection

When a sudden change of the cross section of thebeam pipe occurs, wake field radiation is to be ex-pected. In the wake field experiment jumps occurat the entrance and exit of the test pipes. At areduction of the pipe radius (step in) the pan-cakefield of the bunch is reflected at the aperture. At anexpansion of the radius the pan-cake field has to berecreated. To fulfill the boundary conditions a ra-diation field is needed. In the first case no energy islost, in the latter case twice the field energy of theself-field will be extracted from the beam. Theseeffects have been treated in [23] and [24].

The radiated fields in the beam pipe are

Ez =∑

n

EnJ0

(νn

r

b

)(2.100)

Er = isign(k)∑

n

En

√k2b2/ν2

n − 1 J1

(νn

r

b

)(2.101)

13

0 2 4 6 8−4

−2

0

2

4

6

0 2 4 6 8−0.5

0

0.5

1.0

1.5

0 2 4 6 8−1

−0.5

0

0.5

1

1.5

2

2.5

0 2 4 6 8−1.5

−1.0

−0.5

0

0.5

radial position [mm]

radial position [mm] radial position [mm]

radial position [mm]

Re

Er

[rel

.]R

eE

z [r

el.]

ImE

z [r

el.]

ImE

r [r

el.]

Figure 2.15: Fulfilling the boundary conditions at a ‘step-out’ transition. In this case a step from r=4 mm to 7 mmwas assumed. In the upper left picture the self-field of thebunch has been drawn with the opposite sign to illustrate thematching of the fields, at the metal boundary they cancel tozero. The step in the real part of Er causes a singularity inthe real part of Ez .

with b the radius of the beam pipe and sign(k) =±1 for fields radiated antiparallel respectively par-allel with the beam. The radius of the smaller beampipe is called bs, that of the larger one bl. The con-stants νn are the nth zeros of the Bessel functionJ0.

The functions J0(νnr/b), J1(νnr/b) can betreated as orthogonal bases of a vector space of in-finite dimension. A scalar product can be definedby

[f, g] =

b∫0

rdrf(r) · g(r) (2.102)

The orthogonality of the bases can be verified easily[J0

(νnr

b

), J0

(νmr

b

)]= δnm

b2J21(νn)2

(2.103)[J1

(νnr

b

), J1

(νmr

b

)]= δnm

b2J21(νn)2

(2.104)

The bases are complete for axially symmetric elec-tric fields. Any field pattern then can be written asa linear combination of the Bessel functions whichare

f =∑

n

fnJ0/1

(νnr

b

)(2.105)

fn =2

b2J21(νn)

[f, J0/1

(νnr

b

)](2.106)

f0/1 = (f1, f2, f3, . . .)0/1 (2.107)

In this notation the wake field calculation can bewritten as a system of algebraic equations. Let P

be vector of the additional pan-cake field after thejump , R the field radiated parallel to the beam,and B the backward reflection. Since the pan-cakefield is radial at the jump it has to be compensatedby the radial part of the radiated field. Then theboundary and continuity conditions at a step-outtransition are

Rr = P− Br (2.108)Rz = Bz, (2.109)

The vector P is non-zero at the additional width ofthe larger pipe only (r ≥ bs), the vector B is non-zero only in the smaller pipe. The components of Pare found by

Pn =2

b2l J

21(νn)

[1r, J1

(νn

blr

)]

=2

b2l J

21(νn)

bl∫bs

drJ1

(νn

blr

)(2.110)

The components of B are used in two bases. Twotransformation matrices have to be found

(BR)n,m =2

b2l J

21(νn)

bs∫0

drJ1

(νn

blr

)J1

(νm

bsr

)(2.111)

= 2bs

b2l

J0(νnblbs

)J1(νm)

J21(νm)

νm

bl(ν2n/b2

s − ν2m/b2

l ),

(RB)n,m =2

b2l J

21(νn)

bs∫0

drJ0

(νn

blr

)J0

(νm

blr

)(2.112)

The transformation from the radial to the longitu-dinal field is

(RR)n,n = − i

sign(k)√

k2b2l /ν2

n − 1(2.113)

for the larger pipe and

(BB)n,n =i

sign(k)√

k2b2s/ν2

n − 1(2.114)

for the smaller pipe. With these transformationsthe boundary conditions (2.108), (2.109) can becombined and rewritten

Rr = P−BR ·BB ·RB ·RR · Rr (2.115)⇔ Rr = (1 + BR ·BB ·RB ·RR)−1P (2.116)

The figure 2.15 illustrates the fulfilling of theboundary conditions. Although the agreement isreasonable there is no perfect match at the bound-aries. The coefficients of the modes do not convergefast enough. For the calculation of the longitudinalimpedance this is no problem because the modes ofhigher contribute ∼ 1/n with an alternating sign.

14

The longitudinal impedance can be calculatedfrom

Z =1q

∞∫−∞

dzEz(r = 0)eikz (2.117)

=1q

∑n

En

∞∫−∞

dzei(k−√

k2−ν2n/b2)z (2.118)

For each element of the sum the integrand is non-zero in the intervall (−∞, 0] or [0,∞) respectively,depending on the direction of propagation of thecorresponding wave. The integral can be performedfor the evanescent waves. For the propagatingwaves the integral yields δ(k − kz) which is alwayszero since they all have a phase velocity larger thanc, but assuming an infinitesimal damping the resultis finite

Z(k) = − i

q

∑n

En

k −√

k2 − ν2n/b2

. (2.119)

These calculations have to repeated for every fre-quency (see figure 2.16).

At frequencies far above cut-off the impedanceapproaches a constant value which can be calcu-lated from the energy stored in the additional selffield.

E = 2π

bl∫bs

rdrEδ ×Hδ (2.120)

With a similar treatment the surface roughnesswake field radiation emerging out of the test pipecan be calculated. Only the source terms have tobe modified accordingly.

101

102

103

0

500

1000

1500

wavenumber 1/λ [m−1]

Re(

Z)

[Ω]

101

102

103

0

200

400

600

800

1000

1200

wavenumber 1/λ [m−1]

Im(Z

) [Ω

]

Figure 2.16: Real and imaginary part of the impedance ofa step in the cross section. The dimensions are bs = 4 mmfor the smaller and bl = 35 mm for the larger beam pipeas it is the case for the experiment. The first five cut-offwavenumbers have been marked for the two pipes.

15

Chapter 3

Introduction to Free Electron Lasers

Figure 3.1: Working principle of a SASE FEL [51]. Thetrajectory of the beam is perpendicular to the magnetic field,to become visible in the drawing it has been turned by 90.

The description of the Free Electron Laser (FEL)given here is based on [39] and [26][25]. A moredetailed description can be found in [41] and [53]and the references therein. The basic elements ofan FEL are a bunched beam of highly relativis-tic electrons and an undulator magnet. A planarundulator produces a periodically oscillating dipolemagnet field

By(z) = B0 sin(kuz). (3.1)

A relativistic electron that enters the undulator willbe forced onto a sinusoidal trajectory. The undula-tor is built such that the maximum deflection angleϑ ≤ 1/γ with γ the relativistic factor of the elec-tron. It is therefore justified to assume (for the mo-ment) the forward speed vz to be constant ≈ βc andz = βct. Then it is easy to calculate the deviationx from the straight orbit

x(z) =eB0

γm0ck2u

cos(kuz). (3.2)

Introducing the dimensionless undulator parameter

K =eB0

m0cku≈ 0.934 ·B0[T] · λu[cm] (3.3)

the equation 3.2 can be written as

x(z) =K

γkuβcos(kuz). (3.4)

In this notation K/γ characterizes the deflectionangle in an undulator. The parameter K often ischosen to be in the order of 1.

Calculating the velocities inside the undulatornow it is necessary to drop the assumption of con-stant speed in z-direction

βx =K

γsin(kuz), (3.5)

β2x + β2

z = 1− 1γ2

(3.6)

⇒ βz =

√1− 1

γ2− K2

γ2sin2(kuz)

≈ 1− 12

[1γ2− K2

γ2sin2(kuz)

]= 1− 1

2

[1γ2− K2

2γ2sin2(kuz)

−K2

2γ2+

K2

2γ2cos2(kuz)

]= 1− 1 + K2/2

2γ2︸ ︷︷ ︸β

+K2

4γ2cos(2kuz). (3.7)

Due to the oscillating motion the mean velocity βcis slower than βc. Undulator radiation is character-ized by the coherent addition of the radiation fieldproduced by a single electron at different positionsalong the undulator. FEL radiation in turn is char-acterized by coherent radiation by many electrons.

The electromagnetic wave moves parallel to theundulator axis and is polarized in the horizontalplane

E = exE0 cos(kz − ωt + θ0), (3.8)

with ω = kc and θ0 the phase offset between thewave and the electron. Since the motion of theelectron in the undulator has a component parallelto the electric field of the wave there is an energytransfer between the electron and the wave

mc2 dγ

dt= F · v

=eE0Kc

γsin(kuz) cos(kz − ωt + θ0)

16

=eE0Kc

2γ

sin((k + ku)z − ωt + θ0︸ ︷︷ ︸θ

)

− sin((k − ku)z − ωt + θ0)

].(3.9)

The first sine-term in 3.9 is slowly varying while thesecond has a fast oscillation and averages to zero[26]. Continuous energy transfer is achieved if theponderomotive phase θ is constant

θ = (k + ku)z − ωt + θ0 = const (3.10)dθ

dt= (k + ku)vz − kc =! 0. (3.11)

Using the average velocity derived above

dθ

dt= c

[(k + ku)

(1− 1 + K2/2

2γ2

)− k

].(3.12)

The FEL radiation has a wavelength in the order of100 nm and below while the undulator period is inthe order of cm, therefore ku/k 1

dθ

dt= ck

(ku

k− 1 + K2/2

2γ2

)=! 0. (3.13)

This results in the resonant condition

λ

λu=

1 + K2/22γ2 . (3.14)

This is the formula for the undulator radiation inforward direction. This confirms the close relation-ship between undulator radiation and FEL radia-tion which can be compared to the relation betweenspontaneous and stimulated emission in lasers. TheFEL radiation is referred to as stimulated radiation,while the undulator radiation is spontaneous radi-ation. The resonant gamma is denoted as γ.

3.1 Pendulum Equation

A new variable is introduced

η =γ − γ

γ 1, (3.15)

γ2 =k

ku

1 + K2/22

. (3.16)

Using this variable the time derivative of the phaseangle is

θ = cku

(1− k

ku

1 + K2/22γ2

)= cku

(1− 1

(1 + η)2

)= cku

2η + η2

1 + 2η + η2

≈ 2ckuη. (3.17)

Performing the 2nd time derivative and insertingequation 3.9 one finds

12cku

θ = 11γ

γ =eE0Kc

2γγmc2sin θ. (3.18)

Because of its appearance this equation is referredto as the pendulum equation. The proper deriva-tion of the phase dynamics has to take into accountthe oscillatory trajectory of the electrons instead ofusing only the average velocity β. This can be doneby replacing the undulator parameter K in 3.18 by[39][54]

K → K = K

[J0

(K2

4 + 2K2

)− J1

(K2

4 + 2K2

)](3.19)

The pendulum equation is then changed to

θ =eE0Kc2ku

γγmc2sin θ. (3.20)

In an FEL amplifier with a significant growth ofthe radiated power the value of E0 in 3.20 cannotbe regarded as constant. In this case the inhomo-geneous wave equation for the electric field and theFEL equations have to be solved simultaneously,(

∇2 − 1c2

∂2

∂t2

)~E = µ0

∂ ~J

∂t+∇ρ

ε0. (3.21)

To account for the granularity of the bunches thecurrent and charge density are expressed as sumsover the single electrons [39]

~J = −ec∑

j

~βj(t)δ(~r − ~rj(t)) (3.22)

and

ρ = −e∑

j

δ(~r − ~rj(t)) (3.23)

The electric fields can be divided into two parts:The transverse field of the FEL radiation, and thelongitudinal field due to space charge. Because ofthe short modulation wavelength the space chargeforces cannot be neglected even for ultra-relativisticbeams [39]. Due to the microbunching the dominat-ing part of the longitudinal field is periodic and canbe written as a Fourier series Ez =

∑l El exp[ilθ].

Similar expressions hold for the charge and current.The wave equation for the longitudinal field is then[

∇2⊥ − (k + ku)2 + k2

]El

= − iel

ε0

∑j

[βjk − (k + ku)]e−ilθj (3.24)

For the radiation process the small differences in theelectron velocities are negligible. Using the approxi-mation βj ≈ β ≈ 1−ku/k and (k+ku)2−k2 ≈ 2kku

the equation becomes[∇2⊥−

l2k2(1 + K2)

γ2

]El=i

elk(1 + K2)

ε0γ2

∑j

e−ilθj .(3.25)

17

In a planar undulator the odd harmonics l =1, 3, 5, . . . can be amplified. But here only the fun-damental mode l = 1 is considered. The sum on theright hand side can be abbreviated by the bunchingfactor

< exp(−iθ) > =1

Ne

∑j

e−iθj . (3.26)

In a planar undulator only horizontally polarizedlight is produced. Hence only the wave equationfor the x-component of the radiation field has to beconsidered[∇2⊥ +

(∂

∂z

)2

−(

1

c

∂

∂t

)2]

Ex =1

ε0c2

[∂

∂tJx + c2 ∂ρ

∂x

].

