Investigation of the Transverse and

Longitudinal Beam Parameters

at the

TESLA Test Facility Linac

Dissertation

zur Erlangung des Doktorgrades

des Fachbereichs Physik

der Universitat Hamburg

vorgelegt von

Marc Alexander Geitz

aus Hagen

Hamburg

1999

Gutachter der Dissertation: Prof. Dr. P. SchmuserPriv. Doz. Dr. M. Tonutti

Gutachter der Disputation: Prof. Dr. P. SchmuserPriv. Doz. Dr. A. Gamp

Datum der Disputation: 5. November 1999

Dekan desFachbereichs Physik undVorsitzender desPromotionsausschusses: Prof. Dr. F. W. Buer

Abstract

The operation of a free electron laser in the VUV regime puts stringent demands on thebeam quality of an electron linear accelerator. That is why beam diagnostic techniqueshave been developed to determine the transverse phase space density and the longitudinalcharge distribution of the electron bunches with the required precision. The transversephase space distribution is determined from beam intensity proles using tomographicimage reconstruction techniques. Measurements at the TESLA Test Facility linac yielda normalized transverse emittance of x = 5:5 106m und y = 9:5 106m witha bunch charge of 1 nC. The longitudinal bunch charge distribution is determined byfrequency and time-resolved methods. Fourier and Hilbert-transform spectroscopy ofcoherent transition radiation belong to the former techniques. Longitudinal phase spacerotations to determine the longitudinal bunch charge distribution and streak camera mea-surement using optical synchrotron radiation are applications of time-resolved methods.The dierent techniques yield consistent results and hint at a minimum electron bunchlength of z = 250m.

Zusammenfassung

Der Betrieb eines Freien Elektronen Lasers im ultravioletten Wellenlangenbereich stellthohe Anforderungen an die Strahlqualitat eines Elektronenlinearbeschleunigers. Daherwurden Strahldiagnoseinstrumente entwickelt, die eine Messung der transversalen Phasen-raumverteilung und der longitudinalen Ladungsverteilung der Elektronenpakete mit dererforderlichen Prazision ermoglichen. Die transversale Phasenraumverteilung wird unterVerwendung von tomographischen Rekonstruktionmethoden aus Strahlintensitatsvertei-lungen bestimmt. Die Messungen an der TESLA Test Anlage ergeben, bei einer Ladungder Elektronenpakete von 1 nC, eine transversale Emittanz von x = 5:5 106m und y = 9:5106m. Zur Bestimmung der longitudinalen Ladungsverteilung werden frequenzund zeitaufgeloste Methoden angewendet. Zur Ersteren gehort die Fourier und dieHilbert-Transformations Spektroskopie koharenter Ubergangsstrahlung. Die Ausnutzungdenierter longitudinaler Phasenraumrotationen zur Bestimmung der Elektronenpaket-lange und Streak Kamera Messungen optischer Synchrotronstrahlung sind Anwendungenzeitaufgeloster Methoden. Die verschiedenen Messungen ergeben konsistente Ergebnisseund weisen auf eine minimale Elektronenpaketlange von z = 250m hin.

Contents

1 Introduction 1

2 TESLA Test Facility Linac 32.1 Injectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Thermionic Injector . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.2 RF Photo Injector . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.3 Matching and Dispersive Section . . . . . . . . . . . . . . . . . . . 10

2.2 Linac . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.1 General Linac Layout . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.2 Linac Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.3 Bunch Compression Scheme . . . . . . . . . . . . . . . . . . . . . . 14

3 Properties of Transition Radiation 213.1 Transition Radiation as a Tool for Beam Diagnostics . . . . . . . . . . . . 213.2 Optical Transition Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.1 Ginzburg-Frank Equation . . . . . . . . . . . . . . . . . . . . . . . 213.2.2 Radiated Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.3 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.4 Angular Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.5 Oblique Incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3 Far Infrared Transition Radiation . . . . . . . . . . . . . . . . . . . . . . . 263.3.1 Source Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3.2 Huygens-Fresnel Principle . . . . . . . . . . . . . . . . . . . . . . . 32

3.4 Coherent Transition Radiation . . . . . . . . . . . . . . . . . . . . . . . . . 353.4.1 Radiation Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.4.2 Investigation of the Longitudinal and Transverse Form Factor . . . 37

4 Transverse Phase Space Tomography 414.1 Theory of Phase Space Tomography . . . . . . . . . . . . . . . . . . . . . . 41

4.1.1 The Radon Transform . . . . . . . . . . . . . . . . . . . . . . . . . 414.1.2 The Filtered Backprojection Algorithm . . . . . . . . . . . . . . . . 434.1.3 Discrete Implementation of the Filtered Backprojection Algorithm 484.1.4 Reconstruction Artifacts . . . . . . . . . . . . . . . . . . . . . . . . 514.1.5 Application of the Filtered Backprojection Algorithm to Transverse

Beam Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.1.6 Determination of the Beam Parameters . . . . . . . . . . . . . . . . 564.1.7 Incorporation of Space Charge Eects . . . . . . . . . . . . . . . . . 56

i

4.2 Parabola Fit Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 Determination of the Transverse Beam Parameters 595.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2 Measurements with the Photo Injector Beam . . . . . . . . . . . . . . . . . 60

5.2.1 Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.2.2 Tomographic Reconstruction . . . . . . . . . . . . . . . . . . . . . . 635.2.3 Transverse Beam Parameters and Emittance . . . . . . . . . . . . . 645.2.4 Discussion of the Systematic Errors . . . . . . . . . . . . . . . . . . 655.2.5 Further Emittance Studies using the Photo Injector Beam . . . . . 69

5.3 Measurements with Thermionic Injector Beam . . . . . . . . . . . . . . . . 715.3.1 Parabola Fit Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 73

6 Time Domain Techniques for Bunch Length Measurements 756.1 Streak Camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.1.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 756.1.2 Bunch Length Measurements . . . . . . . . . . . . . . . . . . . . . 77

6.2 Longitudinal Phase Space Rotations . . . . . . . . . . . . . . . . . . . . . . 786.2.1 Theory and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 786.2.2 Bunch Length and Energy Distribution Measurements . . . . . . . 81

7 Fourier Transform Spectroscopy 857.1 Description of the Martin-Puplett interferometer . . . . . . . . . . . . . . . 857.2 Working Principle of the Interferometer . . . . . . . . . . . . . . . . . . . . 887.3 Frequency-dependent Acceptance Function . . . . . . . . . . . . . . . . . . 91

7.3.1 Spectral Transmission and Re ectivity of Wire Grids . . . . . . . . 917.3.2 Spectral Acceptance of the Wire Grid Beam Divider . . . . . . . . . 947.3.3 Diraction in the Interferometer . . . . . . . . . . . . . . . . . . . . 967.3.4 Detector Acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . 997.3.5 Transmission of the Quartz Window . . . . . . . . . . . . . . . . . 1017.3.6 Aluminum Foils of Finite Thickness . . . . . . . . . . . . . . . . . . 1017.3.7 Spectral Acceptance of the Martin-Puplett Interferometer . . . . . . 1047.3.8 Systematic Error of the Acceptance Function . . . . . . . . . . . . . 105

7.4 The Autocorrelation Function . . . . . . . . . . . . . . . . . . . . . . . . . 1067.5 Reconstruction of the Longitudinal Charge Distribution . . . . . . . . . . 1097.6 Determination of the Bunch Length . . . . . . . . . . . . . . . . . . . . . . 113

7.6.1 Analysis of the Autocorrelation . . . . . . . . . . . . . . . . . . . . 1137.6.2 On-line Monitoring of the Bunch Length . . . . . . . . . . . . . . . 114

7.7 Bunch Length Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 1157.7.1 Measurements with the Photo Injector Beam . . . . . . . . . . . . . 1167.7.2 Measurements at the TTF Thermionic Injector . . . . . . . . . . . 121

7.8 Possible Improvements of the Interferometer . . . . . . . . . . . . . . . . . 122

8 Hilbert Transform Spectroscopy 1238.1 Electric Properties of a Josephson Junction . . . . . . . . . . . . . . . . . . 123

8.1.1 The Josephson Eect . . . . . . . . . . . . . . . . . . . . . . . . . . 1238.1.2 DC Characteristic of the Josephson Junction . . . . . . . . . . . . . 124

8.1.3 Josephson Junction with Incident Radiation . . . . . . . . . . . . . 1258.1.4 Hilbert Transform Spectrometry . . . . . . . . . . . . . . . . . . . . 126

8.2 The Josephson Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1278.3 Bunch Length Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 129

8.3.1 The Intrinsic Parameters of the Junction . . . . . . . . . . . . . . . 1298.3.2 The Spectroscopic Measurement . . . . . . . . . . . . . . . . . . . . 1308.3.3 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

9 Conclusion and Outlook 133

A Beam Dynamics and Parameters 135A.1 The Transfer Matrix Formalism . . . . . . . . . . . . . . . . . . . . . . . . 135A.2 Beam Transfer Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136A.3 First Order Beam Matrix Formalism . . . . . . . . . . . . . . . . . . . . . 137

A.3.1 Beam Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140A.3.2 Beam Emittance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

B Derivation of the Ginzburg-Frank formula 141

C Electromagnetic Fields of Relativistic Electron Bunches 149

D Space Charge Forces 159

E Hilbert Transform Spectroscopy 163E.1 Current-Voltage Characteristic without Radiation . . . . . . . . . . . . . . 163E.2 Current-Voltage Characteristic with Radiation . . . . . . . . . . . . . . . 165E.3 The Hilbert Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

F Acceptance Measurement of the Pyroelectric Detectors 171

Chapter 1

Introduction

Electron-positron colliders are essential instruments to investigate the structure of matterand the understanding of the fundamental forces in nature. The presently largest col-lider LEP at CERN stores electrons and positrons with an energy of about 100GeV perbeam and is energy-limited by synchrotron radiation losses. A possible way to go beyondthe energy limit of storage rings is the concept of two linear accelerators in collision.The higher center-of-mass energies are necessary to increase the resolution of high energyphysics experiments and to investigate smaller structures of matter. Detailed studies ofthe top quark and the Higgs particle, the last missing constituent of the standard model,will be a major task of a particle physics experiment at a future linear collider. Thedetection of supersymmetric particles would hint at a unication of the strong, weak andelectromagnetic interaction and at physics beyond the standard model.

The TESLA collaboration proposes a superconducting linear accelerator design to reacha center-of-mass energy of 500GeV in its rst stage. The accelerating structures consistof nine-cell niobium cavities with a eld gradient of 25MV/m and a quality factor of1010. The advantage of the superconducting 1:3GHz cavities is the high conversion e-ciency between the primary and the beam power of about 17%. The perturbing eectsof cavities, like wake elds transforming part of the energy gain into transverse beamoscillations, are weaker than in conventional, normal-conducting accelerators of higherfrequency. The high quality factor of the superconducting resonators facilitates the ac-celeration of highly-charged electron bunches of sub-picosecond length to drive an X-rayfree-electron laser based on the principle of self-amplied spontaneous emission (SASE).

The goal of the TESLA Test Facility (TTF) is to validate the technology of superconduct-ing niobium cavities with the TESLA specications and to demonstrate the accelerationof an electron beam of 8mA average current, up to 8 nC bunch charge, over a macropulselength of 800s. The beam of the TTF linac will be used to drive a SASE free-electronlaser in the VUV wavelength regime.

The measurement of the transverse and longitudinal beam quality is a key ingredient ofthe TTF experimental program and the achievement of the design beam parameters isessential for the success of the test facility. Beam diagnostic techniques have been devel-oped and commissioned to determine the transverse beam emittance, bunch length andenergy spread. The transverse phase space distribution is derived from beam intensity

proles using tomographic image reconstruction techniques. The transverse beam inten-sity distribution is recorded using optical transition radiation. Phase space tomographyproduces an image of the entire transverse phase space distribution without making anyassumptions on the charge distribution. The longitudinal charge distribution is measuredwith time- and frequency-resolved methods. Streak camera measurements using opticalsynchrotron radiation and longitudinal phase space rotations imposed by rf cavities anda magnetic dipole chicane have been applied to measure the longitudinal bunch chargedistribution. Frequency-resolved techniques using coherent transition radiation detectedby a Martin-Puplett interferometer and a Josephson junction have been developed andused for bunch length measurements.

Chapter 2

TESLA Test Facility Linac

The TESLA Test Facility (TTF) linac can be subdivided in three major sections: theinjector, the main superconducting linear accelerator and the experimental area. A de-scription of these accelerator sections, including the transverse and longitudinal beamdynamics, will be outlined.