(3.27)

In the 1-dimensional theory the transverse deriva-tive of the charge density on the right hand side canbe neglected [26].

c2 ∂ρ

∂x ∂

∂tJx (3.28)

The transverse current Jx is given by

Jx = −ecK sin(kuz)∑

j

1

γjδ(z − zj)δ(~x− ~xj). (3.29)

The transverse field is written as

Ex = E0 cos(kz − kct + φ), E =E0

2eiφ

= Eeik(z−ct) + E∗e−ik(z−ct), (3.30)

where E is a slowly varying complex number. Thederivatives are decomposed using

D± =1c

∂

∂t± ∂

∂z(3.31)

D+e±ik(z−ct) = 0,

D−e±ik(z−ct) = ∓2ike±ik(z−ct),(1c

∂

∂t

)2

−(

∂

∂z

)2

= D+D−. (3.32)

Making use of the slowly varying nature of E,| D−E | k | E |, one can approximate

D−

[Eeik(z−ct)

]= −2ikEeik(z−ct). (3.33)

This is called the slowly varying phase and ampli-tude approximation. Additionally there is the exactequality

D+

[Eeik(z−ct)

]= eik(z−ct)D+E. (3.34)

Combining the equations (3.28)-(3.34) equation3.27 becomes

eik(z−ct)(−2ikD+ −∇2

⊥)E (3.35)

+e−ik(z−ct)(2ikD+ −∇2

⊥)E∗ = − 1

ε0c2

∂Jx

∂t

⇒(−2ikD+ −∇2

⊥)E (3.36)

+e−2ik(z−ct) (2ikD+ −∇2

⊥)E∗ = − 1

ε0c2

∂Jx

∂te−ik(z−ct).

Instead of treating single electrons one would liketo handle continuous quantities. Therefore equation3.36 is averaged by means of the following integral

14t

∫ t+4t

t

[. . .]dt

∣∣∣∣∣z=const

,

over a time interval larger than one period of theoscillation and smaller than the coherence lengthλ/c 4t Nuλ/c. Over this interval E can beregarded as constant. In the integral the secondterm of the left hand side of eq. 3.36 averages outdue to its fast oscillation. The averaged equationthen reads(

2ikD+ +∇2⊥)E

= − ikeK

ε0c

1vz4t

cos(kuz)Ne∑j=1

1γj

e−ik(z−ctj)δ(~x− ~xj)

= − ikeK

2ε0γne < exp(−iθj) >, (3.37)

with the electron line densityne = Ne/(vz4t)< δ(~x− ~xj) > and zj ≈ βctj ≈ ctj .

3.2 Dimensionless FEL Equa-tions

In the following dimensionless equations are de-rived. First the independent variables are changedfrom (z, t) → (z, θ) and E(z, t) → E(z, θ).

dE =∂E

∂zdz +

∂E

∂tdt

=∂E

∂zdz +

∂E

∂θdθ

=∂E

∂zdz +

∂E

∂θ(k + ku)dz − ∂E

∂θωdt (3.38)

⇔ 1c

∂E

∂t= −k

∂E

∂θ, (3.39)

∂E

∂z= (k + ku)

∂E

∂θ+

∂E

∂z(3.40)

Using these variables the equation 3.37 changes into(∂

∂z+ ku

∂

∂θ+∇2⊥

2ik

)E = −eneK

4ε0γ< exp(−iθj) > .(3.41)

In 1D theory the transverse derivative is neglected(∂

∂z+ ku

∂

∂θ

)E = −eneK

4ε0γ< exp(−iθj) > . (3.42)

18

The pendulum equations become

dθ

dz= 2ηku,

dη

dz=

eK

2γ20mc2

(Eeiθ + E∗e−iθ

). (3.43)

A fundamental parameter of the FEL is the Pierceparameter ρ [26]. It is defined as follows1

ρ = 3

√e2K2ne

32ε0γ3mc2k2

u

, (3.45)

The advantage of using the Pierce parameter willbecome clear when writing down the main charac-teristics of the FEL at the end of the calculations.The following new variables are introduced

z → z = 2kuρz, (3.46)

η → η =η

ρ, (3.47)

E → a =eK

4γ20kumc2ρ2

E. (3.48)

In general there will be a detuning between the radi-ation field and the resonant ω. Then it is reasonableto assume that the normalized field is oscillating

a ∝ exp(iνθ), (3.49)

corresponding to a frequency detuning of the radi-ation ν = (ω − ω)/ω. The associated normalizeddetuning is

ν =ν

2ρ. (3.50)

With this set of variables the 1-dimensional FELequations can be written

dθj

dz= ηj , (3.51)

dηj

dz= aeiθj + a∗e−iθj , (3.52)(

∂

∂z+ iν

)a = < exp(−iθj) > . (3.53)

The parameter ρ has been chosen such that the co-efficient on the right hand side of 3.53 is one. Twocollective variables are defined

b = < exp(−iθj) > bunching parameter(3.54)

P = < ηj exp(−iθj) > energy modulation (3.55)

1Often an alternative formula is given for

ρ =3

√√√√ 1

8π

I

IA

(K

1 + K2/2

)2γλ2

ΣA, (3.44)

with IA = 17045 A the Alfven current, and ΣA = 2πσ2x the

cross sectional area of the electron beam.

to ease the notation in the following. In terms ofthe collective variables the FEL equations read

db

dz= −iP , (3.56)

dP

dz= a, (3.57)

da

dz= −b− iνa. (3.58)

Equation 3.56 describes the enhancement of mi-crobunching due to the energy modulation of thebunch. Equation 3.57 shows how the energy mod-ulation is caused by the radiation field. Equation3.58 describes the growth of the radiation powerdue to microbunching. The second term on theright hand side of equation 3.57 contains the av-erage < exp(−i2θj) > and averages out. The three

−2 −1 0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

ν

Im(µ

)

Figure 3.2: Growth rate of the FEL radiation versus normal-ized detuning ν = (ω − ω)/(2ρω).

coupled FEL equations can be solved by the ansatz

a = a0 exp(iµz). (3.59)

From equations 3.56-3.58 it follows that

P = − i

µa b =

i

µ2a iµa = −b− iνa. (3.60)

Therefore the assumed solution (3.59) is only pos-sible if

µ3 + νµ2 + 1 = 0. (3.61)

There are procedures to solve such a cubic equation[8]. The solution representing the exponential gainof radiation corresponds to Im(µ) < 0. To findthe solution equation 3.61 is transformed into thereduced form

µ → χ = µ + ν/3, (3.62)

χ3− ν2

3︸︷︷︸p

χ +2ν3

27+ 1︸ ︷︷ ︸

q

= 0. (3.63)

19

The required complex solution only exists if D =(p/3)3 + (q/2)2 > 0

ν > − 33√

4≈ −1.89. (3.64)

For zero detuning ν = 0 the solution is

µ0 =12− i

√3

2. (3.65)

From numerical calculations (see fig 3.2) it can beseen that this corresponds to the maximum growthrate of FEL radiation. The solution for the expo-nentially growing part is then

a(z) =13

(a(0) +

b(0)µ

− iP (0)µ

)e−iµz.(3.66)

The first term corresponds to coherent amplifica-tion, the other two to self amplified spontaneousemission (SASE) generated from random noise onthe electron distributions. The above approxima-tions are only valid for | a |. 1. If | a |≈ 1 satu-ration is reached. The radiation power in this caseis

P =1Z0

EE∗ ≈ 12ρ cγnemc2︸ ︷︷ ︸

beam power

. (3.67)

Some conclusions can be drawn [26]:

The Pierce parameter ρ is a measure for theefficiency of the FEL at saturation.

From the lower graph of figure 3.3 it can beconcluded that the bandwidth of the FEL is4ω/ω ∼ ρ.

Without detuning the power gain length is ap-proximately 1

Imµ0= λu

4π√

3ρ.

The saturation length can be estimated to beLs ∼ λu/ρ.

the coherence length can be estimated from thebandwidth lc ∼ λρ.

An energy loss along the undulator can be simu-lated by changing the detuning ν linearly betweenν0 . . . ν0+4 along the undulator. The effective gaincoefficient is then calculated by

µeff =14

ν0+4∫ν0

µ(ν)dν. (3.68)

The figure 3.3 shows the corresponding gain curvesfor different detuning rates. Note that the detun-ing is growing positively when the energy is lost.

−2 0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

∆=0 ∆=2

∆=4∆=6

∆=8∆=10

∆=12

−2 0 2 4 6 8 10 120

1

2

3

4

5

6

7

8x 10

7

∆=0

∆=2

∆=4

∆=6

ν

ν

<Im

(µ)>

exp(<

Im(µ

)>

Ls)

Figure 3.3: Growth rate of the FEL radiation versus normal-ized detuning assuming a constant energy loss along the un-dulator. The different curves correspond to a certain changeof the detuning along the undulator ∆ = νend − νin. Thelower picture shows the corresponding amplification after 21gain lengths (saturation).

Allowing a maximum detuning of one bandwidth(4 < 1) results in

dν

dz.

1Ls

≈ ρ

λu. (3.69)

Inserting the definition ν = 4ω/(2ρω0) and us-ing ω ∼ γ2 the condition in absolute coordinatesis found

dγ

dz.

ρ2

λuγ. (3.70)

Similar considerations can be made to calculate themaximum energy gradient on the bunch. Duringeach period of the oscillation the radiation slipsahead by one wavelength. Again along the slippagelength the allowed detuning is one bandwidth. Theslippage length is λ/λu · Ls and therefore

dγ

dζ.

ρ2

λγ, (3.71)

with ζ = z − ct the distance from the head of thebunch.

3.3 The TTF FEL

In Table 3.1 the main parameters of the TTF-FEL are summarized. From these numbers thePierce parameter is calculated to be ρ = 2.5 · 10−3.From this a power gain length LG,calc = 50 cmcan be calculated, the measured gain length was

20

Beam energy 240-250 MeVBunch charge 2.7-3.3 nCCharge in radiative part of bunch 0.1-0.2 nCPeak current 1.3± 0.3 kArms energy spread 150± 50 keVrms normalized emittance (6± 2)π mm·mradBunch spacing 0.44/1 µsNumber of bunches in a train up to 70rf pulse repetition rate 1 HzUndulator period λu 2.73 cmUndulator peak field 0.47 TAverage beta function 1.2 mMagnetic length of undulator 13.5 mRadiation wavelength 95-105 nmEnergy in the radiation pulse 30-100 µJ

FWHM radiation pulse duration 50+50−20 fs

Radiation peak power level 1 GWRadiation average power up to 5 mWSpectrum width (FWHM) 1%Spot size at undulator exit (FWHM) 250 µmAngular divergence (FWHM) 260 µrad

Table 3.1: TABLE I. Main parameters of the TESLA TestFacility for FEL experiments (TTF FEL, phase 1)[3].

LG,meas = 67 ± 5cm. The difference is mainlycaused by the energy spread of the bunch. Sincemany of the FEL parameters are inaccessible to thebeam diagnostics, the values in table 3.1 were cho-sen such that simulation of the FEL is consistentwith the measured properties of the radiation pulse.

Using equation 3.70 the maximum allowable en-ergy loss inside the undulator is dE/dz < 2.3 ·10−4 ·E0 ≈ 55 keV/m. This boundary is also marked infigure 3.4. Note that the surface roughness wakefields are not the only source of energy loss in theundulator. Other sources are resistive wall wakefields [44] and the FEL radiation itself.

In the design phase of the TTF-FEL, beam pa-rameters different from those in table 3.1 were as-sumed: A bunch length of σz = 250 µm, a nor-malized emittance εN = 4πmm mrad, and a bunchcharge of 1 nC were planned, yielding a peak currentIpeak=480 A, and the Pierce parameter ρ = 4·10−3.In this scenario the limit for the energy loss wouldhave been dE/dz < 160 keV/m.

The design of the next evolution step of theTTF-FEL (Phase II) foresees a beam energy of1 GeV, a bunch length σz = 50 µm, a peak currentIpeak = 2.4 kA, and a bunch width σx = 67 µm.The undulator magnets will be the same as for thefirst phase, only the total length of the undulatorwill be increased to 30 m. The radiation wavelengthwill be 6.4 nm. This results in a Pierce param-eter ρ = 1.8 · 10−3 and a permitted energy lossdE/dz < 120 keV/m.

For the TESLA X-FEL several beam lines withdifferent parameters are foreseen [51]. As an exam-ple here only a fixed gap undulator with electronbeam energy of 25 GeV is treated. The radiationwavelength is 0.85 A, the undulator period 45 mm.The Pierce parameter is ρ = 3 · 10−4, resulting in apermitted energy loss of dE/dz < 50 keV/m.

100

101

102

100

101

102

103

δ [µm]

peak

ene

rgy

loss

[keV

/nC

/m]

TTF II

TTF ITESLA

σ=25µm σ=250µmσ=50µm

TTF I

Figure 3.4: Expected peak energy loss due to surface rough-ness wake fields and permitted energy loss of different FELs.The peak energy loss due to surface roughness wakes is cal-culated according to the Dohlus model scaling the spectrumof the surface roughness in the wake field experiment.

21

Chapter 4

Longitudinal Phase SpaceTomography

In general the task of computer tomography isto reconstruct a distribution in a space of higherdimension from a set of projections measured in alower dimensional subspace. In most cases the setof projections is generated by rotation of the ob-ject under study. Phase space tomography is anapplication of this technique to particle beams inaccelerators. In the transverse case the phase spaceis the coordinate system of transverse offset and di-vergence. The x and y profiles are measurable withobservation screens or wire scanners. Therefore theprofiles are the natural choice for the projections.The rotation of the phase space can be achievedwith a quadrupole doublet. The profiles can bemeasured via optical transition radiation (OTR)from a metallic screen (see [15]).

The longitudinal phase space is spanned by theenergy offset E and time offset T with respect toa reference particle. With a magnetic spectrometerthe energy profile of the bunches can be measured.Therefore the projections onto the energy axis arethe inputs for the reconstruction. By acceleratingthe bunches at different phases ϕ of the accelerat-ing rf field it is possible to obtain a distortion ofthe phase space but not a rotation. In the linearapproximation the transformation is a shearing(

ET

)′=

(1 E0ω sinϕ0 1

)(ET

).(4.1)

In this thesis the longitudinal phase space of thebunches in the TTF is considered when they enterthe spectrometer at the end of the linac. There isno possibility to project the time profile onto theenergy axis and hence no possibility to obtain afull rotation of 180. This has severe implicationsfor the tomographic reconstruction as will be seenlater. In a synchrotron or with a magnetic chicanethere are however transformations that allow for afull rotation. See for example [40] or [17] for thesecases.

equations

F0

F1j = 1

j = 2j = 3

F2. . .ss

s

Figure 4.1: Illustration of Kaczmarz’s method. By succes-sively performing the projection onto each equation the solu-tion is approached. One iteration is reached when all equa-tions have been considered.

4.1 Algebraic Reconstruction

There are a number of algorithms available to recon-struct the original 2-dimensional distribution froma set of projections. In this context only the ART(algebraic reconstruction technique) algorithm willbe explained as an example for a standard recon-struction technique [31]. For the ART algorithmthe space is divided into a cartesian grid. The con-tent of each bin in the grid is an element of a singlerow vector F . The projections are written into arow vector G. The vectors are related by a matrixA

G = A · F. (4.2)

In most cases A will not be a square matrix. Evenif the measurement is done such that A is a squarematrix, it will be ill posed. This means that withideal data from simulation the reconstruction willthen yield the original distribution, but alreadynoise in the order of 10−3 will cause unaccept-able errors in the reconstruction. Therefore an ap-proximation method is used to perform the inver-sion. The method chosen for the ART algorithm iscalled Kaczmarz’s method. The basic idea of thismethod is sketched in figure 4.1. Each linear equa-

22

original

10 20 30 40 50 60

10

20

30

40

50

60

reconstructed

10 20 30 40 50 60

10

20

30

40

50

60

1020

3040

5060

20

40

60

0.2

0.4

0.6

0.8

1

1020

3040

5060

20

40

60

0

0.2

0.4

0.6

0.8

Figure 4.2: Example for the ART algorithm. The left pic-tures show the assumed distribution. From this distributionthe projections are calculated and input into the reconstruc-tion algorithm. The right pictures show the result of thereconstruction. The main features of the distribution are re-produced with good quality. There are some small artefactswith small amplitude. These are due to the small number of9 projections.

tion can be represented by a straight line in a multi-dimensional vector space. The solution of the equa-tion system is found at the intersection point of alllines. Starting from an arbitrary point in the vectorspace the solution can be approximated by succes-sively performing the projection onto the lines.

Fj = Fj−1 +ω

| aj |2(gj − aT

j Fj−1

)aj (4.3)

j = 1 . . . N

with aj being the jth row of A, gj the jth entry

of G, Fj is the jth corrected version of F after the

jth projection. The parameter ω can be adjustedto control the convergence of the iterations. In gen-eral this method can be used to solve a system oflinear equations. Here it delivers a good approxi-mation for the projections of the two dimensionaldistribution (see fig. 4.2).

A common requirement of all standard algo-rithms is the need for a set of projections cover-ing a full rotation of 180. The ART algorithmis no exception. Figure 4.3 shows a reconstructionbased on a limited set of projection angles. Twomain problems are identified. The resolution of themeasurement is reduced, i.e. the two peaks whichcould be separated before now appear as one, andthe artefacts are enhanced: Especially streaks atthe maximum angles are produced. This problembecomes even more serious for distributions whichalready have a pronounced structure along this di-rection. In the longitudinal phase space this has tobe expected.

original

10 20 30 40 50 60

10

20

30

40

50

60

reconstructed

10 20 30 40 50 60

10

20

30

40

50

60

1020

3040

5060

20

40

60

0.2

0.4

0.6

0.8

1

1020

3040

5060

20

40

60

0

0.2

0.4

0.6

0.8

Figure 4.3: The result of the ART algorithm with a reducedset of projections. In comparison to the previous example theangle of rotation has been reduced to 90. The reconstruc-tion washes out features of the distribution and the artefactsare enhanced.

The analysis of the projections in figure 4.4 showswhy all standard algorithms fail to reconstruct theoriginal distribution in the “reduced angle” prob-lem. Although artefacts appear in the two dimen-sional reconstruction the projections are reproducedvery well in terms of a least square fit and no fur-ther improvement can be expected. By taking theprojections alone the reconstruction algorithm verylikely only finds a relative minimum of the errorminimization resulting in severe artefacts.

There are some possible directions in which onemight look for a cure. The artefacts in the recon-struction seem to be predictable. The number ofwiggles in the distribution is given by the numberof projections and the strongest artefacts are in thedirection of the maximum projection angle. There-fore a spatial filter might improve the reconstruc-tion by deconvoluting the predicted pattern fromthe distribution.

The main contribution of the artefacts appear atlow amplitudes of the projection signal. A non-linear weight of the deviations might improve thereconstruction, the logarithmic function seems tobe a good candidate for that. The third idea isthe maximum entropy method explained in the nextsection. The ART algorithm will not be used anyfurther in this work.

4.2 Maximum Entropy Algo-rithm

The maximum entropy method is a general tech-nique for data analysis. It provides proceduresbased on the least possible prejudice on the mea-

23

0 50 100 150 200 250 300 350 400 450 500−5

0

5

10

15

20

Figure 4.4: Input data for the ART algorithm. The originaland the reconstructed projections are shown. They are puton top of each other and hardly any difference can be seen.Only small deviations can be seen just above the baseline.

surement errors.In the case of phase space tomography, the maxi-

mum entropy method can be utilized to reduce arte-facts in the reconstructed distribution. The entropycan be interpreted as a measure for the amount ofsubstructures in the distribution. The entropy ismaximum for the distribution with the least struc-ture. If there are no constraints this would resultin a uniform distribution.

Interpreting the distribution as a collection ofparticles the entropy has a second interpretation.If there is no further knowledge about the systemeach arrangement of particles in phase space will beassigned the same probability (least prejudice). Thedensity function f is a global description of the dis-tribution and does not distinguish the microscopicdetails of the arrangement of particles. Therefore itis possible to calculate for each density function fthe number of particle arrangements to reproduceit. A measure for this number is the entropy of thisdensity function. Then it is immediately clear thatthe density function with the largest entropy hasthe largest probability to be realized.

For the phase space tomography the task is tomaximize the entropy while at the same time theprojections are reproduced. These are the con-straints for the optimization problem. The proce-dure is first described for the general case of a rota-tional transformation of the phase space. Later itcan be generalized for non-linear transformations.The algorithm described here was developed byG. Minerbo [30]. A description is also found in[11]. See figure 4.5 for an example of this algo-rithm. A comparison with figure 4.3 shows that themaximum entropy method is far superior to the al-

Figure 4.5: Simulated reconstruction with the MENT (max-imum entropy) algorithm. On the left an assumed distri-bution is shown. From this distribution the projections arecalculated and fed into the algorithm. On the right the cor-responding reconstruction is shown. No severe artefacts areobserved, the resolution is as good as can be expected frompurely geometrical arguments (see section 4.2.2). For thisexample 7 projections with a maximum angle of ±45 wereused.

gebraic reconstruction technique if only a limitedangular range is accessible to the measurement.

Let the original distribution F be defined in the(x, y)-plane. A number of J projections is taken,the projection number j consisting of M(j) binswith the content Gjm. For each projection let sbe the axis in the projection plane and t the axisperpendicular to it. The projection data are writtenas

Gjm =

sjm+1∫sjm

ds

∞∫−∞

dtF(s cos θj − t sin θj , s sin θj + t cos θj),

m = 1, . . . , M(j), j = 1, . . . , J, (4.4)

where θ1, . . . θj , . . . , θJ are the projection angles,and

sj1 < sj2 < . . . < sjM(j) (4.5)

are a set of abscissas for the jth view. They neitherhave to be equally spaced nor have they to be thesame for all projections. It is assumed that thedistribution is confined in a limited area D and thatthe sj cover the whole range of the distribution.

The integral 4.4 can be rewritten by introduc-ing the characteristic function χjm of the interval[sjm, sjm+1)

χjm(s) =

1, sjm ≤ s < sjm+1,0, otherwise. (4.6)

In this way the integration can be extended acrossthe whole area D

Gjm =∫ ∫

Ddx dyF(x, y)χjm(x cos θj + y sin θj).

24

The original distribution F is of course unknown.Therefore in the calculations it is replaced by thereconstructed distribution f which is iteratively im-proved starting from a uniform distribution

Gjm =

∫ ∫D

dx dy f(x, y)χjm(x cos θj + y sin θj). (4.7)

The distribution f can be treated as a probabilitydistribution. The entropy is defined as

η(f) = −∫ ∫

Ddx dyf(x, y) ln[f(x, y)A], (4.8)

where A is the area of the domain D. It can beshown that η is proportional to the logarithm ofthe probability of the distribution f [14, 22]. Thetask is now to find the maximum of η subject to theconstraints in eq. 4.7.

This is a variational problem. To find the solutionLagrange multipliers Λjm are introduced, one foreach constraint 4.4. Then the Lagrangian is formed[8]

Ψ(f, Λ) = f(x, y) ln[f(x, y)A] (4.9)

+∑

j

∑m

Λjm [Gjm − f(x, y)χjm(x cos θj + y sin θj)] .

The functional derivative of Ψ with respect to f isset equal to zero,

∂Ψ

∂f= 0 (4.10)

= ln[f(x, y)A] +1−∑

j

∑m

Λjmχjm(x cos θj + y sin θj).

This is the Euler-Lagrange equation for this prob-lem.

⇔ f(x, y) =1

Ae

∏j

∏m

exp[Λjmχjm(x cos θj + y sin θj)]

⇔ f(x, y) =1

A

∏j

∏m

Hχjm(x cos θj+y sin θj)

jm , (4.11)

with Hjm = exp(Λjm−1/J). The χjm can be zeroor one. Therefore the Hjm contribute to the prod-uct as H0

jm or H1jm. For given (x, y) only one Hjm

contributes. Therefore the product can be replacedby a sum

f(x, y) =1

A

∏j

∑m

Hjmχjm(x cos θj + y sin θj). (4.12)

The optimization problem is solved by finding theΛjm respectively the Hjm.

The coefficients Hjm are determined by substi-tuting eq. 4.12 into eq. 4.4

Gjm =1

A

∫∫Ddx dy

∏k

∑n

Hknχkn(x cos θjk − y sin θjk),

(4.13)

where θjk = θj − θk. The non-linear Gauss-Seidelmethod is used to solve this system of equations.

The H0jm are initialized with 1. If Gjm = 0, the

corresponding Hjm are set equal zero and elimi-nated as an active variable. The solution is foundby recursively applying

Hi+1jm =

AGjmHijm∫∫

D dx dy∏k

∑n

Hiknχkn(x cos θjk − y sin θjk)

.

(4.14)

The integrand is piecewise constant over polygons.Thus the double integral can be performed exactlyin a finite number of steps. Due to the introductionof the characteristic function it is not necessary tocalculate any logarithm or exponential function.

4.2.1 The Implementation

As mentioned above the integrand is constant overpolygons. So the task for the implementation is tofind the correct polygons. Every polygon can bemade from a set of triangles. The initial division ofthe space is made up by the bins of the respectiveprojection. These rectangles are divided into a setof triangles. If only linear transformations have tobe expected, two triangles are sufficient. This is thecase for the transverse phase space. In the longitu-dinal phase space the curvature of the rf has to beaccounted for. The easiest way to do so is divid-ing the bins into smaller rectangles over which thetransformation can be regarded as linear. The rect-angles are then cut into two halves to obtain againtriangles.

The triangles are then fed into a recursive func-tion which maps the corner points of the triangle tothe initial grid in front of the transformation andmaps it again to a new projection. In this way non-linear transformations can be perfomed as well. Inthe transformed grid the intersections with the newbins are calculated and the triangles are divided ac-cordingly. This function calls itself recursively untileither all projections have been processed or the tri-angle leaves the valid space. The area of the finaltriangle is multiplied with the corresponding Hjm

and the added to the total sum.

4.2.2 Limits of Applicability

The off-crest acceleration induces a distortion of thelongitudinal phase space, which in first order can bedescribed as a shearing

4E = a · 4T , (4.15)a = E0ω0 sinϕoff .

Assuming a phase shift ϕoff = ±45 and a energygain E0 = 110 MeV,

a =4E

4t≈ 570 keV/ps (4.16)

25

Figure 4.6: The resolution that can be achieved by a to-mography performed with a set of limited angles. The leftpictures show an assumed distribution which is very narrowin time while the right picture shows a reconstructed distri-bution. The best time resolution is achieved at the edges ofthe distribution. Towards the center of the distribution 4Eincreases and so does 4T . Here it may be deduced from theinverse of the height of the maximum. Note: If the bin sizewould have been adapted to the time resolution the recon-structed distribution would show the same rectangular shapeas the original.

Two peaks in the phase space separated by 4Tare shifted by a · 4T against each other in energy.At the same time the peaks are widened. For gaus-sian peaks with σE and σT this can be expressedas

σ′E =√

σ2E + a2σ2

T . (4.17)

This results in a degradation of the time resolu-tion because in the projection it is impossible toachieve the same separation of the peaks in the en-ergy projection as it would be possible in the timeprojection (this can be seen in figure 4.6). The en-ergy resolution of the spectrometer depends on thetransverse emittance of the beam, the β-function,and the dispersion at the location of the diagnosticscreen. The β-function at the OTR screen in thespectrometer is smaller than 0.5 m, the dispersionis 1 m. At 200 MeV and a normalized emittanceεN = 3 π mm mrad (slice) one expects

σx =√

βε ≈ 62 µm ⇒ σE ≈ 12 keV (4.18)

In the experiment a resolution of

σE ≈ 25 keV (4.19)

could be verified, probably dominated by the en-ergy spread of the beam itself. Two δ-peaks can be

distinguished when they are separated by 2σ

4T = 2σEa≈ 85 fs. (4.20)

For extended structures a degradation of the timeresolution is expected. Two gaussian peaks withwidth σT can be separated if

4T = 2

√σ2E

a2+ σ2

T (4.21)

In general the achievable resolution depends onthe structure of the distribution under study. Struc-tures in time can be resolved if there is at leastone projection that delivers sufficient separation inenergy. Thus the resolution depends on the meangradient of the distribution along the energy coor-dinate. It should be stressed that this limitation isderived from geometric arguments only, there is lit-tle additional influence from the MENT algorithm.The real distribution may be narrower than the re-constructed one but from the available energy pro-jections it is not justified to assume any narrowerdistribution unless there is some additional infor-mation.