2.1 Injectors

The TTF linac has been operated with two types of injectors, a thermionic injectorand an rf photo injector. The thermionic gun has been used to produce a high qualitybeam of moderate bunch charge to permit a successful operation of the superconductingaccelerating sections. In a second step, an rf gun photo injector has been installed toproduce a high charged, high quality beam such as required for the TESLA collider anda VUV free electron laser. Both injectors achieve the TTF design average beam currentof 8mA over a macropulse length of 800s and a repetition rate of 10Hz.

2.1.1 Thermionic Injector

The thermionic injector [1] consists of a triode gun with a heated cathode of 8mm diameter(on ground potential), a grid (typically 3V voltage) and an anode providing 30 kV ofaccelerating voltage. The gun produces short electron pulses, t < 1 ns, of 2:3108 electronsby a fast 216:7MHz modulation of the cathode voltage from 0V to about 100V. Themicropulse repetition is a 1=6 sub-harmonic multiple of the linac accelerating frequencyof 1:3GHz. The average beam current is 8mA and extends over a macropulse length of800s. The thermionic injector timing scheme [2] is displayed in Fig. 2.1.

The beam is pre-accelerated by a 1m long 220 kV electro-static column consisting of aseries of eld gradient rings separated by glass insulating discs [2]. The 250 keV energyelectron pulses are then injected into a copper cavity operating at 216:7MHz with amoderate voltage of about 100 kV. The so-called \sub-harmonic buncher cavity" is usedfor the longitudinal compression of the electron bunches. Following the sub-harmonicbuncher the beam passes a superconducting, 9 cell, 1:3GHz cavity for further bunchingand acceleration to an energy of at least 10MeV. The resulting bunch length is z =0:5mm.

_ _ _ _ _ _

Average Beam Current: 8 mA

800 µs

100 ms

Time

Q = 37 pC

4.6 ns

800 µs370 ps (FWHM)

QI =100 mApeak

Figure 2.1: Bunch timing structure of the TTF thermionic injector. Pulses of 2:3 108electrons are emitted with a repetition rate of 216:7MHz over a macropulse length of800s. The macropulse repetition rate is 10Hz.

Beam Parameters: The beam parameters at the exit of the capture cavity, predictedby the PARMELA [3, 2] program are listed in Tab. 2.1 for a 10MeV beam.

Beam Parameter Prediction by PARMELA

rms [106m] 4.8

11, 33 [106m2] 0.24

12, 34 [106m] 0.23

55 [106m2] 0.24

66 [106] 53.3

Table 2.1: Design beam parameters of the TTF thermionic injector as determined by thecode PARMELA. The beam energy is E = 10MeV and the bunch charge is Q = 37 pC.

The beam parameters are understood as elements of the beam matrix (-matrix) [4],which is the covariance matrix of the bunch charge distribution. The diagonal elementsdenote the square of the horizontal and vertical rms beam width (11 and 33), the squareof the horizontal and vertical rms angular divergence (12 and 34), the square of thebunch length (55) and the square of the fractional energy deviation (66). The o-

diagonal elements denote the covariance, hence the coupling, among the respective phasespace coordinates. The beam matrix formalism, which is used throughout this thesis, isoutlined in appendix A.

Bunch Compression:

The 1-cell sub-harmonic buncher cavity is used to compress the longitudinal bunch chargedistribution at a low beam energy of E = 250 keV. The buncher cavity induces a head-tailenergy modulation on the bunch, hence a velocity modulation since = v=c is only about0:55, where the faster electrons trail the slower electrons. In a following drift section of2:66m length, the trailing electrons catch up with the leading electrons and a longitudinalbunch compression is achieved. The phase of the buncher cavity has to be optimized toobtain the minimum bunch length within the rst cell of the nine-cell capture cavity,where the beam energy is increased rapidly into the MeV range ( ! 1). The capturecavity has a non-negligible in uence on the optimum bunching because of the low in-jection energy. Above 10MeV the longitudinal beam dimension is invariant in a linearaccelerator since 1.

To evaluate the optimum phase of the sub-harmonic buncher cavity, let us consider oneelectron at the head, at the center and at the tail of the bunch. The energy of the electronsafter passing the buncher is

(k)f = (k)i +eU

mc2cos(k) + 0

(2.1)

where denotes the relativistic Lorentz factor, U the accelerating gradient, the accel-erating phase and m the electron mass. The index f denotes the nal, i the initial beam

95 105 115 1250

4

8

12

16

20

Buncher Phase [ ]

σ [p

s]t

o

minimum length

85kV 87.5kV90kV 92.5kV95kV 97.5kV

operatingrange

Figure 2.2: Bunch length as a function of the buncher cavity phase for various settings ofthe buncher voltage. The initial bunch length is t = 300 ps [2]. Notice that the buncherphase yielding the minimum bunch length varies linearly with the voltage.

energy. The superscript (k) indicates either of the three electrons, 0 the rf phase of thecenter electron. The energy gain yields a particle velocity = v=c of

(k)f =

vuuuut (k)f

2 1 (k)f

2 =

r (k)i + eU

mc2cos((k) + 0)

2 1

(k)i + eU

mc2cos((k) + 0)

: (2.2)

The particles experience a repelling electromagnetic force as they approach each other.The longitudinal electric eld of a single particle at the longitudinal position 0 travelingwith velocity c along a straight line is (see appendix C)

Ez =e

420 21

( 0)2 (2.3)

where = z ct. The single particle electric eld is weighted with a Gaussian chargedistribution to describe the space charge forces of the entire particle bunch. Figure 2.2shows the bunch length in the rst cell of the capture cavity as a function of the bunchercavity phase for dierent settings of the buncher voltage. The initial bunch length ist = 300 ps [2]. Higher buncher voltages yield shorter bunches. The minimum obtainablebunch length, however, is restricted by longitudinal space charge forces, causing the de-viation from the parabolic characteristic of t versus 0 at voltages U 95 kV. At theseoperating voltages the minimum achievable bunch length depends linearly on the phaseof the buncher cavity. At voltages U > 95 kV, the space charge forces tend to (longitudi-nally) de-focus the bunch before reaching the capture cavity. The minimum bunch lengthis obtained at buncher phases other than indicated by the line of minimum length.

2.1.2 RF Photo Injector

The electron bunches are produced by the photo-electric eect using a UV laser pulseinteracting with a Cs2Te photo cathode. The electron bunch is accelerated rapidly bystrong electromagnetic elds (E acc = 3550MV/m, f = 1:3GHz) in the gun cavityyielding a beam energy of 4MeV. The rf power (P 3MW) is fed into the gun cavityby a side-coupled waveguide. The required focusing is provided by two solenoid coilssurrounding the gun cavity. The primary solenoid focuses the beam envelope close tothe cathode at low beam energies. The secondary solenoid applies further focusing on thebeam at the downstream end of the gun cavity. A third solenoid is used to compensate themagnetic eld on the cathode. An advantage of the twin coil arrangement is the possibilityto correct alignment errors. Figure 2.3 shows a sketch of the 1 1=2 cell, room-temperaturegun cavity. The rapid acceleration, the fringe eld focusing and the helical path of theelectrons in the solenoid eld counteract the emittance blowup by space charge forces.The electron beam is then injected into the superconducting capture cavity to increasethe beam energy to 16MeV [5].

The UV laser of the rf photo injector is driven by a mode-locked Nd:YLF oscillator( = 1:47m) producing a 3ms long output pulse train of 18:5 ns spaced bunches. APockels cell selects laser pulses with a repetition rate of 1MHz and transmits these pulsesinto three linear Nd:YLF ampliers. The pulses of up to 200J energy are frequencyconverted twice to green light by a LPO crystal and to UV by a BBO crystal. The UV

e-

Laser

Cavity

PrimarySolenoid

SecondarySolenoid

Wave guidecoupler

Photocathode

Laserport

+ Mirror

Figure 2.3: The 1-1/2-cell rf gun copper cavity surrounded by three solenoid coils. Therf feed, the photocathode and the UV laser beam is indicated.

beam is sent to the photo cathode in the rf gun to produce the electron bunches [6, 7]. Thedesign laser pulse length in the resonator is about z = 10 ps. The frequency conversionreduces the pulse length by a factor of 2 yielding a UV laser pulse length of z = 5ps,262 nm wavelength and up to 25J energy [8]. The Cs2Te photo cathode has a quantumeciency of about 0:5% which is stable over months [9]. The laser pulse energy, focusedonto a spot of 10mm diameter on the cathode, is sucient to produce bunch charges ofabout 10 nC [5] well above the design bunch charge of 8 nC. The macropulse length is800s.

Figure 2.4 [7] shows the timing scheme for the two operating modes of the rf gun photoinjector. The collider mode (left) requires a bunch charge of 8 nC at 1MHz repetition rateand a FWHM bunch length of 18 ps. The macropulse length is 800s with a repetitionrate of 10Hz. The timing parameters are similar to the parameters of a future high energyphysics collider based on the TESLA design. The free electron laser mode (right) requiresa bunch charge of 1 nC at 9MHz repetition rate and a FWHM bunch length of 10 ps.Macropulses of 800s length repeat with a rate of 10Hz. The beam produced in the freeelectron laser mode is a high quality, low emittance electron beam to demonstrate theSASE principle at a VUV free electron laser.

Beam Parameters: The PARMELA [3, 7] program has been used to optimize the gunparameters to obtain the collider- and free electron laser mode beam parameters at theexit of the capture cavity. The results are listed in Tab. 2.2.

_ _ _ _ _ _

0.11 µs

800 µs10 ps (FWHM)

Average Beam Current: 9 mA

800 µs

100 ms

Time

Q

_ _ _ _ _ _

1 µs

800 µs18 ps (FWHM)

Average Beam Current: 8 mA

800 µs

100 ms

Time

Q

Q = 8nC Q = 1nC

I = 500 Apeak I = 100 Apeak

Figure 2.4: Timing structure for the two modes of the TTF photo injector operation.Left: collider mode. Right: free electron laser mode. Pulses of 5 1010 / 6:3 109 electronsare emitted with a repetition rate of 1MHz / 9MHz over a macropulse length of 800s.The macropulse repetition rate is 10Hz.

Beam Parameter Collider (PARMELA) FEL (PARMELA)

rms [106m] 15 1

11, 33 [106m2]

12, 34 [106m]

to be determined to be determined

55 [106m2] 4 4

66 [106] < 4 < 4

Table 2.2: Design beam parameters of the TTF photo injector as determined by the codePARMELA. The beam energy behind the capture cavity is E = 18:2MeV and the bunchcharge is Q = 8nC for the collider mode and Q = 1nC for the free electron laser mode.

Emittance Conservation Scheme

The transverse emittance dilution of an electron beam in an rf gun is caused by spacecharge forces and eects of the gun rf eld. For convenience, the space charge eldof a particle bunch can be separated in linear and higher-order terms. Carlsten [10]demonstrated that the correlated emittance growth due to the linear part of the transversespace charge eld can be compensated by appropriate linear focusing close to the rf gun.The phase advance of a longitudinal slice of charge is larger at the center than at the headof the bunch because of the larger charge density. The result is the opening of a \phasespace fan", where each longitudinal slice of charge has rotated by a dierent angle in

phase space. The projected emittance along the bunch is hence larger than the emittanceof the individual slices [11]. Using a lens of appropriate strength, the front slice of chargecan be made more convergent than the center slice. During the following drift section,the front slice is focused to a smaller waist than the center slice and is therefore subjectto stronger space charge forces. It experiences a larger phase space rotation and catchesup with the rotation of the center slice. The result is a closure of the phase space fan anda compensation of the correlated emittance growth.

The residual transverse emittance of the rf gun beam, other than the thermal emittance, isdetermined by the following mechanisms which cannot be compensated. Non-linear spacecharge forces become non-negligible for tightly focused, divergent and convergent beams.Non-linearities of radial rf eld components in the gun cavity cause an emittance dilutionfor large radius beams. Lastly, the time dependence of the accelerating rf eld leadingto energy and focusing strength variations over the bunch length, causes a non-reversibleemittance growth [12, 7].

The emittance conservation scheme is based on the focusing of the primary and thesecondary solenoid coil. The strength of the solenoids was adjusted experimentally tominimize the emittance produced at the end of the injector. Secondly, the focusing in- uences the emittance growth caused by non-linear space charge forces, which are strongfor small beam sizes, and by non-linear radial rf elds, which are strong for large beamsizes. There is a signicant dependence of the emittance on the laser spot size at the

0 1 2 3 4 5 60

1

2

3

4

5

6

Injector Position [m]

Bea

m E

nvel

ope

[mm

]

CaptureCavity

Primary Solenoid

Secondary Solenoid

Figure 2.5: The normalized beam emittance along the TTF photo injector as predictedby the PARMELA program.

photo cathode and on the laser pulse length. A large laser spot yields a large electronbunch size and therefore a large emittance dilution due to non-linear rf elds but lowspace charge forces. A long laser pulse yields a long electron bunch and therefore a largeenergy and focusing dierence along the electron bunch.

The envelope of the photo injector beam as simulated by the PARMELA [3] program isshown in Fig. 2.5. The beam is focused by the gun solenoids and the envelope reachesa maximum in front of the capture cavity at z = 1:3m. The beam is accelerated andfocused in the cavity, z = 1:3m to z = 2:3m, yielding a convergent envelope in thematching section, z > 2:3m.

Bunch Compression

The bunch length produced by the rf gun depends on the laser pulse length and thecompression caused by the rf eld within the rst centimeters of the gun cavity. Byproper choice of the rf phase a velocity modulation can be impressed on the electrons inthe bunch leading to a reduction of its length within the gun cavity. Leading electronsmust see a lower electric eld than trailing electrons to obtain pulse compression. Veryshort bunches can be obtained at the price of sacricing a large fraction of the bunchcharge. This, however, is not desirable for the TTF linac because of the peak currentrequired for the FEL. After passing the capture cavity, the beam energy has increasedto 16MeV and the longitudinal extension of the charge distribution remains invariantbecause approaches unity.

2.1.3 Matching and Dispersive Section

The beam is transfered from the capture cavity to the rst acceleration section of the mainlinac through a room-temperature beam line consisting of a quadrupole doublet (Q1-INJ)and two quadrupole triplets (T1-INJ, T2-INJ). The lattice of the matching section can beseen in Fig. 2.7 for the beam produced by the thermionic injector and Fig. 2.8 for the beamproduced by the rf gun. A dipole magnet between the capture cavity and the main linaccan be used to de ect the beam into a dispersive section in order to adjust the injector

module # 2 module # 1 capturecavity

gun

bc 2

(∆E = 120 MeV)(∆E = 160 MeV)

(∆E = 10 - 15 MeV)

(∆E = 4 MeV)

Figure 2.6: Schematics of the TTF linac showing the injector, the accelerating module#1, the magnetic bunch compressor, module #2 and the high energy analysis area. Thearrows indicate three diagnostic stations used for emittance measurements. The energyincrease in the accelerating sections is shown.

parameters to produce a beam of minimum energy spread. Several diagnostic stationsequipped with transition radiation screens are used together with a focusing quadrupolemultiplet, either in the straight section (emittance measurement) or in the dispersivesection (energy prole measurement) to determine the beam parameters experimentally.The last triplet of the matching section is used to match the beam to the design optics ofthe main linac. A schematics of the TTF linac, including the matching and the dispersivesection, is shown in Figure 2.6.

2.2 Linac

2.2.1 General Linac Layout

In its present conguration the main linac consists of two modules of accelerating cavities(ACC1 and ACC2). The superconducting niobium cavities consist of nine cells with aresonant frequency of 1:3GHz. Eight cavities form an accelerating module. The cavitiesare installed in a cryostat which serves as vacuum insulation, heat shield and mechani-cal support structure. The cavities installed in cryostat #1 and #2 achieve an averagegradient of 15MV/m and 20MV/m respectively with a loaded quality factor of aboutQ = 2:2 106. The rf power (about 160 kW per cavity at 20MV/m) is supplied by a singleklystron and fed into the cavities by coaxial input couplers. After a eld lling time of400s, the rf eld is kept stable in gradient and phase for 800s [13]. The cavity electricelds are measured by a pick-up probe inside the cavity resonator. The eld gradient isproportional to the amplitude of the probe signal and the phase information is obtainedby frequency down-conversion (mixing) of the signal using a frequency-stabilized localoscillator. Focusing is applied by a superconducting quadrupole doublet (D1-ACC1 andD1-ACC2) at the end of each rf module.

A magnetic bunch compressor section (BC2) is installed between the two acceleratingmodules. The section consists of a quadrupole triplet (T1-BC2), four dipole magnets de- ecting the beam through a symmetric chicane and a second quadrupole triplet (T2-BC2).Diagnostic stations are used to display either the energy and energy spread of the beam(screen in the dispersive section of the chicane compressor) or the transverse emittance(screen in front of module #2) together with the second focusing triplet [14].

Behind the second accelerating module, the beam is passed to the end of the linac througha room-temperature transfer line. The transfer line has been replaced by a collimationsection [15] and an undulator magnet [16] during the construction period in summer1999. The experimental area at the end of the linac consist of a quadrupole triplet(T1-EXP), and several diagnostic stations equipped with transition radiation screens toperform transverse emittance scans. A spectrometer dipole magnet de ecting the beamhorizontally into a dispersive section is used to produce synchrotron radiation. The TTFspectrometer, consisting of the dipole magnet and a transition radiation screen, yields aresolution of E=E = 2 104 and allows an absolute determination of the beam energywith an accuracy of about 1%. Two de-focusing quadrupole magnets in the dispersivesection are used to increase the transverse beam size before sending the beam into thebeam dump.

Accelerator P

osition [m]

σ [mm]x,y

T1INJ

T2INJ

D1ACC2F/D

D1ACC2F/D

D1BC2F/D

D1ACC2F/D

RF

Module

AC

C1

RF

Module

AC

C2

Figu

re2.7:

Horizon

talandvertical

beam

envelop

esalon

gtheTTFlinac

instage

1.The

design

beam

param

etersof

theTTFtherm

ionicinjector

arematch

edto

thelinac

FIDA

latticebythematch

ingquadrupole

triplets.

Thephase

advan

ceof

thelinac

latticeis

90 .Theacceleratin

gcav

ities(E

acc=

15MV/m

)are

indicated

bythegray

boxes,

the

quadrupole

multip

letsbytheblack

lines.

2.2.2

LinacOptics

Thelinac

latticehas

aFIDA

structu

re:focusin

gquadrupole,

interleave,

de-fo

cusin

gquadrupole

andan

acceleratingsection

.Theperio

dlen

gthis12:2

m.

Stage1:Optics

with

theTherm

ionicInjecto

rBeam

Figu

re2.7

show

sthematch

ingandthelinac

section.Thematch

ingsection

consists

oftwo

quadrupole

triplets

T1-IN

JandT2-IN

J.Thelinac

sectioniscom

posed

oftheacceleratin

grf

modules

ACC1andACC2.

Focusin

gisapplied

bythesupercon

ductin

gquadrupole

doublets

D1-A

CC1andD1-A

CC2.

Aquadrupole

doublet

(D1-B

C2)

andadrift

sectionis

located

betw

eentherf

modules.

Behindthesecon

drf

module,

space

isleft

fora

third

acceleratingsection

ACC3.

Presen

tlythere

isabeam

linewith

aroom

-temperatu

requadrupole

doublet

D1-A

CC3.

Thebeam

envelop

e p11andp33iscom

puted

with

the

XBEAM

[17]program

.Thedesign

beam

param

etersbehindthecap

ture

cavity

(Table

2.1)are

match

edto

thelinac

latticebythelast

twoquadrupole

triplets.

TheFIDAlattice

isadjusted

toaphase

advan

ceof

90 in

both

plan

esyield

ingasymmetric

oscillationof

thebeam

envelop

es[18].

Notice

thered

uction

ofthebeam

sizedueto

theenergy

gainof

thebeam

.Thered

uction

isdueto

thefact

that

themom

entum

ratioof

transverse

and

longitu

dinalmom

entum

isdecreased

bytheacceleration

.Addition

alfocusin

gisapplied

bythestan

dingwave

cavities

[19].Thequadrupole

gradien

tsare

scaledwith

beam

energy.

Thebeam

istran

sferedto

theendofthelinac

throu

ghabeam

linewith

outacceleration

.Thebunch

compressor

chican

ewas

not

installed

durin

gtheoperation

ofthetherm

ionic

injector.

Stage2:Optics

with

theRFGunBeam

Figu

re2.8

show

sthematch

ingandthelinac

sectionoperated

with

therfgunbeam

.The

magn

eticlattice

isidentical

tothelattice

show

ninFig.

2.7excep

tfor

thebunch

compres-

D1INJ F/D

T1INJT2INJ

T1BC2

D1ACC1

T2BC2

D1ACC2

D1ACC3

Figu

re2.8:

Horizon

talandvertical

beam

envelop

esalon

gtheTTFlinac

instage

2.The

initial

beam

param

etersused

forthepresen

tedcom

putation

have

been

determ

ined

by

transverse

phase

space

tomograp

hyoutlin

edinthisrep

ort.These

param

etersare

match

edto

thelinac

latticebythematch

ingquadrupole

triplets

[21].

sor(BC2)

section.Thequadrupole

doublet

D1-B

C2isrep

lacedbytwoquadrupole

triplets

T1-B

C2andT2-B

C2in

frontof

andbehindthemagn

eticchican

e,which

isnot

inclu

ded

inthepresen

tcom

putation

.Thechican

edipole

magn

etswould

cause

addition

alvertical

focusin

gdueto

theedge

focusin

gofthefrin

gemagn

eticdipole

eld

s.Thecom

putation

ofthelinac

latticeisbased

ontherst-ord

erbeam

transfer

codeCOMFORT[20]

inclu

ding

linearized

space

charge

forces(see

appendix

D).

Thelinac

latticehas

been

develop

ed[21]

bycon

siderin

gaset

ofcon

straints.

Asm

allbeam

envelop

ein

thebunch

compressor

sectionisneed

edto

transfer

thebeam

throu

ghthenarrow

gapinsid

ethedipole

magn

ets.In

frontofthecollim

atingsection

(z>60

m),theenvelop

eshave

tobecon

vergentandof

equalhorizon

talandvertical

sizeto

insure

theprop

erfunction

ingof

thecollim

ator.The

beam

param

etersin

frontof

D1IN

J,

x[10

6m]

5.5

y[10

6m]

9.5

11[10

7m

2]3.3

33[10

7m

2]3.9

12[10

8m]

5.234[10

8m]

1.3

obtained by measurement, have been used. The phase advance of the FIDA lattice is asclose as possible to 90.

2.2.3 Bunch Compression Scheme

Longitudinal compression of relativistic electron bunches can be obtained by an o-crestacceleration in a cavity followed by a magnetic chicane bunch compressor. The o-crestacceleration produces a correlated energy spread in the bunch with the higher energyelectrons trailing the lower energy electrons. The higher energy electrons then travel ona shorter path through the magnetic chicane than the lower energy electrons yieldinga bunch length reduction. The compressor section and the longitudinal phase spacedistributions of the electron bunch along the compressor section is shown in Fig. 2.9.

O-Crest Acceleration:

The o-crest acceleration in an rf cavity is described by a longitudinal transfer matrixMA

MA =

1 0M65 M66

acting on the vector

l

!(2.4)

where l is the longitudinal position and the fractional energy deviation of an electron.Let 0 be the rf phase corresponding to the reference electron located at the center ofthe bunch and the rf phase of an electron located z behind the center. If the cavityoperates with a voltage Uacc, the energy deviation fE between the two electrons behindthe accelerating section will be

fE = eUacc (cos cos 0) + iE (2.5)

where iE denotes the initial energy deviation. For = 0, the Taylor expansion ofEq. (2.5) yields

fE = iE eUacc sin0 ( 0) 1

2eUacc cos 0 ( 0)

2 (2.6)

The fractional energy deviation at the nal location is derived by dividing Eq. (2.5) withthe nal energy Ef = Ei + eUacc cos 0 (Ei denotes the initial beam energy). We obtain

fEEf

=iE

Ei + eUacc cos 0 eUacc sin0

Ei + eUacc cos 0( 0)

1

2

eUacc cos 0

Ei + eUacc cos 0( 0)

2 +O( 0)

3:

(2.7)

The transfer matrix elements M65 and M66 can be associated with the zeroth and rstorder term of Eq. (2.7), hence

M66 =Ei

Ei + eUacc cos 0and M65 =

eUacc sin0

Ei + eUacc cos 0

2

(2.8)

where the relation (0) = 2l= has been used. The second order term of the Taylorexpansion (2.7) accounts for the non-linearity of the rf wave within the dimension of

Magnetic bunch compression

2

1

1 2 3 4Position inside bunch compressor [m]

σ [

mm

]z

Dipole

Dipole Dipole

Dipole

E

t

rf cavity

long. position

Ene

rgy

σ = 250 µmz

long. position

Ene

rgy

long. position

Ene

rgy

long. position

Ene

rgy

d

α

Figure

2.9:

Longitudinal

bunch

compressionbyan

o-crest

acceleration

inan

rfcavity

follow

edbyamagnetic

chicanecompressor.