Figure 4.7: Reconstruction of a bunch as expected duringthe wake field experiment. On the left the simulated bunchis depicted, on the right the tomographic reconstrution isshown. The horizontal axis is the time in ps, the verticalaxis the energy in MeV. The head of the bunch is to theright. The projections used in this example are plotted infigure 4.8, the corresponding projection angles can be foundin figure 4.12.

The limitations of the tomography can be seenalso in the reconstruction of the longitudinal phasespace distribution that is expected in the TTF (seealso chapter 5). Figure 4.7 shows a simulated dis-tribution with a bunch at maximum compressionand synchronous mode wake fields imposed on it.The structures parallel to the energy axis are muchbetter resolved than the structures orthogonal toit. This results in an ensemble of separated peaksin the reconstruction although the original distri-bution is continuous. The projection data and the

26

Time Profile

energy [MeV]-4 -2 0 2 4

0

0.01

0.02

0.03

0.04

0.05

0.06

Phase -33

energy [MeV]-10 -8 -6 -4 -2 0 2 4

0

0.005

0.01

0.015

0.02

0.025

0.03

Phase 33

energy [MeV]-4 -2 0 2 4

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Phase -14

energy [MeV]-6 -4 -2 0 2

0

0.005

0.01

0.015

0.02

0.025

0.03

Phase 14

energy [MeV]-5 -4 -3 -2 -1 0 1 2 3

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Phase 0

Figure 4.8: Projections used for the reconstruction in figure4.7. The top left graph without content is a place holderfor the missing time profile. The other graphs are markedwith the corresponding off-crest rf phase. The correspondingprojection angles can be found in fig. 4.12. The profiles areproduced by both the original and reconstructed phase spacedistribution.

corresponding projection angles can be found in thefigures 4.8 resp. 4.12.

4.2.3 Combining IndependentSources of Information

To overcome the problems explained in the last sec-tion it may be useful to combine data from differentsources. Here it is appropriate to supplement theenergy spectra with a time spectrum derived froman independent interferometric measurement. Adifficulty is that the relative time offset of the mea-sured distributions is usually unknown. Thereforethe directly measured time profile may be shiftedwith respect to the time profile reconstructed fromthe energy distributions. Figure 4.9 shows the re-sult of a combination of simulated projections. Theprocedure of reconstruction is such that first a to-mography is performed without the time profileyielding the distribution in figure 4.7. The recon-structed time profile and the directly measured timeprofile are then matched such that the points ofmaximum weight coincide in time. In the case of thedistribution in figure 4.9 the offset was only ≈200 fs.

The complete set of projections is shown in figure4.11.

Figure 4.9: Reconstruction of the bunch from figure 4.7. Inthis case the longitudinal profile of the bunch has been addedto the projection data. The origin of the coordinates hasbeen aligned by matching the points of maximum weight ofthe longitudinal profile and the reconstruction without. Seethe projections in figure 4.11.

time [ps]-6 -5 -4 -3 -2 -1 0

0

0.02

0.04

0.06

0.08

0.1

0.12

Time Profile

original

interferometertomography

Figure 4.10: Reconstruction of the bunch from figure 4.7.The bunch is moving to the right. The reconstructed profile(dashed line) closer follows the original profile (solid line)than the interferometer data (dotted line).

The frequency response of the time measurementis generally different than that of the energy mea-surement. Especially when using interferometricdata this may be the case, since the interferome-ter suffers from low frequency cut-offs. In the sim-ulation this is modelled by a low frequency cut-offfilter. Afterwards the low frequency amplitudes areenhanced to ensure positive values in the bunchshape. This is necessary because the tomographyalgorithm requires non-negative projection data.Additionally there is some uncertainty about thereal profile since the phase information of the formfactor cannot be measured directly but has to be de-duced with the aid of the Kramers-Kronig-relation.

The figure 4.10 shows the results for the timeprofile. The simulated interferometer data producea very narrow bunch profile. Even with this nar-row profile the combined tomography is able to re-produce the original profile which is considerablylonger. This is due to the fortunate situation that

27

time [ps]-4 -2 0 2 4

0

0.02

0.04

0.06

0.08

0.1

0.12

Time Profile

energy [MeV]-4 -2 0 2 4

0

0.01

0.02

0.03

0.04

0.05

0.06

Phase -33

energy [MeV]-10 -8 -6 -4 -2 0 2 4

0

0.005

0.01

0.015

0.02

0.025

0.03

Phase 33

energy [MeV]-4 -2 0 2 4

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Phase -14

energy [MeV]-6 -4 -2 0 2

0

0.005

0.01

0.015

0.02

0.025

0.03

Phase 14

energy [MeV]-5 -4 -3 -2 -1 0 1 2 3

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Phase 0

Figure 4.11: Projections used for the reconstruction in figure4.9. The top left graph shows the time profile, which isdistorted by a low frequency cut-off in the interferometer.The other graphs are marked with the corresponding off-crest rf phase, see figure 4.12 for the corresponding projectionangles.

the energy measurements and the interferometerare complementary to each other. The short bunchhead is resolved by the interferometric measurementwhile the long tail of the bunch is well reproducedin the energy profiles. This situation holds as longas none of the projections explicitely excludes partsof the distribution by delivering zero or negativevalues.

As can be seen from figure 4.9 as well as fromfigure 4.10 the reconstruction of the phase spacestill does not fit the original completely. This canbe traced back to the fact that there is still a gapin the set of projection angles.

−10 −5 0 5 10−10

−5

0

5

10

time [ps]

ener

gy [M

eV]

−10 −5 0 5 10−15

−10

−5

0

5

10

time [ps]

ener

gy [M

eV]

−10 −5 0 5 10−15

−10

−5

0

5

10

time [ps]

ener

gy [M

eV]

−10 −5 0 5 10

−10

−5

0

5

10

time [ps]

ener

gy [M

eV]

−10 −5 0 5 10

−10

−5

0

5

10

time [ps]

ener

gy [M

eV]

−10 −5 0 5 10−10

−5

0

5

time [ps]

ener

gy [M

eV]

time projection φ = −33

φ = 33 φ = −14

φ = 14 φ = 0

Figure 4.12: Sketch of the projection angles used in figure4.7 and 4.9. The lines in the phase space are parallel to thepaths of integration. The projection angles are plotted inthe same order as the profiles in the figures 4.8 and 4.11.

28

Chapter 5

The TESLA Test Facility Linac

The TESLA Test Facility Linac is a supercon-ducting linear accelerator for electrons. The radiofrequency (rf) of the nine-cell cavities is 1.3 GHz.Since the machine serves as a test facility for theTESLA collider its setup is subject to changes. Herethe setup will be described as it was used during thewake field experiment.

While a maximum electron energy of approxi-mately 340 MeV can be reached, the nominal work-ing point is about 235 MeV. The electron bunchesare produced by photoemission from a Cs2Te pho-tocathode. Ultraviolet light pulses are required forthe photoemission. They are produced by frequencyquadrupling the light from a mode-locked Nd:YLFlaser. The light pulses have an approximately gaus-sian shape with σt ≈ 8.5 ps. The photocathode ismounted inside a normal conducting rf cavity op-erating with a peak field of 35 MV/m. This pro-vides immediate acceleration of the electrons andthus a quick compensation of the repulsive Coulombforces by attractive magnetic forces. Additional fo-cusing is provided by a solenoid field inside the cav-ity. The strength of the solenoid coils and the ac-celeration phase of the rf gun are adjusted to op-timize the longitudinal and transverse emittances.For optimum conditions the normalized emittanceis εN = 3.0 ± 0.2 mm mrad [37] and the rms bunchlength is σz = 3.2± 0.2 mm (10.7± 0.7 ps) [18, 46].

Approximately 1 m behind the gun the first su-perconducting acceleration cavity boosts the elec-tron energy to 16.7 MeV. A beam line follows withseveral quadrupoles and diagnostic screens used to

laserbunch compressor

undulator

240 MeV16 MeV4 MeV 120 MeV

boostercavity

laser drivenelectron gun photon beam

diagnostics

superconducting accelerating cavities

e - beamdiagnostics

- e - beamdiagnostics

-collimator

Figure 5.1: Schematic layout of the TESLA Test Facility (TTF). Although separated by the bunch compressor the two moduleswith superconducting accelerating cavities are driven by a single klystron.

measure the transverse emittance, and a spectrome-ter dipole to analyse the energy distribution. Due tothe nonlinear curvature of the accelerating field inthe gun and the booster cavity the bunches acquirean energy modulation of 500 keV (rms). The resid-ual energy spread is measured to be 25 keV (rms).That is the energy width of each temporal slice inthe longitudinal phase space which cannot be com-pensated by any time dependent energy modula-tion. It is dominated by dynamic effects during theacceleration in the gun, presumably the initial en-ergy spread of the electrons leaving the cathode ismuch smaller (in the order of eV). The measuredvalue is close to the resolution of the spectrometerin the first section1. The beam dynamics in thegun have been simulated with comparable results[45]. Other simulations, however, yield consider-ably smaller values [36].

The beam passes then a first module consist-ing of eight superconducting cavities, followed bya magnetic chicane for bunch compression. The ac-celeration voltage of the module is approximately110 MV. By adjusting the rf phase in the first mod-ule a time to energy correlation is imposed on thebunch. The path length in the bunch compres-sor depends linearly on the particle momentum,4z/l = −αc4p/p, lαc = 0.227 m (see figure 5.2).In combination with the off crest acceleration this

1In the meantime the resolution of the measurement sys-tem has been improved. Newer measurements of the energyspread yield numbers below 5 keV (rms). These results couldnot be included in this work. The main results, however, arenot affected.

29

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−0.1

0

0.1

0.2

0.3

0.4

0.5

∆s ∆s

δ δ

z/m

orbi

t, R

16, R

56 /m

hor. dispersion R16

reference trajectory

beam orbit straigth

long. dispersion R56

dipole 1

dipole 2 dipole 3

dipole 4

Figure 5.2: Schematic of the bunch compressor [45]. Thelongitudinal and transverse dispersion functions are plottedas function of the position in the bunch compressor. Themomentum compaction is αc = R56/l.

can be used for bunch compression. The optimumlongitudinal compression by about a factor of 5 isachieved with an off-crest phase of φ = 12. Choos-ing different phases the bunches can be shaped tobe more suitable for the wake field experiment. Ata later point this will be explained in more detail.

In the bunch compressor and in the spectrome-ter dipole synchrotron radiation is produced. Theradiated spectrum ranges from the cut-off of thebeam pipes in the cm-wave regime up to ultravi-olet light. Wavelengths comparable to the bunchlength or longer are radiated coherently (coher-ent synchrotron radiation, CSR). By shortening thebunches the coherent part of the spectrum is ex-panded towards higher frequencies and the totalpower is increased dramatically. Due to the curvedtrajectory of the beam in the magnetic chicane theradiation emitted at one point can interact with thebeam at another point. Similar to wake fields thiswill lead to a modulation of the energy distribu-tion of the bunch. But unlike wake fields CSR actsahead of the source particle. The energy shift ofthe electrons due to CSR is proportional to the dif-ferentiated charge distribution, therefore it is mosteffective at the head of the bunch where the steepestcharge density gradients can occur.

Behind the bunch compressor the second acceler-ation module raises the electron energy to the finalvalue. During the experiment the maximum energywas 235 MeV. Owing to a shortage of equipmentthe two acceleration modules are driven by only oneklystron. To maintain the stability of the beam pa-rameters at the entrance of the bunch compressorthe rf control only stabilizes the first module. Thesecond module receives the same input rf power asthe first. The accelerating field depends on the dy-namic response of the cavity resonators to the rfinput, which in turn depends on the detuning andquality factor of the resonators. The cavity inputcouplers and the waveguide tuners have been ad-

justed to obtain the same quality factors for all16 cavities within a range of 5 %. The frequen-cies of the cavities are adjusted with an accuracyof ±50 Hz. The field stability in the second modulecan only be maintained within 1% while it is betterthan 10−3 in the first module. The reason for theserather large variations may be cavity detuning dueto mechanical vibrations (microphonics).

0 500 1000 1500 20000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

frequency [GHz]

long

. im

peda

nce

[arb

. uni

ts]

spoiler 1absorber 1spoiler 2absorber 2sum

Figure 5.3: Frequency dependence of the impedance of thecollimation system. The varying radii of the collimator el-ements have been considered. The sum of all componentsyields one resonant frequency at 830 GHz. The assumedeffective dielectric constant is ε = 1.55.

The main components of the free electron laserare three undulator modules with an upstream col-limation system. The task of the collimator is toprotect the permanent magnets of the undulatorfrom radiation damage caused by beam halo. Twostages of collimation with 90 betatron phase ad-vance in between are used. Each stage consists ofa so called spoiler with narrow aperture (minimumdiameter 6 mm, see figure 5.4) and an absorber ofwider aperture which removes secondary particlescreated in the spoilers. The production of geomet-rical wakes at these structures has been reduced bytapering the structures. The holes were made byelectro-erosion, the surface roughness is consider-ably larger than in extruded tubes. In the spoilersa roughness of 5 µm (rms) was measured, on theabsorbers 3 µm (rms) [43]. Therefore it is expectedthat surface roughness wake fields are excited. Forthe calculation of the wake frequencies the taperedstructures of the collimators have to be taken intoaccount. The longitudinal impedance can be deter-mined by integrating the wake functions excited byeach part of the collimator with the correct phase

φ(z) =

L∫z

(kz(z′)− k)dz′

Ztot‖ (k) =

L∫0

dzZ0

π[b(z)]2exp(−iφ(z)). (5.1)

30

The figure 5.3 shows the resulting impedance of thecollimator assuming an effective dielectric constantε = 1.55 to model the surface roughness wake fields.This value has been chosen to reproduce the mea-sured resonant frequency of 830 GHz (see chapter7).

pump port

pump port

pump port

pump port

pump port

toroid T6

toroid T5

BP

M1

BP

M2

quadrupole Q1

quadrupole Q3

quadrupole Q2

quadrupole Q4

fast shutter

valve

spo

iler 1

spo

iler 2

abso

rber 1

abso

rber 3

abso

rber 2

BP

M_B

OT

R screen

steerer H1/V

1

steerer H2/V

2

steerer H3/V

3

beam

Spoiler 1:

Absorber 1:

Absorber 3:

Spoiler 2:

Absorber 2:

beam

copper

aluminum

insulationvacuum

water cooling channel

water cooling channels

bellow

contact surface

z = 65248z = 67830

Figure 5.4: Side view of the collimator and cross sections ofthe spoilers and absorbers [45].