Theimposed

energy-positioncorrelation

(higherenergy

electronstrailinglower

energy

electrons)andthedispersion

ofthechicane

accountforthebunch

compression.Show

nistheevolution

ofthelongitudinalphasespace

distribution

alongthebeam

lineandtherm

sbunch

lengthalongthechicane.

the particle bunch. This eect can be included into the transfer matrix formalism byexpanding the transfer matrix to

~M =MA SA0 TA

(2.9)

such that the linear transfer matrix MA forms the upper left square matrix of ~M ,

SA = 12

eUacc cos 0

Ei + eUacc cos 0

2

(2.10)

denotes the second order contribution mapping l on the fE=Ef and TA = M265. ~M , ex-

pressed in the basis of (l;E=E; l2), yields a 3 3 square matrix and can be used for thebeam matrix formalism outlined in appendix A.

The photo injector is adjusted to produce a beam of minimum energy before injectioninto the rst accelerating rf module. Then there is no correlation between l and andthe longitudinal part of the beam matrix can be expressed as a diagonal matrix as

i =

0@ 55 0 0

0 66 00 0 255

1A (2.11)

where the term 255 accounts for the non-linearity of the rf eld in the cavity. The beammatrix is transformed by

f = ~M i ~MT (2.12)

hence

f =

0@ 1 0 0M65 M66 SA0 0 M2

65

1A0@ 55 0 0

0 66 00 0 255

1A0@ 1 M65 0

0 M66 00 SA M2

65

1A

=

0@ 55 M6555 0M6555 M2

6555 +M26666 + S2

A455 SAM

265

255

0 SAM265

255 M4

65255

1A : (2.13)

The inspection of the matrix (2.13) shows that the o-diagonal elements (f )56 and (f )65do not vanish forM65 6= 0, yielding a correlated longitudinal phase space distribution afteran o-crest acceleration in an rf cavity. The bunch length

p55 does not vary during the

o-crest acceleration. The nal energy deviation receives contributions from the initialenergy spread M2

6666, the initial bunch length M26555 and the non-linear term S2

A455

which are added quadratically. Figure 2.9 shows the correlated phase space distributionyielding an optimum bunch compression in the following chicane section.

Magnetic Chicane

The magnetic chicane consists of four rectangular dipole magnets de ecting the beam inthe horizontal plane. The dipole magnets are of the same strength but the polarity of thesecond and third dipole is inverse to the polarity of the rst and the fourth. Dipole 1 de- ects the beam by an angle of = 18. After passing the second dipole magnet the beamaxis is transposed 35 cm parallel to the axis of the accelerator magnets 3 and 4 de ect

0 1 2 3 40

0.1

0.2

0.3

Position inside bunch compressor [m]

Hor

izon

tal D

ispe

rsio

n [m

]

0 1 2 3 4Position inside bunch compressor [m]

Long

itudi

nal D

ispe

rsio

n [m

]

0

0.1

0.2

Figure 2.10: Left: horizontal dispersion of the magnetic chicane compressor. The max-imum dispersion is Dx = 0:35m in the center of the chicane. Right: longitudinal dis-persion of the magnetic chicane compressor. The maximum longitudinal dispersion isM56 = 0:18m.

the beam back to the axis (see Fig. 2.9). Figure 2.10 shows the horizontal dispersion as afunction of the position inside the bunch compressor. The dispersion is increasing towardsthe middle of the compressor reaching a maximum value of Dx = 35 cm. The transversedispersion has to be entirely suppressed behind the compressor to avoid that electronsof dierent energy travel on dierent trajectories through the accelerating module #2yielding a blow-up of the transverse emittance.

The transfer matrix ~MC of the magnetic chicane is given by

~MC =MR(; ) MD(L1) MR(; ) MD(L2) MR(; ) MD(L1) MR(; ) : (2.14)

where MR(; ) and MD(L1;2) denote the transfer matrices of a rectangular bendingmagnet with bending angle and pole face rotation angle and a drift space of lengthL1;2 respectively. The denition of these transfer matrices is given in appendix A. Thelongitudinal part of ~MC yields

MC =1 M560 1

(2.15)

where the matrix element M56 denotes the longitudinal dispersion of the chicane com-pressor. M56 is shown as a function of the position in Fig. 2.10. Signicant variationsof M56 take place in the second and third dipole magnet. M56 describes the compressionstrength of the magnetic chicane since it maps the initial energy spread onto the nallongitudinal position with respect to the reference particle.

The beam matrix (2.13) can be transformed through the magnetic chicane yielding in thebasis of (l;E=E)

~f =1 M560 1

55 M6555M6555 M2

6555 +M26666 + S2

A255

1 0M56 1

(2.16)

where

~55jf = 55(1 +M56M65)2 + 66M

256M

266 + 255S

2AM

256 (2.17)

0 10 20

10

20

30

Gradient [MV/m]

Pha

se [

]

0 10 20

10

20

30

Gradient [MV/m]

Pha

se [

]0 10 20

10

20

30

Gradient [MV/m]

Pha

se [

]

0 10 20

10

20

30

Gradient [MV/m]

Pha

se [

]

o ooo

(a) (b)

(c) (d)

σ =z

σ =z

σ =z

σ =z

σ =z

1mm

0.5mm

0.25mm

0.16mm

0.12mm

Final rms Bunch Length

Figure 2.11: Lines of constant rms bunch length z behind the bunch compressor as afunction of the gradient and phase of the rf acceleration in Module #1. The four subplotsdepict the optimum phase gradient setting for dierent injector beam parameters. (a)iz = 1:25mm, iE = 2 103. (b) iz = 1:25mm, iE = 8 103. (c) iz = 1:75mm,iE = 2 103. (d) iz = 1:75mm, iE = 8 103.

~56jf = 55M65 +M2

65M56

+ 66M

266M56 + 255S

2AM56 (2.18)

~66jf = M26555 +M2

6666 + S2A

255 (2.19)

The nal bunch length ~z therefore is

~z =q55(1 +M56M65)

2 + 66M256M

266 + 255S

2AM

256 : (2.20)

The linear contribution of 55 can be minimized by choosingM65 such thatM56M65+1!0. The contribution of the initial energy spread 66 to the nal bunch length can bedecreased by using stronger accelerating eld gradients, hence an higher energy gain,yielding a smaller M66. The term 255S

2AM

256 can be decreased only be choosing a smaller

initial bunch lengthp55. The latter contribution limits the achievable longitudinal

compression in the magnetic chicane.

Figure 2.11 shows a contour plot of the achievable bunch length as a function of thegradient and the phase of the rf cavities in front of the magnetic chicane. Injector bunchlengths of iz = 1:25mm and iz = 1:75mm and injector energy spreads of iE = 2103 andiE = 8 103 are considered. The rf de-phasing to obtain optimum bunch compression isdecreasing with increasing eld gradient. The vertical lines indicate the nominal operatinggradient of the TTF linac, which is 15MeV/m. At this eld gradient, a bunch length of(a) z = 110m, (b) z = 200m, (c) z = 190m and (d) z = 250m can be obtained.

Acceleration by RF Module #2

The energy spread at the experimental area can be evaluated by transforming the beammatrix through the second rf module. We obtain the beam matrix

f =

0@ 1 0 0R65 R66 UA

0 0 R265

1A0@ ~55 ~56 0

~56 ~66 00 0 ~255

1A0@ 1 R65 0

0 R66 00 UA R2

65

1A (2.21)

=

0@ ~55 R65~55+R66~56 0R65~55+R66~56 R65(R65~55+R66~56)+R66(R65~56+R66~66)+U

2A~

255 U2

AR265~

255

0 U2AR

265~

255 R4

65~255

1A

The nal energy spreadp66 can be expressed in terms of the initial beam parameters

66 = R265~55 + 2R65R66~56 +R2

66~66 + U2A~

255 (2.22)

= R265

n55(1 +M56M65)

2 + 66M256M

266 + S2

AM256

255

o+

2R65R66

n55M65 +M2

65M56

+ 66M

266M56 + S2

AM56255

o+

R266

n55M

265 + 66M

266 + S2

A255

o+ (2.23)

U2A

n55(1 +M56M65)

2 + 66M256M

266 + S2

AM256

255

o2:

The second accelerating module is generally operated at an rf phase 0 = 0 yielding themaximum energy gain. In this case we obtain R65 = 0 and Eq. (2.23) reduces to

66 = R266

n55M

265 + 66M

266 + S2

A255

o+

U2A

n55(1 +M56M65)

2 + 66M256M

266 + S2

AM256

255

o2:

(2.24)

p66 is obtained by scaling the the rms energy deviation behind the magnetic chicane

with R66. The non-linear contribution of the rf acceleration is added quadratically.

A measurement of the longitudinal bunch charge distribution can be performed by rotatingthe longitudinal phase space distribution using the magnetic chicane and the second rfacceleration. In particular, the longitudinal phase space distribution is rotated by =2transforming a temporal slice in front of the chicane into a slice of constant energy behindthe second rf module. This condition is fullled forM56R65+R66 ! 0 [22] and Eq. (2.23)reduces to

66 = 55R265 + U2

A

n55(1 +M56M65)

2 + 66M256M

266 + S2

AM256

255

o2(2.25)

55R265 : (2.26)

Equation (2.26) is a linear relation between the longitudinal bunch charge distributionand the bunch energy deviation and is the basis for the bunch length measurement. Theparameters of the second rf module R65 and R66 have to fulllM56R65+R66 ! 0. Detailswill be presented in chapter 6.

Chapter 3

Properties of Transition Radiation

3.1 Transition Radiation as a Tool for Beam Diag-

nostics

Transition radiation is emitted whenever a charged particle passes from one dielectricmedium into another. In either medium the charge carries an electromagnetic eld dis-tribution, which changes rapidly at the boundary because of the dierent permittivities.In this process, part of the eld traveling with the charge is radiated away as transitionradiation. The radiation is emitted both into the forward and the backward hemisphereof the boundary.

This chapter is dedicated to outline the basic properties of transition radiation, such as theradiation spectrum, angular distribution, polarization and the radiation source dimension.Attention will be drawn to optical transition radiation (OTR), 400 nm 800 nm,and far-infrared transition radiation, 0:1mm 10mm.

Optical transition radiation can be used to record the transverse and the longitudinalcharge distribution of an electron bunch. The light emitted at the boundary is imagedonto a CCD. The radial intensity distribution is a direct measure of the transverse bunchcharge distribution. The temporal distribution of the light pulse can be used to determinethe longitudinal bunch charge distribution.

Far-infrared transition radiation is a powerful tool to measure the longitudinal bunchcharge distribution of electron bunches in the pico- and sub-picosecond range. The bunchlength and the radiation wavelength are of equal magnitude and coherent transition ra-diation is emitted. The coherent transition radiation power spectrum carries informationabout the bunch length and the shape of the longitudinal bunch charge distribution.

3.2 Optical Transition Radiation

3.2.1 Ginzburg-Frank Equation

Figure 3.1 shows a bunch of particles in medium 1 with permittivity 1 approaching aboundary to medium 2 with permittivity 2. The particle bunch moves on a straight line

v

ε2ε1

z

medium 2medium 1

z = 0

= 1 = ε

Figure 3.1: Conguration for the derivation of the Ginzburg-Frank Equation.

with uniform velocity ~v, directed perpendicular to the boundary. The boundary itselfis assumed to be of innite extent. The radiation energy U emitted into the backwardhemisphere when a single electron of charge e crosses the boundary between vacuum(1 = 1) and a medium (2 = ) is described by the Ginzburg-Frank formula

U =Z 1

0

Z 2

0

Z

0d! d sin d U1(!; ) with (3.1)

U1 =e2

430c

2 sin2 cos2 ( 1)21 2 +

p sin2

2(1 2 cos2 )2

1 +

p sin2

2 cos

p sin2

2 : (3.2)

U1(!; ) denotes the spectral energy distribution, ! the angular frequency and thepolar angle of the radiation. The velocity of the electron is given by = v=c. Notethat U1 does not depend on the radiation frequency !. This feature makes transitionradiation an excellent tool for beam imaging, since no frequency-dependent correction forthe emitted radiation power is necessary. For metallic screens we let ! 1, which is avalid approximation for frequencies well below the plasma frequency of the metal, yielding

U1 =e2

430c

2 sin2

(1 2 cos2 )2: (3.3)

Equation (3.3) is the Ginzburg-Frank formula as it is commonly applied to describe tran-sition radiation for optical beam diagnostics. The derivation of Eq. (3.3) [23] is outlinedin appendix B.

3.2.2 Radiated Energy

The total energy per unit frequency emitted from a metallic screen is determined byintegration of Eq. (3.3) over the solid angle d = sin dd

dU

d!=

Zd d sin U1() (3.4)

0 200 400 600 800 1000γ

0.2

0.16

0.12

0.08

0.04

dU dωdω

[ev]

∫

Figure 3.2: Energy in the visible range emitted by single particle transition radiation asa function of the Lorentz factor .

=e22

220c

Z

0d

sin3

(1 2 cos2 )2(3.5)

= e2

820c

4 +

(1 + 2)

log

(1 )2

(1 + )2

!: (3.6)

The emitted energy by a single electron in the visible wavelength band 400 800 nm isshown in Fig. 3.2 as a function of the relativistic Lorentz factor . The total radiated en-ergy is increasing with increasing electron energy, however, the increase is most signicantbelow 100. The emitted radiation energy per electron is evaluated by

U =Zopt

dU

d!d! (3.7)

where frequencies in the optical range are taken into account (4:7 10151/s ! 2:4 10151/s). For = 100, the total emitted radiation energy is U = 0:13 eV per electron.The photon energy corresponding to ! = 3:8 1015 1/s is E = h! = 2:5 eV, hence thenumber of photons can be approximated by 0.05 per electron, i. e. twenty electrons areneeded to produce one photon. A bunch of 5 1010 electrons generates therefore 5 108photons in the visible range, a number which is easily detectable with standard CCDdevices [24].

3.2.3 Polarization

The polarization of transition radiation can be understood in terms of its generationprocess. Part of the charge eld conguration in vacuum is transfered to the radiationeld while the bunch is undergoing transition. The eld conguration is polarized inthe so-called \radiation plane", which is dened by the radiation momentum ~k and thedirection of particle motion ~z. The polarization of the electric eld ~ER emitted withmomentum ~k is shown in Fig. 3.3.

v

E

z

medium 2medium 1

θ

Figure 3.3: Polarization of transition radiation.

3.2.4 Angular Distribution

The angular distribution of backward transition radiation is shown in Fig. 3.4. Themaximum energy is radiated at an angle of 0 = 1=( ). This result is obtained by thedierentiation of the Ginzburg-Frank formula with respect to yielding

dU1

d=2q;!(0)

430

4c sin 2

(1 2 cos2 )3

n1 2 2 sin2

o: (3.8)

Equation (3.8) vanishes for the maximum energy radiated, hence

sin2 0 =1 2

2) 0 =

1

: (3.9)

No energy is emitted at = 0, because of the radial polarization of the elds travelingwith the electron bunch. Notice, however, that a signicant part of the energy is radiated

0 0.02 0.04 0.06 0.08 0.10

0.2

0.4

0.6

0.8

1

θ [rad]

dU /

dΩ d

ω [a

.u.]

θ = 1 / (β γ)0

Figure 3.4: Angular distribution of single particle transition radiation.

at angles larger than 0. The optical system used for the imaging of optical transitionradiation therefore has to have a larger aperture to collect sucient intensity.

3.2.5 Oblique Incidence

The transition radiator at the TESLA Test Facility is a thin 40 nm aluminum layer evap-orated onto a 25m Kapton foil stretched in a frame. The foil is mounted at an angle of45 with respect to the accelerator axis so that the backward lobe of transition radiationis re ected at 90 out of the vacuum chamber. The theoretical description of transitionradiation has to be extended to describe this situation.

(1) Oblique Incidence

The particle is moving on a straight line with uniform velocity in the xz-plane, as shown byFig. 3.5. is the angle between the electron trajectory and the z axis. The derivation of

k

v

z

medium 2medium 1

θ ψ

y

x

yz

x

ϕ

k’

Figure 3.5: Coordinates for oblique incidence of the particle bunch. denotes the anglebetween the particle velocity ~v and the z-axis in the xz-plane, the angle between thebackward lobe of the transition radiation and z. The azimuthal angle of ~k0 (projection of~k in the xy-plane) with respect to the x-axis is denoted by .

the radiation eld is more elaborate than for normal incidence. The more general situationrequires the introduction of a polar angle and an azimuthal angle . The \radiation-plane" which is dened by ~k and ~v becomes the preference plane. Radiation polarizedparallel and orthogonal to the radiation plane is denoted by Ek and E? respectively.

(2) Metallic Interface

The formulae describing the spectral energy distribution radiated by a single electron ofcharge e are [25]

Uk1 =

e2

430c

cos (sin cos sin )

(1 sin cos sin )2 2 cos2 cos2

!2

(3.10)

U?1 =

e2

430c

2 cos cos sin sin

(1 sin cos sin )2 2 cos2 cos2

!2

(3.11)

where Uk1 and U?

1 denote the radiated spectral energy density of the eld componentpolarized parallel and perpendicular to the radiation plane and = v=c. The metallicinterface is of innite permittivity ( ! 1). The Ginzburg-Frank formula (3.3) can be

recovered from Eq. (3.10) and (3.11) for normal incidence = 0. Uk1 then agrees with the

Ginzburg-Frank formula, whereas U?1 vanishes. Note that transition radiation vanishes

for a grazing incidence angle of =2.

Figure 3.6 shows the angular distribution of Uk1 (left column) and U?

1 (right column) as afunction of the polar angle . The beam is incident at an angle of = 45. The subplotsshow the distribution for dierent azimuthal angles . The lobe is centered at = . Theenergy density of Uk

1 is decreasing rapidly for larger azimuthal angles indicating that

Uk1 contributes mostly in the reference plane dened by (~v; ~z). U?

1 , on the other hand,

vanishes for = 0, rises to the magnitude of Uk1 at = 1= and decreases for larger

azimuthal angles. The angular distribution of U?1 contains only a single maximum.

3.3 Far Infrared Transition Radiation

The description of transition radiation in the millimeter- and sub-millimeter wavelengthrange needs a modication of the Ginzburg-Frank formula. The size of the transitionboundary is no longer large compared to the transverse extension of the electromagneticelds. Diraction at a transition screen of nite size leads to a widening of the angulardistribution and a reduction of the spectral acceptance of the detection system.

3.3.1 Source Dimension

The transverse source size of transition radiation is the projection of the electromagneticelds carried by the charge distribution onto the transition boundary. The Maxwellequations can be solved using a Fourier transformation of the longitudinal coordinate = z ct to the longitudinal wave number k

Er;z(; r) =Z 1

1~Er;z(k; r) exp (ik)dk (3.12)

while the transverse coordinates are not transformed. This procedure permits a descrip-tion of the transverse eld as a function of the wave number k.

The frequency domain description of the electric and magnetic elds, ~Ez, ~Er and ~B

(the other components vanish for reasons of symmetry) is evaluated as the bunch ofcharges passes by the observer. A ring charge distribution of radius a modulated witha longitudinal distribution whose Fourier transformation is denoted as ~(k) is evaluated.The elds have to obey boundary conditions at the surface of the beam pipe with radius b,which is assumed to be perfectly conducting. The following results can be derived [26, 27]

θ [rad] θ [rad]−0.9 −0.8 −0.70

0.5

1

−0.9 −0.8 −0.70

0.5

1

−0.9 −0.8 −0.70

0.5

1

−0.9 −0.8 −0.70

0.5

1

θ [rad]θ [rad]

−0.9 −0.8 −0.70

0.5

1

−0.9 −0.8 −0.70

0.5

1

θ [rad]θ [rad]

−0.9 −0.8 −0.70

0.5

1

−0.8 −0.7

−0.9 −0.8 −0.70

0.5

1

θ [rad]θ [rad]

−0.9 −0.8 −0.70

0.5

1

0

0.5

1

−0.9 −0.8 −0.7θ [rad]θ [rad]

−0.9 −0.8 −0.70

0.5

1

−0.9 −0.8 −0.70

0.5

1

dU/d

Ω d

ωθ [rad]θ [rad]

dU/d

Ω d

ωdU

/dΩ

dω

dU/d

Ω d

ωdU

/dΩ

dω

dU/d

Ω d

ω

dU/d

Ω d

ωdU

/dΩ

dω

dU/d

Ω d

ωdU

/dΩ

dω

dU/d

Ω d

ωdU

/dΩ

dω

ϕ = 0 mrad ϕ = 0 mrad

ϕ = 0.5 mrad ϕ = 0.5 mrad

ϕ = 1 mrad ϕ = 1 mrad

ϕ = 2 mrad ϕ = 2 mrad

ϕ = 3 mrad ϕ = 3 mrad

ϕ = 4 mrad ϕ = 4 mrad

U||1

U||1

U||1

U||1

U||1

U||1

U⊥1

U⊥1

U⊥1

U⊥1

U⊥1

U⊥1

Figure 3.6: Transition radiation dU=dd! as a function of the polar angle for the obliqueincidence angle of = =4. The left column shows the polarization component parallel,the right column orthogonal to the radiation plane. The subplots show dU=dd! for aset of azimuthal angles .

010

2030

4050

0

0.2

0.4

0.6

0.8 1

Radius [m

m]

30 GH

z 150 G

Hz

500 GH

z1 T

Hz

2 TH

z 3 T

Hz

γ =100

~r⋅Er (k,r) [a.u.]

010

2030

4050

0

0.2

0.4

0.6

0.8 1

Radius [m

m]

30 GH

z 150 G

Hz

500 GH

z1 T

Hz

2 TH

z 3 T

Hz

γ =400

~r⋅Er (k,r) [a.u.]

30 GH

z 150 G

Hz

500 GH

z1 T

Hz

2 TH

z 3 T

Hz

γ =200

010

2030

4050

0

0.2

0.4

0.6

0.8 1

Radius [m

m]

~r⋅Er (k,r) [a.u.]

010

2030

4050

0

0.2

0.4

0.6

0.8 1

Radius [m

m]

30 GH

z 150 G

Hz

500 GH

z1 T

Hz

2 TH

z 3 T

Hz

γ =800

~r⋅Er (k,r) [a.u.]

Figu

re3.7:

Fourier

componentsof

therad

ialelectric

eld

r~Eras

afunction

ofrad

iusfor

frequencies

betw

een30

GHzand3THz.

Theeld

distrib

ution

sare

norm

alizedto

1at

r=0.

~Ez (k

;r)=

ik

20

2 q~(k) 24 8<:

K

0 ka I0

kr

,r<a

I0

ka K

0 kr

,r>a 9=;

+I0

ka !

K0

kb

I0

kb I0

kr ! 35

:

(3.13)

~Er (k

;r)=

k

20 q~(k) 24 8<:

K

0 ka I1

kr

,r<a

I0

ka K

1 kr

,r>a 9=;

+I0

ka !

K0

kb

I0

kb I1

kr ! 35

:

(3.14)

and

~B==c

~Er .I0 ,K

0 ,I1andK

1denote

themodied

Bessel-fu

nction

sof

zerothand

rst

order,

therelativ

isticLoren

tzfactor.

Thebunch

charge

isgiven

byq.

Details

ofthederivation

areoutlin

edin

appendix

C,where

asolu

tionof

theprob

leminclu

dinga

vacuum

cham

ber

ofnite

conductiv

ityispresen

tedas

well.

Theeld

sare

risingprop

or-

tional to I0 and I1 inside the ring charge but are decreasing proportional to K0 and K1

outside. The second term denotes the contribution of the cylindrical beam pipe to thetotal eld conguration. If we assume a line charge distribution (a = 0), the primaryelds will drop with K0(kr= ) or K1(kr= ), while the beam pipe causes a rise of theelds with I0(kr= ) or I1(kr= ). Notice that the contribution of the vacuum chamber canbe neglected for Fourier components corresponding to the optical range (k 107 1/m),because it is entirely suppressed by the factor K0(kb= ). For further considerations, itis sucient to consider the radial component of the electric eld. The longitudinal elec-tric eld component can be neglected compared to the transverse electric eld, because~Ez ~Er= . The azimuthal magnetic eld is related to the radial electric eld by thefactor of =c.