The aluminum undulator vacuum chamber wasmade by extrusion. The inner diameter of the beampipe is 9.5 mm. There are some wake fields to beexpected from this structure as well. Especiallythe integrated effect may be strong given a totallength of 15 m. The main contributions to the ex-pected wake fields are caused by the resistivity andthe roughness of the vacuum chamber. The sur-face roughness in the undulator is measured to be600 nm (rms) [16], so the harmonic wake occursat a frequencies between 1.6 − 5 THz, dependingon the model. Therefore the roughness wake fromthe undulator like the resistive wall wake influencesmainly the head of the bunch. In the wake field ex-periment these wakes are disregarded by measuringonly in the tail of the bunch.

The setup for the wake field experiment ismounted behind the undulator. It consists of a setof test pipes with varying surface treatment and ra-dius. By means of a linear drive the different beampipes can be introduced into the beam axis. Thelength of the beam pipes is 855 mm. The radii varyfrom 3 mm to 5 mm. One meter downstream thetest pipe chamber there is a movable screen made oftwo polished silicon wafers with an aluminum coat-ing acting as a mirror. A special holder allows theuse as a closed mirror or with a central slit of upto 10 mm width. This screen can be utilized to

deflect radiation out of the vacuum chamber intoa far infrared interferometer. At the end of thelinac there is a spectrometer dipole to analyze theenergy distribution of the bunches. A screen be-hind the dipole is used to take images of the energydistribution. Observing the 25 keV energy spreadalready known from the injector it could be verifiedthat the resolution obtained in this spectrometeris better than 10−4. This spectrometer is used tomeasure the wake field induced energy modulationof the bunches.

5.1 Simulation of the Longitu-dinal Phase Space

A computer code has been developed to simulatethe longitudinal phase space starting from the en-trance of the first accelerating module to the spec-trometer. The effects of various accelerator compo-nents on the phase space are analyzed by analyticalexpressions. To ease the calculation of projectionsthe phase space at the end of the linac is repre-sented on a cartesian grid. A matrix is preparedwhich contains the entries for the charge density.A second matrix contains the corresponding coor-dinates of the bins. Instead of shifting the contentsof the density matrix according to the transforma-tions along the linac, the coordinate matrix is trans-formed. The components of the linac are treated inbackward direction, i.e. the physically last com-ponent is treated first. The density matrix then isfilled with values calculated by an analytical expres-sion for the charge density calculated as a functionof the transformed coordinates E and t.

The method works if it can be guaranteed thatthe area of the bins is not changed during the trans-formations. For linear operations such as rotationand shearing this is immediately clear. For non-linear operations such as cosine modulations thiscould be verified as well (see figure 5.5). Note thatthe transformations required in the simulation actonly on one coordinate at a time.

The initial phase space distribution is assumed tobe slightly non-gaussian. The following distributionagrees within the errors with the measured bunch

rotation shearing non−linear

Figure 5.5: A few examples for the transformations of bins.The nonlinear transformations are exagerated. In all casesthe area of the bins is invariant.

31

length and energy profiles

ρ = ρ0 exp(− E2

2σ2e

)· exp

(−| t |

2.5

2σ2.5t

). (5.2)

Here E and t are the energy resp. time offsets fromthe reference point in the phase space, σe =25 keV,and σt =10.7 ps [45, 46]. Due to their large lengththe bunches acquire an energy modulation from therf curvature of the accelerating field (see figure 5.6)

4E = E0 (cos(ωrf t− φ0)− cos(φ0)) . (5.3)

In the bunch compressor the particles travel on dif-ferent trajectories depending on their energy. Thisinduces a longitudinal dispersion

4t =lαc

cE , (5.4)

which, in combination with the off-crest accelera-tion in the first module, leads to a shortening of thebunches (see figure 5.7).

With shorter bunch length the effects of coher-ent synchrotron radiation have to be taken into ac-count. Due to the curved trajectory in the dipolemagnets the synchrotron radiation moves ahead ofthe source particle. The influence of CSR on thebeam energy can be expressed by [42]

dEdz

= − qe

2πε0(3c)1/3ρ2/3

t∫−∞

dt′

(t− t′)1/3

∂λ(t′)

∂t′, (5.5)

with λ the normalized line charge density of thebunch charge. The main contribution to the CSReffects has to be expected from the third and fourthdipole of the bunch compressor and the spectrom-eter dipole. The bunches have been treated as ifthey were already fully compressed when enteringthe third dipole of the bunch compressor. The effectof CSR in the spectrometer dipole cannot fully be

−5

0

5

∆E [M

eV]

−20 −10 0 10 20−1

−0.5

0

0.5

1

Time [ps]

Wak

e [k

V]

−20 −10 0 10 20Time [ps]

−5

0

5

∆E [M

eV]

−20 −10 0 10 20−1

−0.5

0

0.5

1

Time [ps]

Wak

e [k

V]

−20 −10 0 10 20Time [ps]

Figure 5.6: Simulated phase space at the entrance of the bunch compressor. In the figures 5.6-5.9 the left picture correspondsto a bunch at maximum compression and the right picture to a more moderate compression used in the wake field experiment.At the entrance of the bunch compressor only the energy profiles differ, the time profiles are equal.

seen in the spectrometer because the electrons havealready passed a certain distance in the dipole be-fore they change their energy. The effective lengthhas been approximated by half of the real length.The corresponding simulation results are shown infigure 5.8.

In the second module the bunches are normallyaccelerated on crest. During the wake field ex-periment the rf phase of the second module waschanged in two ways. For the energy profile mea-surements the phase was adjusted for maximumcontrast of the peak structure, see chapter 7 formore details. A phase of 14 turned out to be themost suitable value. During the tomography mea-surements the phase was shifted in sequences be-tween −35 . . . 35. The phase shifter would haveallowed ±45 but the beam transport becomes lessefficient towards larger off crest phases.

The surface roughness of the collimator beampipes is of the same order of magnitude as theroughness in the test pipes of the wake field ex-periment (5 µm on the spoilers, 10 µm in the testpipes). In both cases a harmonic wake is expected

4xe = W0 cos(ω0xt) exp(−xt/τ). (5.6)

Because the resonant frequency of the wake is apriori unknown it has been derived from the exper-imental data for both the collimator and the wakefield experiment. Similarly the time constants ofthe wakes have been adjusted to reproduce the ex-perimental data. The figure 5.9 shows the simulatedphase space after passage of the 4 mm reference piperesp. the 4 mm sandblasted pipe. The summary ofparameters can be found at the end of chapter 7.

32

−5

0

5

∆E [M

eV]

−10 −5 0 5Time [ps]

−20 −10 0 10 20−1

−0.5

0

0.5

1

Time [ps]

Wak

e [k

V]

−5

0

5

∆E [M

eV]

−10 −5 0 5Time [ps]

−20 −10 0 10 20−1

−0.5

0

0.5

1

Time [ps]

Wak

e [k

V]

Figure 5.7: Simulated phase space at the exit of the bunch compressor. No wake or coherent synchrotron radiation has beentaken into account.

−5

0

5

∆E [M

eV]

−10 −5 0 5Time [ps]

−20 −10 0 10 20

−150

−100

−50

0

50

Time [ps]

Wak

e [k

V]

−5

0

5∆E

[MeV

]

−10 −5 0 5Time [ps]

−20 −10 0 10 20

−80

−60

−40

−20

0

20

40

Time [ps]

Wak

e [k

V]

Figure 5.8: Simulated phase space at the exit of the bunch compressor with coherent synchrotron radiation taken into account.

−2

−1

0

1

2

3

4

5

∆E [M

eV]

−10 −5 0 5Time [ps]

−15 −10 −5 0 5−300

−200

−100

0

Time [ps]

Wak

e [k

V]

−2

−1

0

1

2

3

4

5

∆E [M

eV]

−10 −5 0 5Time [ps]

−15 −10 −5 0 5

−400

−200

0

200

Time [ps]

Wak

e [k

V]

Figure 5.9: Simulated phase space of a bunch passing the 4 mm reference pipe of the wake field experiment (top) and passingthe 4 mm sandblasted pipe (bottom).

33

Chapter 6

Experimental Tomography

6.1 Setup

The main accelerator components for the longitu-dinal phase space tomography are an accelerationmodule and the energy spectrometer with a straightsection in between. By changing the rf phase in themodule the longitudinal phase space of the bunchesis deformed, and using the spectrometer the re-sulting energy profiles can be measured. Providedthat there are no energy dependent effects on thebunches along this section, the phase space distri-bution in front of the spectrometer can be recon-structed.

Besides nonlinear corrections in this setup only ashearing of the phase space can be achieved. Formaximum resolution the shearing has to be maxi-mized. In the TTF the phase offset is limited by thefact that the two acceleration modules are drivenby a single klystron. The phase and amplitude ofthe rf in the first module are kept constant whilethe phase of the second module can be shifted bya phaseshifter in the waveguide. The adjustmentrange of this phase shifter is 90. When the rf phaseis shifted by ϕoff from the crest the energy is in firstorder modulated according to

dE = E0ω0 sin(ϕoff )dt. (6.1)

The maximum energy shift of two points in thephase space against each other is then

4E = E0ω0 [sin(ϕmin + 90)− sin(ϕmin)]4t,

=√

2E0ω0 sin(45 − ϕmin)4t, (6.2)

with ϕmin the lowest phase of the rf and4t the sep-aration of the two points in time. This is optimumfor a range of the phase between ±45 with a totalenergy shift of 4E/4t = E0ω0

√2 ≈ 570 keV/ps.

A second constraint is the beam transport alongthe accelerator. Due to the phase shift the energy ofthe beam changes. At φoff = 45 the acceleratingvoltage in the second module changes by 30% andthe total electron energy changes by 15%. Withoutan independent klystron for the second acceleration

x position [pixel]0 100 200 300 400 500

rela

tive

sens

itivi

ty

0

0.2

0.4

0.6

0.8

1

1.2Sensitivity

Figure 6.1: Sensitivity of the diagnostic screen in the energyspectrometer versus position on the screen. The parametersof the lens system have to be deduced from the observedsignals because a zoom lens was used. The focal length wasapproximately 90 mm, the visible width of the screen was37 mm. The aperture diameter was 16 mm corresponding toan f-number of 5.6.

module this cannot be compensated by changingthe amplitude of the rf-field. The focusing onto thediagnostic screen has to be adjusted for the differ-ent energies to achieve the optimum spectrometerresolution. With additional adaption of the opticsthe beam transport can be optimized, but duringthe measurements this turned out unnecessary.

Behind the bunch compressor the energy distri-bution of the bunches has a total width of ap-proximately 10 MeV. With a total bunch length of≈ 10 ps and an additional modulation of 570 keV/psfor the tomography the full acceptance of the mea-surement system has to be at least 15 MeV. Thedispersion at the position of the screen was 1 m,the total beam energy is 235 MeV.

To allow for a range of 15 MeV the OTR screenwould have to cover 65 mm, whereas the installedscreen is only 40 mm wide. At the same time the fo-cusing of the beam would vary considerably acrossthe screen yielding a poor energy resolution of thespectrometer. To overcome these difficulties themeasurement was performed such that a numberof images was taken for each projection. There-fore two quadrupoles in front of the dipole and the

34

dipole were scanned simultaneously in a predefinedsequence.

A schematic of the spectrometer section is foundin figure 6.7. The focusing is adapted with thequadrupoles Q2 and Q3, while Q4 and Q5 are ad-justed for zero field. The beam position monitorsBPM1 and BPM2 are used to find a reproducibleorbit as input to the spectrometer.

During the operation of the TTF some jitter ofthe beam energy has been observed due to the miss-ing rf control of module 2. Therefore the horizontaloffsets between the images have to be adjusted indi-vidually. To ease this operation an overlap betweensubsequent images is desirable. Thus a large activearea of the diagnostic screen is required. The activearea of the screen is determined by the adjustmentand acceptance of the optical system.

The screen is observed by a camera equipped witha zoom lens. To protect the camera from highenergy photons and electrons it was necessary toshield it with lead. Therefore the direct view onthe accelerator was blocked, and the light was de-flected by a mirror onto the camera. The mirrorhad a diameter of 75 mm in order not to restrictthe acceptance of the system. The distance of thecamera lens from the screen was 450 mm, its focallength approximately 90 mm.

The detected optical transition radiation is verydirectional. The maximum intensity is found atan opening angle 1/γ. Due to the narrow open-ing angle of the transition radiation the active areaof the screen equals the effective aperture of thelens system. The optical system has to be adjustedsuch that the active area is centered on the screen.In absence of an alignment system the adjustmentonly could be verified using OTR itself. The cam-era could be moved horizontally via remote control.The vertical adjustment was done manually.

In the spectrometer dipole synchrotron radiationis produced. The OTR screen acts as a mirror forthe synchrotron radiation and deflects it into thecamera. Since its origin is far out of the focal planeit is not focused but appears as a brightening of thebackground. Since the origin of the synchrotron ra-diation is distributed along the orbit of the electronsthe radiation cannot be subtracted directly. To sup-press it the camera was equipped with a polarizingfilter transmitting the vertical polarization only andhence suppressing the synchrotron radiation whichis polarized horizontally.