Figure 3.7 shows the Fourier components of the radial electric eld multiplied with 2r(Jacobian of cylindrical coordinates) for frequencies between 30GHz and 3THz. The ex-pressions are normalized to 1 at r = 0 to permit a comparison of the radial dependences.The screens at the TTF linac have a diameter of d = 35mm. The eld componentspropagating at larger distances from the charge distribution miss the screen and do notcontribute to the radiation. Therefore the radiated energy decreases towards lower fre-quencies, because an increasing fraction of the eld misses the screen. The radius b of thebeam pipe is also of in uence; the low frequency eld components do not vary signicantlywithin the beam pipe, hence the minimum acceptance is (d=2)2=b2. The fraction of theeld components contributing to the radiation process can be evaluated by

D =

R d=20 dr 2r ~ErR b0 dr 2r ~Er

: (3.15)

Spe

ctra

l acc

epta

nce γ = 100

γ = 200γ = 400γ = 800

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3f [THz]

Figure 3.8: Spectral acceptance D(f) of far infrared transition radiation due to the exten-sion of the electromagnetic elds traveling with the charge, Eq. (3.15), is plotted versusthe frequency of the electric eld Fourier component. Four dierent beam energies aredenoted by the Lorentz factor .

0 2 4 6 80

0.2

0.4

0.6

0.8

1

x

x ⋅ K

(x)

1

x

1e

Figure 3.9: The function x K1(x). The transverse source dimension is dened as x K1(x) = 1=e.

The quantity D is plotted versus the frequency of the Fourier component in Fig. 3.8. D(f)can be interpreted as a spectral acceptance function. It clearly shows the suppression oftransition radiation at low frequencies (f 3THz). For high frequency and low electronenergy one gets D(f) 1, since the electric eld traveling with the charge is connedwithin the area of the transition radiator.

For a quantitative description of the source dimension we consider

r ~Er(k; r) kr

K1

kr

! x K1 (x) (3.16)

with x = kr= . Figure 3.9 shows the function x K1(x). The transition radiation sourcedimension Rs is dened by the argument x where the function x K1(x) drops to 1=e.Hence

xK1(x) =1

e) kRs

= 1:66 ) Rs =

1:66 k

: (3.17)

The transverse dimension of the electromagnetic elds is proportional to the beam energy and the wavelength of the Fourier component ( = 2=k). With increasing Lorentzfactor the radial electric eld amplitude extends to larger radii, which can be under-stood in terms of the Lorentz transformation of the eld. The increase of the radiationsource dimension with respect to the radiation wavelength can be understood in terms ofdiraction.

The transverse extension of the electromagnetic eld limits the applicability of the opticalimaging technique for determining the bunch size. For low emittance bunches and large factors the transverse eld extension may exceed the whole bunch size and then ameaningful measurement is no longer possible. The transverse beam size r = r=

p is

decreasing proportional to the square root of the beam energy, when the strength of the

−1

0

1

x [mm]−1

0

1

y [mm]

0

0.2

0.4

0.6

0.8

1

E [

a.u.

]r

~

−0.20

0.2

−0.20

0.2

0

0.2

0.4

0.6

0.8

1

x [mm]y [mm]

E [

a.u.

]r

~

BeamBeam

Figure 3.10: Transition radiation eld Eq. (3.13) of a uniform charge distribution of radiusa = 230m, = 100 (left) and a = 83m, = 800 (right). The wavelength is 500 nm.Left: the shape of the uniform charge distribution can be recovered by the electromagneticelds. Right: the shape of the uniform charge distribution is smeared out.

magnetic lattice is scaled with . The source dimension of transition radiation, on theother hand, increases proportional to beam energy. It is evident, that transition radiationof xed wavelength can be used for the imaging of the transverse charge distribution toa maximum beam energy only [28], the so-called critical electron energy c. Equatingthe radial beam size r with the source dimensions of the radiation elds Rs = 1:66 c=kyields

c =

k2r

2

2:76

!1=3

: (3.18)

Assuming = 500 nm (optical transition radiation) and a normalized emittance of 5 106m, the critical Lorentz factor is c = 924 corresponding to an electron beam energyof about 470MeV. It is evident that tight focusing of the electron beam on the transitionradiator should be avoided.

The resolution limit of optical transition radiation can be visualized by calculating theradiation source of a uniform transverse charge distribution of radius a

(r) =1

a2

1 , r < a0 , r > a

: (3.19)

The eld of a ring charge distribution Eq. (3.13) is weighted with Eq. (3.19) and integrated.The evaluation, shown in Fig. 3.10, is performed numerically on a rectangular grid for twodierent beam energies using a normalized beam emittance of 5 106m. The transversebeam widths are a = 220m at = 100 and a = 80m at = 800. The left graph showsthe transverse electric eld distribution ( = 500 nm) for = 100 < c, the right graphfor = 800 > c. Below c the eld distribution is almost identical with the transversecharge distribution (3.19), above c the eld distribution is much wider and the chargedistribution cannot be resolved.

x

y

ξ

η

R

r

A

B

α

Transition Radiation Screen

Observation Screen

Figure 3.11: Determination of the distance from a point A located on the transitionboundary to a point B on an observation screen. This screen may also be the entranceaperture of the detection device.

3.3.2 Huygens-Fresnel Principle

The Ginzburg-Frank formula has been derived by solving the Maxwell equations for theelectromagnetic elds traveling with a charge and for the radiation eld propagating intothe backward hemisphere of the transition boundary. The transition boundary is assumedto be of innite extent to justify the ansatz of plane waves for the radiation eld. Thecharacteristics of the radiation eld are derived by the application of boundary condi-tions, that is the equality of the tangential electric and magnetic eld components onboth sides of the boundary, at the time the charge undergoes transition. The Ginzburg-Frank formula is therefore applicable only if the transition boundary is large comparedto the transverse extension of the electromagnetic elds traveling along with the bunchcharge, as is the case for optical transition radiation.

An alternative way to compute the spectral energy density of transition radiation is basedon the Fresnel-Kirchho scalar diraction theory. The Huygens-Fresnel principle statesthat every point on a primary wavefront (which is, here, the eld distribution on thetransition boundary) can be considered as a continuous emitter of spherical secondarywave amplitudes [29]. To describe the directionality of the secondary emissions, an incli-nation factor (1+cos ())=2 is introduced which has its maximum in the forward directionnormal to the primary wavefront and vanishes for = . The eld amplitude at a givenpoint is determined by the superposition of all secondary wave amplitudes.

The following numerical method is applied to derive the characteristics of transition ra-diation. The Fourier component of the electric eld ~Er (Eq. 3.14) is evaluated at a pointA = (x; y) on the transition boundary, yielding the value ~EA. Next, a point B = (; ) onan observation screen separated by a distance R from the transition boundary is selected.The distance r between A and B is (see Fig. 3.11)

r =q(R2 + (x )2 + (y )2) : (3.20)

0 0.02 0.04 0.06 0.08 0.10

2

4

6

8

θ [rad]

Spe

ctra

l ene

rgy

dens

ity [a

.u.] Huygens-

Fresnel

Ginzburg-Frank

Figure 3.12: Angular distribution of transition radiation as obtained from the Ginzburg-Frank formula and the numerical recipe using the Huygens-Fresnel principle. Parameters: = 200, = 2m and d = 35mm.

and is converted into a phase dierence AB = r=, where is the wavelength. Theamplitude of the eld at B is

~EB =Z Z

~EA (x; y)1 + cos ()

2

exp(iAB)q(R2 + (x )2 + (y )2)

dx dy : (3.21)

The integration is performed over the entire transition radiator yielding the sum of thesecondary wave amplitudes at the observation screen. The radiation intensity pattern isthen derived by calculating the absolute value of the amplitudes squared. The calculationhas to be performed for every Fourier component of Eq. (3.14). The Fresnel-Kirchhodiraction theory is a scalar theory. In case of transition radiation, however, the polar-ization of the electric eld vector has to be taken into account. The electric eld of thesecondary wave is polarized in the same direction as the electric eld of the bunch chargeat the transition boundary. Before being added at point B, the elds of the secondarywaves are decomposed into their horizontal and vertical polarization components.

Figure 3.12 compares the angular distribution of transition radiation as obtained withthe prediction of the Ginzburg-Frank formula and with the numerical calculation usingthe Huygens-Fresnel principle. The Lorentz-factor is = 200, the radiation wavelengthis = 2m and the diameter of the transition radiator is d = 35mm. There is an ex-cellent agreement between the result of the numerical calculation and the prediction ofthe Ginzburg-Frank formula. The conditions for the applicability of the Ginzburg-Frankformula are fullled, because the transition boundary is much larger than the transverseextension of the electromagnetic elds.

Figure 3.13 shows the angular distribution of transition radiation for dierent wavelengths.

0 0.04 0.08 0.12 0.160

0.25

0.5

0.75

1

θ [rad]

Inte

nsity

λ = 15µmλ = 50µmλ = 150µmλ = 500µmGinzburgFrank

γ = 100

Figure 3.13: Computed angular distribution of transition radiation using the Huygens-Fresnel principle. The prediction of the Ginzburg-Frank formula is indicated by the circles.The diameter of the transition screen is d = 35mm.

For wavelengths 50m the opening angle distribution is in good agreement with theGinzburg-Frank formula. For larger wavelengths the angular distribution becomes widerand an interference pattern appears. The deviation of the angular distribution from theGinzburg-Frank formula is explainable by diraction because the electromagnetic elddistribution incident on the transition screen at that wavelength has a greater transverseextension than the screen itself.

Diraction of the electromagnetic elds at the boundary yields a suppression of the far-infrared transition radiation energy density. A parabolic mirror (dM = 100mm diameter,fM = 200mm and R = 200mm) is used to collect the far-infrared radiation and to deliverit to a detector. The parabolic mirror denes an acceptance angle of max = 0:14 rad.The parts of the radiation eld which are missing the parabolic mirror due to diractiondo not reach the detector and cannot contribute to the measurement. Figure 3.14 showsan acceptance function indicating the suppression of low frequency eld components (f 3THz) due to diraction. The quantity

A(f) =

Rmirror r dr d

~ER(k; r)R10

R 20 dr d r ~ER(k; r)

(3.22)

is evaluated at the given distance R as a function of the radiation frequency f = kc=2.Expression (3.22) yields a second spectral acceptance function. It describes the collectioneciency of the detection system. The metallic screen inserted into the vacuum chambercan, in a sense, be considered as a high-pass lter for far-infrared transition radiation.The suppression of low frequencies has to be corrected for when using this radiation forbeam diagnostics.

γ = 100γ = 200γ = 400γ = 800

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

f [THz]

A(f

)

Figure 3.14: Fraction of the electromagnetic elds collected by a parabolic mirror at adistance R = 200mm from the transition boundary. The maximum acceptance angle ismax = 0:14 rad. The diameter of the transition screen is d = 35mm.

3.4 Coherent Transition Radiation

The spectral energy density of transition radiation per unit solid angle is described for asuciently large transition boundary by the Ginzburg-Frank formula (3.3)

U1 =e2

430c

2 sin2

(1 2 cos2 )2:

Equation (3.3) yields a proper description of transition radiation in the optical frequencyrange (! 31015 s1) where a bunch of N particles radiates N times the power of a singleparticle. However, at frequencies corresponding to the bunch passage time (! = 2=t,typically 1012 s1) the phase relations of the single particle emission processes have tobe considered leading to coherence eects. The situation is depicted in Fig. 3.15. If thetransition radiation wavelength exceeds the bunch length, the individual electrons emitcoherently while at shorter wavelength this phase relation is lost and the interferenceeects average out. The coherence can be described analytically by

U1 =N2e2

430c

Tq;!2 Lq;!2 2 sin2

(1 2 cos2 )2(3.23)

where Lq;! denotes the Fourier transform of the longitudinal part of the charge distri-bution and Tq;! the Fourier transform of the transverse part of the charge distribution.The result can be obtained directly by following the derivation outlined in appendix Bfor an extended charge distribution. Eq. (3.23) describes coherent transition radiationfor a bunch of N particles described by a charge distribution (x; y; z) = L(z)T (x; y)crossing a transition boundary perpendicularly in uniform motion. The latter descriptionof coherent transition radiation holds for a transition boundary exceeding the transverse

Coherent radiation

Incoherent radiation Sum signal

λ/2

λ/2

σz

Figure 3.15: Upper graph: phase relation and the resulting eld amplitude for the coherentradiation process. The wavelength is larger than the bunch length z. Lower graph: atshort wavelengths the individual eld amplitudes average out.

extension of the electromagnetic elds carried by the charge distribution. In this sectionwe neglect the nite size of the radiator in order to emphasize coherent transition radi-ation. It should be kept in mind that the formulae derived in this section have to becorrected by the spectral acceptance functions Eq. (3.15) and Eq. (3.22).

The example of a three dimensional Gaussian charge distribution

G(r; ; ) =Ne

2r

1p2 z

exp

r2

22r

!exp

2

22z

!where = z vt (3.24)

leads to

U1 =N2e2

430c

2 sin2

(1 2 cos2 )2exp

!

2

v2

2z + 22R sin

2!

(3.25)

3.4.1 Radiation Power

The spectral energy density depends on the number of particles per bunch squared (N2),so the emitted radiation power is N times larger (typically N = 1010) than in the non-coherent optical regime. The coherent peak power can be evaluated for the collider-modeand FEL-mode machine parameters. The machine parameters (in autumn 1999) are

Collider-mode FEL-mode

N 5 1010 6:25 109E [MeV] 330 330t [ps] 1.8 0.86

f [GHz] 88.5 185

The single particle spectral energy density at = 660 is U = 8 1017 eVs. The radiatedcoherent energy can be derived using

U = N2 dUsp

d!

!

!(3.26)

where the term !=! denotes the frequency band under consideration. Under the as-sumption of a Gaussian longitudinal bunch charge distribution of variance t the centralfrequency is ! = 1=t and the frequency interval ! can be evaluated using

Z 1

0d! exp

!22t

=

p

2t: (3.27)

The energy radiated in the coherent frequency band amounts to U = 1:6 102 J for thecollider-mode and U = 5:1 104 J for the FEL-mode. The peak radiation power is

P =U

6t(3.28)

where the bunch passing time of a Gaussian bunch is assumed to be 6 times the rmsbunch length containing more than 99% of the charges. The peak power radiated by co-herent transition radiation is P = 1440MW for the collider-mode and P = 99MW for theFEL-mode. Coherent transition radiation from short and high charged electron bunchesis therefore an extremely intense source of millimeter and sub-millimeter wavelength radi-ation. Notice that the values quoted for the emitted peak power are reduced by spectralacceptance functions Eq. (3.15) and Eq. (3.22). The expected peak power radiated outof the beam pipe is therefore attenuated by a factor of 20 to 80 depending on the beamenergy and radiation wavelength.

3.4.2 Investigation of the Longitudinal and Transverse Form

Factor

An application of coherent transition radiation is the determination of the longitudinalcharge distribution L [30, 31, 32, 33, 34]. The investigation of Eq. 3.23 shows that thetransverse component of the charge distribution has to be minimized. This can be done byfocusing the beam onto the transition radiator, i. e. T ! (r)=2, hence Tq;! ! 1. Theresulting coherent radiation spectrum depends on the Fourier transform of the longitudinalcharge distribution only (see Eq. 3.23), hence

U1(!; ) N2e2

430

Lq;!2 4c sin2

(1 2 cos2 )2: (3.29)

In accordance with the nomenclature in nuclear physics, the Fourier transformation of thecharge distribution is called the bunch form factor. Figure 3.16 shows a set of longitudinalcharge distributions L and their corresponding coherent power spectra jLq;!j2. The uppergraph depicts three Gaussian longitudinal charge distributions of dierent rms width andtheir corresponding frequency spectra. The shorter the bunch is, the higher frequenciescontribute to the coherent radiation spectrum. The lower graph depicts three dierentlongitudinal bunch charge distributions of equal rms width but of dierent shape. In

charge distribution

−2 0 2

0.2

0.6

1

time [ps]

char

ge [a

.u.]

0 0.5 1

0.1

0.6

1

frequency [THz]

pow

er [a

.u.]

0 0.5 1

0.2

char

ge [a

.u.]

frequency [THz]time [ps]

0.2

0.6

1

−2 0 2

frequency spectrum

pow

er [a

.u.]

0.6

1

0.2

0 0.5 1

Figure 3.16: Upper graph: Gaussian longitudinal charge distributions of dierent varianceand their corresponding frequency spectra. Lower graph: dierent longitudinal bunchcharge distributions and their frequency spectra.

frequency domain, the three distributions can be distinguished by the visibility of thesecondary and higher maxima of the spectra. For frequencies lower than the rst minimumit is dicult to distinguish the spectra of the three bunch charge distributions. To arst approximation, the rms bunch length can be determined from the rms width of thelongitudinal form factor in the range of the central maximum.

In uence of the Transverse Charge Distribution The nite acceptance angle max

of the optical imaging system permits the detector a limited side-view on the bunchcharge distribution. The projected component of the transverse charge distribution hasto be taken into account for the energy density of coherent transition radiation. For aGaussian bunch charge distribution we obtain

U1 N2e2 exp

!

2

v2

2z + 22r sin

2 max

!: (3.30)

The transverse form factor Tq;! exp (!222t sin2 max=v

2) limits the determinationof the longitudinal form factor Lq;! exp (!22z=v

2). If, however, the condition z r sin max is fullled, the in uence of the transverse bunch charge distribution is negligi-ble. The parabolic mirror at the TESLA Test Facility has a maximum acceptance angle ofmax = 0:14 rad, leading to the condition that r 3:9mm (Collider-mode, t = 1:8 ps)and r 1:8mm (FEL-mode, t = 0:86 ps). A normalized emittance of Nx = 5 106mleads to a beam size of x = 91m ( = 660) which is a lot smaller than the latterboundaries.

0 100 200 300 4000

0.5

1

Frequency [GHz]fo

rm fa

ctor

0 100 200 300 4000

0.5

1

Frequency [GHz]

form

fact

or

0 100 200 300 4000

0.5

1

Frequency [GHz]

form

fact

or0 100 200 300 400

0

0.5

1

Frequency [GHz]

form

fact

or

0 100 200 300 4000

0.5

1

Frequency [GHz]

form

fact

or

0 100 200 300 4000

0.5

1

Frequency [GHz]

form

fact

or

ρ ||q,ω ρ ||

q,ω

ρ⊥q,ω ρ⊥

q,ω

ρ⊥q,ωρ ||

q,ω ρ⊥q,ωρ ||

q,ω⋅ ⋅

Figure 3.17: In uence of the transverse form factor on the product form factor deter-mined during a measurement using coherent transition radiation. Left column: A largetransverse bunch size causes the signicant decrease of the transverse form factor towardshigher frequencies and dominates the product of the longitudinal and transverse form fac-tors. The product form factor is signicantly dierent from its longitudinal part and willlead to a larger bunch length. Right column: A small transverse beam size (nearly con-stant transverse form factor) has only little in uence on the product of both form factors.The product form factor correctly represents the longitudinal charge distribution.

The in uence of the transverse form factor as described by Eq. (3.30) is shown in Fig. 3.17:let us assume a Gaussian bunch of z = 500m and r = 10mm (left column). The lon-gitudinal form factor does not vanish for frequencies below 300GHz while the transverseform factor vanishes already at 50GHz. The spectroscopic measurement of the bunchform factor, however, cannot distinguish the two and detects their product (or the entirethree-dimensional form factor of the bunch charge distribution). In the present scenario,the transverse part of the form factor is suppressing the longitudinal part and the prod-uct pretends a bunch length larger than it actually is. The problem can be solved byfocusing the beam as strongly as possible onto the transition radiator to minimize thetransverse beam dimension. The right column shows the form factors transverse beamsize of r = 600m. The product kq;! ?q;! is now dominated by the longitudinal formfactor and a bunch length measurement using coherent radiation is possible.

Chapter 4

Transverse Phase Space Tomography

The transverse phase space distribution of an electron beam can be explored by applyinga quadrupole scan in combination with tomographic image reconstruction techniques [35,36]. For this purpose a set of quadrupoles is used to rotate the phase space distribution inwell-dened angular steps between a reconstruction point zi before the quadrupoles andan observation point zf behind the quadrupole magnets. The transverse beam densitydistribution at the observation point is recorded by means of optical transition radiationand a CCD camera. The horizontal and the vertical beam proles at dierent phase spacerotation angles allow a reconstruction of the initial horizontal and vertical phase spacedistribution at zi.

4.1 Theory of Phase Space Tomography

This section introduces the \ltered backprojection algorithm", a standard algorithmfor tomographic image reconstruction. The application of the ltered backprojection totransverse beam dynamics and the procedure of the measurement will be presented. Thesteps of the analysis are illustrated by simulation results.

4.1.1 The Radon Transform

Take a two-dimensional Gaussian distribution g(x; y) which is rotated in space aroundthe z-axis. The rotation by an angle is described by a 2 2 matrix MR

MR =

cos sin sin cos

; (4.1)

where the coordinates x and y of the distribution g transform according to u

v

!=MR

x

y

!: (4.2)

The rotation MR conserves the area of the initial distribution since det(MR) = 1. HenceZ~g(u; v) du dv =

Zg(x; y) dx dy : (4.3)

x

y

u p (u)~

φ

−0.1−0.05

0

050

100150

2000

0.050.1

1

φ [°]

p (u

) [a

.u.]

φ~

u [mm]

φ

Figure 4.1: The Radon transform of a two-dimensional Gaussian distribution. Uppergraph: contour plot of the distribution g(x; y) and the rotated coordinate system (u; v).The distribution is projected onto the u-axis yielding ~p(u). Lower graph: The Radontransform of the Gaussian distribution. 18 projections are displayed as a function of therotation angle .

The rotated distribution ~g is then projected onto the u-axis yielding ~p(u). The projectionis not a simple geometric projection, it has to be understood in terms of an integrationof the distribution ~g along a \line of projection". We obtain

~p(u) =Z~g(u; v) dv (4.4)

or by using an area integral over the entire distribution

~p(u) =Z~g(~u; v)(u ~u)d~udv : (4.5)

The rotation can now be included into the projection by inserting Eq. (4.3) into Eq. (4.5)yielding

~p(u) =Zg(x; y)(u x cos y sin) dx dy ; (4.6)

where ~u = x cos+ y sin has been used. The index denotes the projection angle . Theset of projections ~p(u), 0 , is called the Radon transform of the distributiong(x; y) [37].

Figure 4.1 shows the initial distribution g, the coordinate systems (x; y) and (u; v) coupledby the rotation matrixMR and a projection ~p(u) at an angle . The Radon transform isoften displayed in a two-dimensional plot where the projections are shown as a functionof the rotation angle.

4.1.2 The Filtered Backprojection Algorithm

The reconstruction gR(x; y) of the initial distribution g(x; y) is obtained formally [38] by

gR(x; y) =Z

0p(x cos+ y sin) d (4.7)

where x cos + y sin = u according to Eq. (4.2). The backprojection described byEq. (4.7) is illustrated in Fig. 4.2. The proles of a Gaussian charge distribution, eval-uated at four rotation angles (0, 45, 90 and 135) are shown at the fringe of thelower contour plot (left column). The reconstructed distribution is computed by project-ing these proles back through the reconstruction plane at their specic projection angleeach of them yielding a ridge of Gaussian cross-section. The integral sums the contribu-tions from the individual projections and yields the reconstructed distribution gR. Thestar-like, wavy structure at the outer rim of the reconstructed image is due to the limitednumber of projections. Increasing the number of projections averages these artifacts out.Figure 4.3 shows the results of the reconstruction for projections every 20 and 10. TheRadon transform consisting of 18 projections in equi-angular intervals yields a reconstruc-tion without noticeable wavy artifacts.

However, even with arbitrarily ne angular subdivision the backprojection described byEq. (4.7) is not the exact inverse of the Radon transform. A careful inspection of Fig. 4.2and Fig. 4.3 shows that the width of the center peak of the reconstructed distribution islarger than the original width. The problem is further illustrated by Fig. 4.4: supposethe initial distribution consists of two Gaussian peaks of the same height but of dierentwidth . The projections then contain two Gaussian peaks of dierent heights. Since thebackprojection is a linear operation, the reconstructed image will have two peaks of dier-ent heights as well. This deciency can be cured using a lter which enhances structuresof large curvature corresponding to high \spatial frequencies" f . The projection ~p(u)is convoluted with a spatial lter function w(u) combining a frequency ramp-functionjf j which is cut o at a maximum frequency fcut corresponding to half of the samplingfrequency. The lter function is

w(u) =1

82

Zjf jL(f) exp (iuf) df : (4.8)

Figure 4.2: The backprojection of a Gaussian distribution. The original distribution isshown on the left side. The lower left graph illustrates the projections of the distributionin angular steps of 45. The graphs on the right side show the backprojection. The prolesare projected back through the area of reconstruction under their respective projectionangle each of them yielding a ridge of Gaussian cross-section. (The ridge might be thoughtof a sheet of paper placed onto the area of reconstruction which is folded such that thecross-section matches the beam prole.) These backprojections add constructively toproduce the reconstructed image.

We obtain the following expression for the ltered projection ~pF (u)

~pF (u) =Z~p(u

0)w(u u0) du0 : (4.9)

The ramp function is responsible for the enhancement of large spatial frequencies [38]. InFig. 4.4 it can be observed that the convolution of the projection with the lter functionyields proles of equal height but of decreased width. A side-eect of the convolution is theappearance of negative components in the projection ~pF (u). These negative components

Figu

re4.3:

Theback

projected

Radon

transform

isshow

nfor

adieren

tnumber

ofpro-

jections.