Figure 6.1 shows the experimentally determinedrelative sensitivity of the screen. The relative sen-sitivity is measured by shifting the beam in smallsteps across the screen. With this the spectrom-eter was calibrated at the same time. The activewidth of the screen corresponds to 370 pixels onthe CCD. The calibration of the spectrometer was

171 pixels per 1% energy shift. Thus the energy ac-ceptance was approximately 2%. From the sensitiv-ity measurement it was deduced that the apertureof the camera lens was 16 mm corresponding to anf-number of 5.6 at a focal length of 90 mm. Mean-while a high-resolution macro lens was obtainedwith an f-number of 2.8. It was used to measurethe energy profiles shown in figure 7.4. The cor-rect horizontal shift of the images while scanningthe spectrometer can be found by calculating thecross-correlation of two subsequent images. Themaximum of the correlation function is found atthe correct offset. It is sufficient to apply the cross-correlation to the projection of the image. Afterfinding the correct offset the images were scaled tocorrect for intensity fluctuations.

The signal to noise ratio in figure 6.1 is poor. Asubstantial contribution to the noise in the energyprofiles comes from x-rays hitting single pixels of theCCD. Obviously the shielding of the camera was in-sufficient. By eliminating isolated pixels from theimage the noise was drastically reduced. The elim-ination of isolated pixels was done parallel to thedirection of projection. In this way no degradationof the resolution is expected.

6.2 Measurements

The measurements were done with the bunch com-pressor and acceleration module 1 adjusted for max-imum compression of the bunches. This is achievedby setting the phase of the rf field to −12. Thecorrect setting is verified with a far infrared de-tector measuring coherent transition radiation fromthe bunches.

In the TTF an unexpected phenomenon is ob-served: When the accelerator is adjusted for veryshort bunch lengths the energy spectrum of thebunches is split into several peaks. The investiga-tion of this effect was the first motivation to per-form the longitudinal tomography. Figure 6.2 showsthe result of the first tomography to study this ef-fect. Note that this measurement was done beforethe camera alignment was optimized and withoutscanning the spectrometer. Therefore the accep-tance and resolution was lower than in later mea-surements.

Simultaneously with the tomography the longi-tudinal profile was measured by interferometry ofthe coherent transition radiation.The interferome-ter has been described in [15]. Since then the setuphas been considerably improved by replacing thepyroelectric detectors with Golay-cell detectors. Inthis way a smooth response of the interferometer isachieved. In the interferometer the autocorrelationfunction of the CTR is obtained. The Fourier trans-

35

-2

-1

0

1

2

3

4

5

∆ E

[MeV

]

-6 -4 -2 0 2 4

20

30

40

50

60

70

80

Time [ps]

Pop

ulat

ion

[arb

.uni

ts]

0 50 100Population [arb.units]

Figure 6.2: Longitudinal phase space of a TTF bunch atmaximum compression [19]. The lines indicate the sensitivearea. By better aligning the camera and scanning the spec-trometer the sensitive area could be enlarged substantially(compare chapter 7).

form of the autocorrelation function is the square ofthe form factor of the bunch (see figure 6.4). There-fore the interferometer is principally unable to de-termine the phase of the bunch form factor. Toreconstruct the longitudinal profile from the far in-frared spectrum of the CTR the Kramers-Kronigrelation is used to compute the phase as a functionof frequency. Due to the symmetry of the autocorre-lation function it is impossible to decide which is thehead or the tail of the reconstructed bunch profile.This has to be deduced from additional information(simulation or measurement). Below approximately60 GHz the interferometer shows a cut-off of the in-tensities, this has to be taken into account whencomparing the result of the interferometry and thetomography (figure 6.3). An considerable enhance-ment of the low frequency intensities is required ascan be seen in figure 6.4.

Meanwhile a streak camera with a temporal res-olution of 200 fs is available. It has been usedto measure the duration of the synchrotron radi-ation pulse created in the spectrometer dipole. Theresults are shown in figure 6.5. The tomographyyields a shorter tail than the streak camera. Inpart this may be explained by different beam con-ditions in the two measurements, which were done

−10 −8 −6 −4 −2 0−0.2

0

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time [ps]

Cha

rge

Dis

trib

utio

n (C

urre

nt)

[a.u

.]

(a)

(b)

(c)

Figure 6.3: Longitudinal profile of bunches in the TTF mea-sured with tomography (a) and interferometry (b) [51]. Thedata for the two methods was taken in the same week inApril 2000. The fully reconstructed phase space is shownin figure 6.2. The interferometer data suffer from a low fre-quency cut-off which cuts away the tail of the bunch as canbe seen on the lowest curve. For a better comparison lowfrequency components have been extrapolated (c).

with a delay of two years. More important maybe the limited acceptance discussed above. Notethat this is a problem which has been fixed by themeasures described above (see figure 6.6). Unfor-tunately since then no measurement has been doneat maximum compression of the bunches, thereforea bunch profile at medium compression measuredduring the wake field experiment is shown.

36

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0 0.5 1 1.510

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inte

nsity

[arb

. uni

ts]

Figure 6.4: Autocorrelation function and power spectrumof the coherent transition radiation produced by a bunch atmaximum compression. To obtain the reconstructed bunchshape shown in figure 6.3 low frequency components wereenhanced (dashed-dotted line).

Tomography

Average

Figure 6.5: Longitudinal profile of bunches in the TTF mea-sured with a streak camera [46] in March 2002. In theseplots the head of the bunch is on the left side. The upperplot (A) shows single shot profiles, the lower plot (B) showsthe average and the result from tomography. The shortertail measured by tomography may in part be due to differ-ent beam conditions or due to reduced acceptance.

time [ps]-10 -8 -6 -4 -2 0 2 4 6

curr

ent [

arb.

uni

ts]

0

1

2

3

4

5

Figure 6.6: Longitudinal profile of bunches in the TTFduring the wake field experiment, measured with tomogra-phy. The corresponding two dimensional phase space canbe found in figure 7.6. The pedestal between approximatelyt = 2 . . . 4 ps can be explained by the poor time resolutionfor structures parallel to the energy axis (see figure 4.7 forreference).

T

T

T

T

T

T

l = 1226.9 mmρ = - 3515 mmθ = - 20°δ = 1° 14’ 30’’2G = 66.3 mm

455

3629.6

720

265.4

87310.488030.4 91660

92115

BPM3

SpectrometerDipole B1

94075

9622097340

2145

1120

455.2

93000

(94062.5)

(96207.5) (97327.5)

85830.4

1480 1030

700

Q5(Q1EXP3)

BPM1 BPM2Q4Q1 +H1

Q2 Q3 +H2 W

W418Toroid T7

ToroidT8

Q6(Q2EXP3)

1071

4485

Experimental area 1 (EXP1) Experimental area 3 (EXP3)Exitwindow

PM2EXP3PM1EXP3

PM1EXP1

Beam Dump

10 m0 m

TTF optics at experimental area 1 and 3 (E = 235 MeV)

βyβx

βy

βx

Dx

Figure 6.7: Beam line close to the spectrometer dipole andthe optical functions [45]. The solid lines show the horizon-tal and vertical beta functions and dispersion during stan-dard operation of the linac, when the beam cross sectionwas made especially large at the exit window. For spectrom-eter operation the optics shown by dashed lines can be used.The gradient of quadrupole Q3 has to be changed by 20%to switch the optics. During the tomography measurementsthe quadrupoles Q2 and Q3 are used to adapt for differentenergies.

37

Chapter 7

The Wake Field Experiment

For the surface roughness wake field experimenttest beam pipes with a known roughness of the in-ner surface have been prepared. The wakes excitedby the test pipes have been analyzed using two dif-ferent observables, the energy modulation of thebeam and the radiated electromagnetic fields. Fora clear separation of the surface roughness wakesfrom other effects such as coherent synchrotron ra-diation in the bunch compressor and undulator orwake field effects in other elements of the accelera-tor a difference measurement was performed.

A special ultra-high vacuum chamber [29] wasconstructed to house an ensemble of beam pipeswith radii between 3 and 5 mm and with dif-ferent surface preparations (smooth, sandblasted,grooved, see table 7.1). The chamber was mountedbehind the undulator. Via a linear movement eachbeam pipe could be positioned on the beam axis.Due to the limited space in the accelerator thelength of the beam pipes was limited to 855 mm.To increase the effects the test pipes were preparedwith an enhanced surface roughness in compari-son with the standard undulator vacuum chambers.The pipes were composed of two half cylinders ma-chined into two flat aluminum plates. In this way acontrolled surface preparation by sand-blasting orgrooving was possible. The surface roughness hasbeen measured with a tracer type measuring de-vice featuring a resolution of 0.02 µm. The wakefields created by the two narrow longitudinal gapsare known to be negligible [10].

Seven beam pipes were prepared. One beam pipewith inner radius 8 mm was foreseen for the nominallinac operation. Its aperture was large enough to al-low for transmission of beam pulses with the max-imum possible average power. Three beam pipeswith an inner radius of 4 mm have been prepared.The first of these has a smooth surface and servedas a reference for all measurements. After machin-ing the respective half pipes their inner surface werecleaned with NaOH. Their surface profile was mea-sured to have an rms height of 1.6 µm. The secondpair of half pipes has been sandblasted after ma-

chining and finally cleaned with NaOH. Their sur-face profile was measured to have an rms height of10 µm. The third pair of half pipes was treatedby sparking erosion to achieve regular grooves onthe surface with a period of 150 µm and a depthof 60 µm. There were two additional beam pipestreated by sand-blasting with radii of 3 mm and5 mm, and one beam pipe with grooves on the sur-face had a radius of 5 mm.

7.1 Energy profiles

Figure 7.1 shows the energy profiles as obtainedwhen the beam passes the reference pipe and asandblasted pipe, respectively. The acceleratorwas adjusted for a moderate compression of thebunches, 6.5 off-crest in module one instead of12 as required for optimal compression. In thisway the bunches were shaped such that they hada steep rising edge (∼ 100 fs) and a long, slowlydecaying tail (∼ 10 ps). Then a 14 off-crest accel-eration in the second module generates a correlatedenergy-position distribution in the tail of the bunch(see figure 7.2). The synchronous mode wake fields,which are mainly produced by the sharp front peakof the bunch, can then be observed via the imposedenergy modulation in the long tail. There is someresemblance to the pump-and-probe technique inlaser physics. Note that coherent synchrotron ra-diation in the bunch compressor, as well as wakefields caused by resistive walls and cross sectionalchanges, act mainly on the sharp front peak of thebunch but have little influence on the long tail.

The figure 7.1 shows only the tail of the bunches,the head is left of the border of the plot due toits lower energy. A clear difference of the profilescan be observed. When passing the smooth refer-ence pipe the bunches show a wide and smooth en-ergy spectrum. Superimposed is a slight structurewhich possibly can be assigned to wake field effectsupstream of the experiment. This question will bediscussed later in more detail.

The solid curve shows the energy distribution

38

pipe number 1 2 3 4 5 6 7radius 8 mm 4 mm 5 mm 4 mm 3 mm 5 mm 4 mmpreparation dummy reference sandbl. sandbl. sandbl. grooves groovesδ rms 1.4 µm 1.4 µm 10 µm 10 um 10 um (60 µm) (60 µm)

Table 7.1: Parameters of the beam tubes. The parameter δ is the rms depth of the roughness. For the grooved pipes the depthof the grooves is printed instead of the rms height.

−0.5 0 0.5 1 1.5 2 2.5 3

0.1

0.2

0.3

0.4

0.5

0.6

∆ E [MeV]

Pop

ulat

ion

[arb

.uni

ts]

Figure 7.1: Difference measurement of smooth and roughbeam pipes [20]. The dashed curve shows the energy pro-file after passing the reference pipe, whereas the solid curveshows the energy profile when passing a pipe of same geom-etry but sandblasted surface.

when the beam has passed the sandblasted beampipe of the same radius r = 4 mm. In this casea regular peak structure is visible which can be as-signed to a harmonic wake potential: each peak canbe identified with a zero crossing of the wake poten-tial with negative slope (see figure 7.2). It shouldbe emphasized that the only difference between thetwo cases is the different surface roughness of thetwo pipes. Figure 7.4 demonstrates that the regu-lar peak structure becomes much more pronouncedwhen the rough pipe of 3 mm radius is inserted.

A precise determination of the time structure ofthe distribution is achieved by varying the rf phaseof the second acceleration module. This has no im-pact on the longitudinal bunch profile nor on thewake fields. By measuring the resulting changes inthe energy separation of the peaks it is possible toresolve their separation in time without making anyassumptions about the initial energy distribution.The method works as follows. Consider two peaksin the energy profile which are separated in energyby Esep and in time by τ . When the rf phase φ inmodule 2 is changed by ∆φ the change in separationenergy is

∆Esep = ωτEmodule (sin(φ + ∆φ)− sin(φ)) (7.1)

with Emodule being the maximum energy gain inthe module and ω the rf angular frequency. Fromthe measured values ∆φ and ∆Esep the time sep-

−10 −5 0 5−150

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Wak

e [k

V]

−10 −5 0 5Time [ps]

−3

−2

−1

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1

2

3

4

∆E [M

eV]

Figure 7.2: Simulation of a bunch in longitudinal phase spacein the presence of a synchronous mode wake field. The up-per left part shows the bunch in phase space, the lower leftthe projection onto the time axis, and the upper right theprojection onto the energy axis. The lower right plot showsthe harmonic wake field. The periodic energy shift togetherwith the time-energy correlation generates the peaks in theenergy distribution.

aration τ of the two peaks can be derived with anaccuracy of better than 120 fs. Then fw = 1/τ isthe frequency of the harmonic wake.

Using this method it was verified that the peaksseen behind the rough test pipes (figures 7.1 and7.4) have indeed equidistant spacing in time, im-plying that they are caused by a harmonic modu-lation of the particle energies. The experimentallydetermined wake frequencies for the different roughtest pipes are summarized in table 7.1 and plot-ted in figure 7.3 as a function of the pipe radius.Good agreement with the 1/

√r behaviour of equa-

tion 2.36 is found. The fact that the time separationof the peaks changes with the pipe radius rules outthe vague possibility that the observed regular peakmight be due to an initial modulation of the bunchwhich is only enhanced by the rough pipes.

The observed harmonic wake frequencies agreewith the dielectric layer model prediction for a di-electric constant εeff ≈ 1.55 while the numericalcalculations in ref. [52] prefer εeff ≈ 2, corre-sponding to ≈ 20% lower frequencies. In the surfaceroughness model of ref. [48] higher wake frequen-cies are predicted (εeff ≈ 1.27) but it should beremarked that the small angle approximation forthe irregularities, used in this paper, is not fully

39

justified for the sandblasted beam pipes of the ex-periment.

2 2.5 3 3.5 4 4.5 5 5.5 6300

400

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600

700

800

900

1000

1100

radius r [mm]

mod

e fr

eque

ncy

f W [G

Hz]

Novokhatski et al

Stupakov et al

Dohlus

Figure 7.3: The frequency of the synchronous mode plottedversus the radius of the beam pipe.

It has been shown that the frequencies of theharmonic wakes can be determined in a model-independent way, applying the method describedabove. Their strengths can be derived with a longi-tudinal phase space tomography, which is an ex-tension of this method. For an estimate of thewake field amplitude a numerical simulation of thewhole experiment is carried out with the programdescribed in chapter 5. The wake field effect inthe rough test pipe is imposed as a damped har-monic wave using the frequency determined above.The simulation model yields the following valuesfor the maximum energy shift which an electron inthe tail of the bunch experiences during its pas-sage through one of the 800 mm long roughenedtest pipes: 39 keV for r = 5 mm, 60 keV for r = 4mm, and 105 keV for r = 3 mm. The amplitudesof the wake functions required to achieve these val-ues agree well with the predictions Z0c/(πb2) fromthe wake field models. The damping constants are4.8 ps for r = 5 mm, 4.0 ps for r = 4 mm, and3.4 ps for r = 3 mm.

To account for possible surface-roughness wakesin the collimator and undulator section, which maybe the origin of the peak structure observed behindthe smooth reference test pipe, another dielectric-layer wake is used in the simulation whose frequencyand amplitude is adjusted to yield a reasonable de-scription of the energy profile measured with thereference pipe. Simulation parameters like the ini-tial charge distribution and the rf phases in modules1 and 2 are allowed to vary within the experimen-tal uncertainty. The dashed curves in figure 7.4 arethe predictions of the model simulation for an opti-mized parameter set. The main parameters are anaccelerating voltage of 107 MV in both modules, an

−0.5 0 0.5 1 1.5 2 2.5 3 3.5 0

0.2

0.4

∆E [MeV]

d) 4 mm (ref)

c) 5 mm

0

0.2

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Po

pu

latio

n [a

rb. u

nits]

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a) 3 mm

Figure 7.4: Energy profiles in the tail of the bunch measuredbehind different test tubes.a), b), c): sandblasted tubes of3, 4, resp. 5 mm radius, d): smooth reference tube of 4 mmradius. The solid curves show the profiles measured withthe spectrometer at the end of the linac. The dashed curvesshow the simulated distributions. The simulation includesthe off-crest acceleration, bunch compression, and surfaceroughness wake fields generated in the test beam pipes andin the collimator upstream of the setup. The spectrometerresolution is also taken into account.

40

off-crest phase of −6 in the first, and −14 in thesecond module. With the latter the contrast in theenergy profiles behind the 3 mm sandblasted testpipe was optimized. The agreement with the mea-sured profiles is quite satisfactory indicating thatthe basic physics processes are well understood.

It should be noted that the determination of thewake frequency, the damping constants and themaximum energy shift is independent of any spe-cific wake field model.

7.2 Tomography

For a more precise understanding of the wake fieldeffects the longitudinal phase space of the buncheshas been analyzed via tomography (see chapter 4).During the tomography measurements the setupwas chosen slightly different from the one describedin the previous section. Instead of setting the phaseof the first acceleration module lower than requiredfor maximum compression here it is set higher.Again the result is a moderately compressed bunchbut the tail now lies at low energies. The peaksin the energy distribution correspond to the pointswhere the wake potential crosses zero with positivegradient. Due to the limited angle of observationin the tomography they appear as peaks in the re-constructed phase space as well. By determiningthe distance of the peaks in time domain the fre-quencies of the wakes can be measured. The errorof this time measurement is smaller than 120 fs. Itis the same as for the energy profile measurementdescribed in the last section. In figure 7.5 the recon-structed phase space is shown as it appears whenthe beam passes the 4 mm reference beam pipe.In figure 7.7 the phase space with 1 nC is shownagain with the contours of a simulated bunch plot-ted on top. Obviously there is a regular structureon the bunch. In particular note the enhanced den-sity at (0 ps, 2.4 MeV). At energie deviations below0 MeV the dark spots correspond to zero crossingsof the wake potential with positive slope. Above2 MeV the dark spots are found where the wakepotential crosses zero with negative slope. The fre-quency of the corresponding harmonic wake fieldis determined to be 830 ± 60 GHz. Having estab-lished the main features of surface roughness wakefields in the last section, it appears that the regularstructure on the bunch most probably is caused bysurface roughness wake fields. The reference pipeitself is ruled out as a source for this wake: Thesurface structure is too smooth and changing to apipe of 8 mm radius does not change the structureon the energy profile.

Owing to their narrow aperture the undulatorvacuum chamber or the collimator section might

Figure 7.5: Reconstructed phase space behind the referencebeam pipe. The left plot corresponds to a bunch charge of0.75 nC, the right plot to a charge of 1.0 nC. At energiedeviations below 0 MeV the dark spots correspond to zerocrossings of the wake potential with positive slope. Above2 MeV the dark spots are found where the wake potentialcrosses zero with negative slope. Between 0 and 2 MeV thedistribution is not reconstructed correctly (see chapter 4).From the distance between the spots a wake frequency of830±60 GHz can be deduced. With increasing bunch chargethe wake get stronger. This can be seen in the right by thestronger tilt of the dark spots.

have caused the wake in question. The sur-face roughness of the undulator chamber is .0.7µm (rms) at a pipe radius of 4.5 mm yieldingwake frequencies of 1.6-5 THz depending on themodel. Should the undulator by some unknownreason excite wakes with a resonance frequency of830 GHz, the wake field would have to be muchstronger than observed because the effect would beintegrated along the large length of 15 m. Thereforethe undulator can be ruled out as a source of thisstructure.

In chapter 5 it has been shown that a harmonicwake at 830 GHz can be explained by surface rough-ness wakes produced in the collimator. Thereforethe collimator is most likely the origin of the reg-ular structures observed in the longitudinal phasespace.

Figure 7.6 shows the reconstructed phases spaceof a bunch having passed a sandblasted beam pipeof 4 mm radius (equal to the reference pipe). Infigure 7.8 it is shown again together with the con-tours of a simulated phase space distribution. Theoverall form of the distribution is that of a sickle asit is expected from simulations of the beam trans-port through the bunch compressor (see chapter 5).Superimposed is an energy modulation of the elec-trons. Within the limitations of the tomographyit can be identified as being caused by a harmonicwave (see figure 4.7 for comparison). The ampli-tude of the modulation is much larger than in figure7.5. The energy modulation is larger than the en-ergy difference between the zero crossings, leading

41

Figure 7.6: Reconstructed phase space having passed thesandblasted pipe with radius b=4 mm. At energie deviationsbelow 0 MeV the dark spots correspond to zero crossings ofthe wake potential with positive slope. Above 2 MeV thedark spots are found where the wake potential crosses zerowith negative slope. Between 0 and 2 MeV the distributionis not reconstructed correctly (see chapter 4). From the dis-tance between the spots a wake frequency of 575 ± 30 GHzcan be deduced.

Figure 7.7: Tomographic reconstruction of the bunch havingpassed the reference pipe. The distribution from figure 7.5has been plotted together with the contours of the simulateddistribution from figure 5.9.

to a double peak structure where the peaks belongto the maximum respectively minimum of the sinefunction. The frequency of the modulating wavecan be determined from the distance of the pointsof higher density in the reconstruction. It is foundto be 575 ± 30 GHz in good agreement with othermeasurements described in the previous and follow-ing sections.

7.3 Microwave Measurements

Surface roughness wakes are special waveguide-modes propagating in the beam pipe. At the exitof the test pipes they are radiated into the largervacuum chamber of the accelerator. At the TTF

Figure 7.8: Tomographic reconstruction of the bunch behindthe 4 mm sandblasted pipe. The distribution from figure 7.6has been plotted together with the contours of the simulateddistribution from figure 5.9.

linac a diffraction radiation screen was used to de-flect the radiated fields through a quartz windowout of the vacuum. In a far infrared interferometerthe wake fields then were analyzed. A detailed de-scription of the interferometer can be found in [15].The diffraction radiator screen was divided into twoparts giving the opportunity to open a slit of 10 mmwidth. At the exit of the test beam pipes and at thescreen also coherent diffraction radiation(CDR) re-spectively transition radiation (CTR) is producedyielding radiation in the same frequency range asthe wakes. Therefore the analyzed spectrum con-tains a wide, continuous spectrum from the CDRresp. CTR and single spectral lines from the har-monic wakes.

0 200 400 600 800 100010

−2

10−1

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frequency [GHz]

pow

er s

pect

rum

[arb

. uni

ts]

(a)

(b)

Figure 7.9: Far infrared spectrum with the 4 mm refer-ence pipe (a) and the 4 mm sandblasted pipe (b). Around560 GHz the two spectra clearly deviate. At 550 GHz and750 GHz there are some absorption lines due to water vapour(see figure 7.10). The spectra were taken with the screenopened by 10 mm.

42

To separate the effects the spectra from the ref-erence pipe and the roughened pipes can be com-pared. The figure 7.9 shows the comparison of twospectra obtained with pipes of equal radius of 4 mm.One spectrum corresponds to the reference pipewith smooth surface, the second to a sandblastedbeam pipe with a rms roughness of 10µm. Around550 GHz the power spectrum from the sandblastedpipe shows a clear enhancement. A problem is thatat 560 GHz a narrow gap occurs caused by watervapour absorption. See the figure 7.10 for reference.

200 300 400 500 600 700 800 900 10000

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tran

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Figure 7.10: Atmospheric transmission at far infrared fre-quencies. The transmission has been simulated by the pro-gram atm cso [28, 9]. It plots the zenith atmospheric trans-mission on the summit of Mount Mauna Kea in Hawaii. Theamount of water vapour in the air can be given as inputparameter by quoting the effective column height. For a col-umn height of 185 µm the program delivers similar resultsto [35]. The plot shown here has been obtained for a col-umn height of 30 µm which was estimated to be present ata relative humidity of 60%, a temperature of 22C, normalpressure, and a path length of 2.5 m in the interferometer.

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Figure 7.11: Radiated wake fields from three different sand-blasted beam pipes. The radii of the pipes are (a) 3 mm,(b) 4 mm, and (c) 5 mm. The spectra were taken with thescreen closed. Additionally there is a water absorption lineat 560 GHz. For the reference spectrum two different expo-nential fits were used as can be seen in the left pictures.

−15 −10 −5 0 5 10 15

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[nor

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100 200 300 400 500 600 700 800 900 10000

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Inte

nsity

[nor

m.]

frequency [GHz]

Figure 7.12: Autocorrelation function and correspondingspectrum with the r = 4 mm grooved beam pipe inserted.The corresponding functions from the reference pipe areshown in the same plots with the amplitude scaled such,that the spectra outside the resonance coincide.

In figure 7.11 the spectral lines of three differentsandblasted pipes are shown. In this measurementa worse signal to noise ratio was obtained. This isbecause the measurements were done with the cen-tral slit of the screen opened to 10 mm. In this waythe spectral lines of the wakes should be enhancedin comparison to the CDR spectrum. But due tothe poor signal to noise ratio the reference spectrumdelivers no reliable data at the relevant frequencies.Instead an exponential fit to the spectra from fig-ure 7.9 served as a reference. Two different fits wereused, one only based on the data close to 500 GHz,the other one based on the data from 200 GHz to600 GHz. In figure 7.11 the fits are shown in thecorresponding left graphs.

At first sight the relative strength of the spectrallines is opposite to what can be expected from cal-culation and from the measurements of the beamenergy. Without losses the pulse energy W shoulddrop with the beam pipe radius like W ∝ 1/b2 andincrease with the frequency like W ∝ f4 whereasthe spectral lines tend to be slightly weaker forhigher frequencies. In the lossless case the pulselength τ would be decreasing with the radius τ ∝1/b while from the Dohlus model and the measure-ments presented in table 7.2 a different behaviouris found: τ ∝ b. If this is taken into considera-tion the radius dependence is compensated, so theenergy contained in the wake field pulse should beindependent of b. Within the errors this is found infigure 7.11. The weakness of the middle line mostprobably is caused by the atmospheric absorption.

Much clearer signals have been observed from thegrooved beam pipes. In figure 7.12 an autocorrela-tion function and the corresponding spectrum areshown for the 4 mm test pipe with grooves on the

43

0 10 20 30 40 50

−0.5

0

0.5

1

offset [mm]

Inte

nsity

[nor

m.]

offset [mm]

freq

uenc

y [G

Hz]

0 10 20 30 40 50

100200300400

Figure 7.13: Autocorrelation function with the 4 mm groovedbeam pipe inserted. In the lower plot the spectrum is plottedin columns taking into account the autocorrelation functionbetween the corresponding point on the x-axis and the end.

surface. A strong harmonic modulation of the au-tocorrelation function is observed. The resonancefrequencies are found at 200± 10 GHz for b=4 mmand 177±10 GHz for b=5 mm. At the central peakthe autocorrelation function with wake exceeds thatwithout wake by 50 %. This means that the totalenergy in the wake field radiation pulse is approxi-mately half as much as in the CTR pulse.

Given this ratio the autocorrelation is dominatedby the front peak of the CTR1 scanning the har-monic function of the wake field radiation. In figure7.13 the autocorrelation function is plotted for pos-itive offset in one interferometer arm over a largerange. The corresponding spectra are plotted verti-cally taking into account only part of the autocor-relation function. Therefore the spectra were calcu-lated between a starting offset z1 and the maximumoffset zmax = 55 mm.

S(k, z1) =1

zmax − z1

zmax∫z1

A(z) exp(ikz)dz, (7.2)

with A the autocorrelation function. Moving fromzmin < z1 < zmax the relevant part is further andfurther cut. At approximately z1 ≈ 26 mm theintensity of the 200 GHz harmonic wake starts todrop and vanishes at 30 mm. From the group veloc-ity derived in chapter 2 a pulse length of L ≈ 23 mm=77 ps is expected, which is in good agreement withthis observation.

7.4 Summary

The results of the wake field experiment are listedin table 7.2. The figure 7.14 shows the correspond-

1excited by the front peak of the bunch

0 2 4 6 8 10 120

1

2

3

4

5

6

7

8

9

rms roughness [µm]

surf

ace

indu

ctan

ce [p

H]

ε=2ε=1.55

Dohluscoll.

wake exp.

Figure 7.14: Inductance versus the height of the surfaceroughness. The experimental data from the wake field ex-periment and collimator are shown. The behaviour of theinductance derived from the linear boundary approximation(Dohlus) are shown for roughness spectra similar to (a) thosein the collimator (b) those in the wake field experiment. Thedielectric layer model with ε = 2 and ε = 1.55 (Novokhatskiet al.) is shown.

ing surface inductances for the pipes with stochas-tic surface structure. The pipes with a regular and60 µm deep structures are well described by thedielectric layer model with ε = 2. This supportsthe treatment in section 2.3.1 whereas [2] deviatesby 20%. The harmonic wakes in the sandblastedpipes can be described with an ε ≈ 1.6 yielding fre-quencies right in the middle between the predictedε = 2 and the calculations according to Dohlus. Forthe smaller surface roughness in the collimator theDohlus model fits within 15%.

The amplitudes of the wake functions have thesame value Z0c/(pib2) in all wake models and arein good agreement with the observations. The timeconstants agree best with the predictions by a com-bination of the Dohlus model and the dielectriclayer model (see figure 7.15). From the group veloc-ity of the modes (described for dielectric layers) theradius dependence should be opposite to the obser-vations. Among all models discussed in this thesisthe Dohlus model [12] delivers the best descriptionof surface roughness wakes in case of shallow sur-face structures. Very deep surface structures arebest described by the dielectric layer models withε = 2. Hence these two models can be regarded asthe two limiting cases.

44

preparation r/mm δ rms fw/GHz εeff ampl./kV τ/psreference 4 1.4 µm - - - -sandblasted 5 10 µm 480± 27 1.64 39± 5 4.8sandblasted 4 10 µm 564± 32 1.55 60± 5 4.0sandblasted 3 10 µm 658± 40 1.53 105± 10 3.4grooves 5 60 µm 177± 10 1.94 - -grooves 4 60 µm 200± 10 1.90 - 77collimator 3 5.7 µm 830± 60 1.55 - -

Table 7.2: Summary of results for different beam tubes. The parameter δ is the rms depth of the roughness except for theeroded pipes where it is the depth of the grooves. The wake frequencies fw have been determined from the energy distributionof the electron bunches and verified with the interferometer. The 6th column gives the amplitude of the wake potential and thelast column the decay constant resp. pulse length.

2 2.5 3 3.5 4 4.5 5 5.5 610

0

101

102

surface resistivity

group velocity

Dohlus

Dohlus + group velocity

radius [mm]

time

cons

tant

[ps]

Figure 7.15: Time constants of the surface roughness wake. Predictions made by different models are plotted for comparison.Only the combination of the Dohlus model with a dielectric layer wake model yields reasonable agreement with the measurement.

45

Chapter 8

Conclusion and Outlook

Surface roughness wake fields are a significantconcern for the performance of high gain free elec-tron lasers. The worry is that already micrometer-size structures on the inner surface of the undulatorbeam pipes could severely degrade the energy dis-tribution inside the bunches. The waveguide modesinside the beam pipe are slowed down by a roughsurface and at a certain characteristic frequency en-ergy can be transfered resonantly from the beam toa radiation field. A similar effect happens in beampipes covered with a dielectric layer. Although thesurface roughness wake fields have received a con-siderable theoretical attention, no convincing con-clusion had been found since the energy losses pre-dicted by the theories differed by orders of magni-tude.

To clarify the situation an experiment has beenconducted at the TESLA Test Facility to study thesurface roughness wake fields. By introducing beampipes with an enhanced roughness of the inner sur-face strong wake fields could be excited that alloweda detailed investigation. Two methods were em-ployed to detect the wake fields: The wake fieldradiation was measured in a far infrared interfer-ometer and the influence of the wakes on the beamwas investigated by measuring the energy profilesof the bunches and applying the methods of tomog-raphy to it. In this way it was possible to measurethe longitudinal phase space of the bunches. At theTESLA Test Facility no rotation of the longitudi-nal phase space was possible. To obtain reliableresults a tomography algorithm based on the maxi-mum entropy method was applied. It is a powerfultool to study processes in the longitudinal phasespace like bunch compression, wake fields, and co-herent synchrotron radiation. The suppression ofartefacts achieved by the maximum entropy methodmakes the algorithm attractive also for the trans-verse phase space. There it can lead to a reductionof required quadrupole currents, a relaxation of con-straints on rotation angles, and a reduction of thenumber of required images.

Harmonic wake fields were detected in accordancewith a dielectric layer model. For stochastic surface

structures which were produced by sand-blastingthe observed frequencies can be explained assum-ing an effective dielectric constant in the order of1.6. In numerical studies made for periodic and rel-atively large structures an ε of approximately 2 waspredicted, yielding resonance frequencies 20% lowerthan measured, whereas with a linear approxima-tion of the surface structures resonance frequenciesare calculated which are 20% higher than the mea-sured values. The two different calculation methodscan be regarded as the asymptotic solutions for verylarge respectively very shallow surface structures.The surface structures in the wake field experimentwith an rms height of 10 µm lie in between, the sur-face structures with an rms height of 6 µm in thecollimator are already well described by the linearboundary approximation.

The observed time constants in the order of afew ps of the wakes are a strong hint that the linearboundary approximation according to Dohlus [12]is the correct description for surface roughnesses inthe order of micrometers and below. For the un-dulator vacuum chamber with a measured surfaceroughness of approximately 600 nm this model pre-dicts that the roughness wake is negligible in com-parison with the resistive wall wakes. Therefore itcan be concluded that for the surface roughness thepreparation of the undulator vacuum chambers asit was described in [16] is sufficient for future FELprojects.

It should be remarked that the Dohlus modelwas derived for axially symmetric surface struc-tures only. The transition to an isotropic surfaceroughness was done in analogy to the Stupakovmodel. Deriving this more rigorously may improvethe agreement between calculations and measure-ments.

Instead of regarding the wake fields as a para-sitic effect they may also be utilized in future FELprojects. They may be used as a radiation sourceof their own. Or – due to the variation of the wakefield strength along the bunch – they may be usedto shorten the photon pulse by impeding the FELprocess in parts of the bunches.

46

Appendix A

Derivatives

Group velocity

k =

√2ε

(ε− 1)bδ

(J0(krb)

J2(krb) + J0(krb)− bδ

2εk2

r

)(A.1)

dk

dkr=

12

√2ε

(ε− 1)bδ

ddkr

(J0(krb)

J2(krb)+J0(krb) −bδ2εk2

r

)√

J0(krb)J2(krb)+J0(krb) −

bδ2εk2

r

(A.2)

=ε

(ε− 1)bδk. . . (A.3)

Series expansion of km and kz

According to equation 2.31 km is

km =

√2ε

(ε− 1)bδ

(J0(krb)

J2(krb) + J0(krb)− bδ

2εk2

r

)(A.4)

Inserting the series expansion of the Bessel functions

J0(x) = 1− x2

4+

x4

64− x6

2304. . . (A.5)

J1(x) =x

2− x3

16+

x5

384− x7

9216. . . (A.6)

into A.4 taking into account that ξ = b2k2r

km =

√√√√ 2ε

(ε− 1)bδ

(1− b2ξ

4 + b4ξ2

64 − b6ξ3

2304 . . .

1− b2ξ8 + b4ξ2

192 −b6ξ3

4608 . . .− δ

2bεξ

)(A.7)

Performing the polynomial division

=

√2ε

(ε− 1)bδ

(1− b2ξ

8− b4ξ2

192− b4ξ3

4608. . .− bδ

2εξ

)(A.8)

With kres =√

2ε(ε−1)bδ

= kres

√1− b2ξ

8− b4ξ2

192− b6ξ3

4608. . .− bδ

2εξ. (A.9)

47

The terms with ξ1 are grouped with the abbreviation am = b2

8 + bδ2ε

= kres

√1− amξ − b4ξ2

192− b6ξ3

4608. . .. (A.10)

Now inserting the expansion

√1− x = 1− x

2− x2

8− x3

16. . . (A.11)

One find the series for km

x = amξ +b4

192ξ2 +

b6

4608ξ3 . . .

x2 = a2mξ2 +

amb4

96ξ3 . . .

x3 = a3mξ3 . . .

km = kres

(1− am

2ξ −

(a2

m

8+

b4

384

)ξ2 −

(a3

m

16+

amb4

768+

b6

9216

)ξ3 . . .

)(A.12)

The series for kz is found by solving kz =√

k2m − k2

r with

k2m = k2

res

(1− amξ − b4ξ2

192− b6ξ3

4608. . .

)(A.13)

⇔ k2m − k2

r = k2res

(1−

(am +

1k2

res

)ξ − b4ξ2

192− b6ξ3

4608. . .

)(A.14)

(A.15)

With the abbreviation

ar = am +1

k2res

=b2

8+

bδ

2(A.16)

the expansion of kz is found in analogy to km

kz = kres

(1− ar

2ξ −

(a2

r

8+

b4

384

)ξ2 −

(a3

r

16+

arb4

768+

b6

9216

)ξ3 . . .

)(A.17)

Series expansion of km vs kz

With the knowledge from the last section it is now possible to calculate the Taylor series expansion of km(kz).To do so one has to know the derivatives of the function km(kz)

dkm

dkz=

dkm

dξ

dξ

dkz=

dkm

dξ

dkz

dξ

(A.18)

d2km

dk2z

=ddξ

dkm

dξ

dξ

dkz=

d2km

dξ2dkz

dξ −dkm

dξd2kz

dξ2

(dkz

dξ )3(A.19)

d3km

dk3z

=d3km

dξ3 (dkz

dξ )2 + 3dkm

dξ (d2kz

dξ2 )2 − 3d2km

dξ2dkz

dξd2kz

dξ2 − dkm

dξdkz

dξd3kz

dξ3

(dkz

dξ )5(A.20)

Inserting the coefficients calculated above

dkm

dkz=

am

ar(A.21)

d2km

dk2z

=8

a3rkres

(a2

mar

8+

b4ar

384− ama2

r

8− b4am

384

)

48

=b4 − 48amar

48a3rk

3res

(A.22)

d3km

dk3z

=32

a5rk

2res

[a2

r

4

(3a3

m

8+

amb4

128+

b6

1536

)+3

am

2

(a4

r

16+

a2rb

4

384+

b8

36864

)−3

ar

2

(a2

ma2r

16+

a2mb4

768+

a2rb

4

768+

b8

36864

)−amar

4

(3a3

r

8+

arb4

128+

b6

1536

)]=

amarb4 − 48a2

ma2r

16a5rk

4res

− 48a2rb

4 − 4arb6 − b8

768a5rk

4res

. (A.23)

The Taylor series then reads

km = kres +am

ar(kz − kres) (A.24)

+b4 − 48 amar

48 a3rk

2res

· (kz − kres)2

2kres

+(

amarb4 − 48a2

ma2r

16 a5rk

2res

− 48a2rb

4 − 4arb6 + b8

768 a5rk

2res

)(kz − kres)3

6k2res

Or in analogy (with ar and am swapped)

kz = kres +ar

am(k − kres) (A.25)

− b4 − 48 amar

48 a3mk2

res

· (k − kres)2

2kres

−(

amarb4 − 48a2

ma2r

16 a5mk2

res

− 48a2mb4 − 4amb6 + b8

768 a5mk2

res

)(k − kres)3

6k2res

49

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52

Acknowledgments

At this point I would like to express my heardful thanks to all who helped me make this work a success.

I want to thank my advisor Prof. Dr. P. Schmuser for his aid, encouragement, and advice throughout mywhole time at DESY. He always found the time for helpful discussions, may it be for creating new ideas orstraightening out thoughts.

I thank Prof. Dr. M. Tonutti for introducing me to the TESLA project and for his support during mythesis. He was always prepared to listen to my problems.

At the TTF a young scientist can find all the support and opportunities he can think of. Therefore I wantto thank Dr. M. Leenen, Dr. H. Weise, Dr. J. Rossbach, Dr. A. Gamp, and Dr. D. Trines.

I would like to thank Dr. H. Schlarb for his many hints and explanations about the physics of wake fields.For our many discussions I would also like to thank Dr. S. Schreiber, Dr. S. Simrock, and Dr. G. Schmidt.

For his valuable help with the debugging of the computer code for the tomography I want to thankJ. Scheins.

I always enjoyed the atmosphere and the spirit in the group FDET. My thanks go to all those who gavetheir contribution to this. I. Nikodem deserves special mention who keeps everything running.

For the support during the design and assembly of the experimental chamber I would like to thankJ. Weber, J. Dicke, R. Heitmann, O. Peters, K. Escherich, J. Prenting, W. Benecke, and J. Holz. Andespecially I want to thank the people of MVP, K. Zapfe, D. Hubert(†), G. Wojtkiewicz, D. Ahrendt, andA. Wagner. Very special thanks to B. Sparr. With computer hard- and software I always found help byK. Rehlich, O. Hensler, and G. Grygiel.

I would like to thank the Arbeitsbereich Elektrotechnik VII of the TUHH and M. Seidel for the equipmentto measure the surface profiles.

I want to thank H. Schlarb, J. Scheins, M. Dohlus, and M. Minty for carefully reading my manuscript andfor their remarks and suggestions.

Finally I want to thank my parents for their support, and Tina and Jonas who brighten my day.

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