Left

graph:9projection

s.Righ

tgrap

h:18

projection

s.Thestar-like

artifactsat

thefrin

geoftherecon

structed

image

vanish

asthenumber

ofprojection

sisincreased

.

−0.1

−0.08

−0.06

−0.04

−0.02

00.02

0.040.06

0.080.1

0

1000

2000

3000

4000

5000

6000

−0.1

−0.05 0

0.05

0.1

0.15

0.2

0.25

Fp (u) p(u)

uu

Figu

re4.4:

Theprojection

oftwoGaussian

peak

sof

dieren

twidth

butequal

heigh

tlead

sto

twoGaussian

sofdieren

twidth

anddieren

theigh

t(U

pper

rightgrap

h).Usin

galter

function

thenarrow

erpeak

enhances,

butdoes

not

quite

reachtheheigh

tof

the

largepeak

,becau

sehigh

spatial

frequencies

aresuppressed

bythelow

-pass

lter

L(f).

0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1

L(f)

f / f cut

0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1

L(f)

f / f cut

−2 0 2

−0.5

0

0.5

1

1.5

w(u

)

u [a.u.] −2 0 2

−0.2

−0.1

0

0.1

0.2

0.3

0.4

w(u

)

u [a.u.]

Figure 4.5: The low pass lter L(f) and the lter function w(u) for smoothing parameters = 0 (upper and lower left) and = 1 (upper and lower right). The right lter curvecauses a stronger smoothing, because of the suppression of the secondary maxima.

cancel a positive surplus in the fringe of the area of reconstruction which originates fromthe backprojected proles at other angles.

A spatial frequency low-pass lter is introduced because of the discrete sampling of theprojections by the detection device. Let s be the width of the discrete sampling interval,then the maximum frequency that can be recovered is of the order of 1=2s [38, 39]. Aconvenient low-pass lter function is [40]

L(f) =

(1 f

fcut; jf j fcut

0 ; jf j > fcut; (4.10)

where the parameter fcut denotes the cut-o frequency and [0; 1] the smoothness pa-rameter. For = 0 the low-pass is an edge lter which is, in combination with the rampfunction jf j, most sensitive to curvy structures because the high frequency componentsare included. For > 0 the higher frequency components are damped, which results in asmoothing of the reconstructed image. The lter function now becomes

w(u) =1

82

Z fcut

fcutjf j 1

jf jfcut

!exp (iuf) df (4.11)

=1

42

Z fcut

0f

1

f

fcut

!cos (uf) df (4.12)

which is evaluated by partial integration to

w(u) =f 2cut42

((fcutu) (fcutu)) (4.13)

Figure 4.6: The backprojected ltered Radon transform. The reconstructed peak is of thesame width and height as the original. The blurred reconstruction in the fringe of gure4.2 is corrected by the negative components of the ltered projections.

Figure 4.7: The reconstruction of the object is shown for 9 (left) and 18 (right) projec-tions. The star-like, wavy artifacts at the fringe of the reconstructed image vanish for anincreasing number of projections. In contrast to Fig. 4.3, the projection have been lteredusing Eq. (4.13) prior to the reconstruction. The smoothing parameter = 0 has beenused.

where

(v) =

(cos v1

v2+ sin v

v; v 6= 0

12

; v = 0(4.14)

(v) =

(2 cos vv2

+1 2

v2

sin vv

; v 6= 013

; v = 0: (4.15)

The function w(u) is shown in Fig. 4.5 for dierent smoothing parameters = 0 and = 1. In the following = 0 will be used.

The result of the ltered backprojection for the two-dimensional Gaussian distribution isshown in Fig. 4.6 and Fig. 4.7. The backprojected and the original image match at thecenter of the image. The star-like artifacts disappear for a sucient number of projectionsused for the reconstruction as shown for the reconstructions of projections every 45, 20

and 10.

4.1.3 Discrete Implementation of the Filtered Backprojection

Algorithm

The implementation of the ltered backprojection algorithm is a straight forward dis-cretization of Eq. (4.7), Eq. (4.9) and Eq. (4.13) [38, 41]. The detecting device samplesthe projections of the transverse charge distribution in discrete steps yielding N samplingpoints of equal spacing uk = ku, k = N=2 + 1; :::; N=2. The projections are obtainedat m equally spaced angles j = j= M , j = 0; :::; M 1.

The integral (4.9) is approximated by the sum [38]

~pFj (ku) =N=2X

n=N=2+1

~pj(nu)w(jn kju) u (4.16)

where k = N=2 + 1; :::; N=2. The lter function w(u) evaluated at the sampling pointsuk [40] is

w(uk) =

8>>><>>>:

14

6; k = 0

2k2

; k 6= 0 and even

12k2

; k 6= 0 and odd

: (4.17)

The discretization of the backprojection (A.13) leads to

gR(x; y) =M

M1Xj=0

pFj

x cos

jM

+ y sin

jM

: (4.18)

The area of reconstruction is approximated by a rectangular grid [xl; ym], where l; m =

1:::N . At every point (xl0 ; ym0) the value of uj = x cos

j= M

+ y sin

j= M

at a

given j is evaluated. The uj then have to be compared to the measured uk. A linearinterpolation routine is used to get the best estimate for the uj and hence for ~p

Fj(uj) such

that

~pFj (uj) =(k + 1) uj

u

~pFj (ku) +

uju

k~pFj((k + 1)u); (4.19)

where k u=u < k + 1 and k = N=2 + 1; :::; N=2.

The algorithm described above has been implemented as a Matlab [42] code. The convo-lution and the backprojection routines are linked in C code for faster processing. Figure4.8 shows the performance of the tomographic image reconstruction using the exampleof three asymmetrically arranged Gaussian peaks of dierent width and height. Thepresented simulation covers an angular interval of 180 and calculates 180 projectionswith a resolution of 100 pixels in x and y. The pixel-to-pixel agreement,

Pij(gR(xi; yj)

g(xi; yj))2=P

ij g(xi; yj))2, is better than 1%.

−0.50

0.5−0.5

00.5

1

x [mm]y [mm]

Inte

nsity

−1 0 1−0.5

0

0.5

x [mm]

y [m

m]

0.5

0

−0.50

0.5−0.5

00.5

1

x [mm]y [mm]

Inte

nsity

0.5

0

−1 0 1−0.5

0

0.5

x [mm]

y [m

m]

Figure 4.8: Simulation with an asymmetric distribution consisting of three Gaussian peaksof dierent height and width. The upper graph shows the initial distribution, the lowergraph the result of the reconstruction. The pixel-to-pixel agreement of the reconstructedand the original distribution is better than 1%.

Figure 4.9: Performance of the ltered backprojection algorithm applied to a Radontransformation covering an angular interval of less than 180. Shown is the reconstructionof a two-dimensional Gaussian with 6 projections from 0 to 60 (top), 9 projections from0 to 90 (middle) and 12 projections from 0 to 120 (bottom). The reconstructed imageis distorted, tilted and contains negative contributions sideways to the maximum.

4.1.4 Reconstruction Artifacts

The Radon transform has to cover an angular interval of 180 to avoid reconstruction ar-tifacts. Problems rise if the Radon transform contains a smaller angular interval. Figure4.9 illustrates the problem. A Gaussian phase space distribution is chosen for graphicalclearness. The distribution is reconstructed according to the algorithm of ltered back-projection. The angular interval spans 60 (top), 90 (middle) and 120 (bottom). Thecontour plots on the right show that the reconstructed image is tilted and distorted. Fur-thermore, negative contributions next to the maximum appear. In general the negativecontribution of the ltered proles are needed to cancel the positive surplus of the projec-tions taken at the other rotation angles. If projections at certain angles are missing thiscancelation cannot occur and negative contributions remain at the sides of the angularreconstruction interval. The positive surplus at the fringe of the area of reconstruction

−0.1−0.05

0 0.050.1

−500

50100

1500

0.5

1

x [mm]

inte

nsity

[a.u

.]

φ [ ]o

Figure 4.10: Performance of the ltered backprojection algorithm with a Radon transformsampled with varying angular spacing. As a result, the Gaussian peak becomes distortedand the contributions of the backprojected proles do not cancel in the fringe regions.

within the angular interval is also not compensated due to the missing negative contribu-tion. This eect causes the tilt of the reconstructed distribution.

Figure 4.10 shows artifacts occurring when the projections are not equally spaced. TheGaussian peak is distorted and non-zero contributions occur in the fringe of the area ofreconstruction.

Hence, a reconstruction free of artifacts is obtained only if the Radon transform coversan interval of 180 where the individual projections are recorded at a constant angularspacing. If the latter condition cannot be fullled, the artifacts can be avoided by aninterpolation of adjacent proles.

4.1.5 Application of the Filtered Backprojection Algorithm to

Transverse Beam Dynamics

The task is to determine the transverse phase space distribution i(x; x0; y; y0) at a re-

construction point zi of the beam line. x and y denote the spatial coordinates in thehorizontal and vertical plane, x0 and y0 the horizontal and vertical angular divergence.Two, or more, quadrupoles are used to rotate the phase space distribution. The detectiondevice, an optical transition radiation screen and a CCD camera, are located at a positionzf downstream of zi. The setup is shown in Fig. 4.11. Under the assumption of linearbeam dynamics, the phase space coordinates x, x0, y and y0 are transformed by a 4 4transfer matrix M 0

BB@xx0yy0

1CCAf

=M

0BB@xx0yy0

1CCAi

(4.20)

where the indices i and f denote the position zi and zf respectively. The integrated phasespace density is conserved yielding

Zf(xf ; x

0f ; yf ; y

0f) dxf dx

0f dyf dy

0f=

Zi(xi; x

0i; yi; y

0i) dxi dx

0i dyi dy

0i : (4.21)

The OTR screen projects the four-dimensional phase space distribution onto the spatialaxes x and y yielding a transverse intensity distribution. This distribution can be pro-jected onto either spatial axis to obtain a transverse beam prole p(x) and q(y). Theseproles are identical to a direct projection of the four-dimensional distribution onto thespatial coordinate axes as presented by Fig. 4.12. The projection of the four-dimensionalphase space distribution onto the horizontal x-axis at the position z is

pz(xf) =Zf (xf ; x

0f ; yf ; y

0f) dx

0f dyf dy

0f =

Z~f(xf ; x

0f ) dx

0f : (4.22)

or by using an integral over the entire phase space distribution

pz(xf ) =Z

~f (xf ; x0f)(xf ~xf) d~xf dx

0f : (4.23)

A similar relation holds for the vertical projection qz(y). The beam transfer described

by the matrix M can be included in Eq. (4.23) by

pz;M(xf ) =Z~i(xi; x

0i)(xf M11xi M12x

0i) dxi dx

0i : (4.24)

Equation (4.24) denotes the Radon transform of a phase space distribution ~i(xi; x0i) with

respect to a generalized rotation described by the matrixM . Notice thatM performs notonly a rotation, but also a shearing of the initial distribution. The ltered backprojectionalgorithm is a priori not suited to handle these generalized rotations. A transforma-tion of the Radon transform pz;M(xf ) (Eq. (4.24)) to a Radon transform originated fromCartesian rotations ~pz;(u) (compare with Eq. (4.6))

pz;(xf) =Z~i(xi; x

0i)(xf xi cos x0i sin) dxi dx

0i : (4.25)

has to be derived to make use of the tomographic image reconstruction. denotes theCartesian rotation angle. The arguments of the -function have to be compared to performthe latter transformation. Dening the rotation angle by

cos =M11q

M211 +M2

12

and sin =M12q

M211 +M2

12

(4.26)

1

X

X’

M

X’

0

2I )1I( ,

ZZ

X

y

xz

Figure 4.11: The beam line consisting of a quadrupole doublet and an OTR screen forthe quadrupole scan is displayed in the upper graph. The associated phase space beforethe rst quadrupole magnet and at the screen is shown below. For graphical clearnessGaussian beams are assumed. The beam transfer matrixM is a function of the quadrupolegradients generated by the currents I1 and I2. The detector, usually a CCD camera torecord the transition radiation intensity, is not shown.

q(y) q(y) y’

y

x

y

p(x)

x’

x

p(x)

Figure 4.12: The transverse intensity distribution on the OTR screen is a two-dimensionalprojection of phase space onto the spatial coordinates x and y. The beam proles obtainedfrom the projection of the OTR intensity distribution on the x and y axes and from theprojection of the phase space distribution (~f (x; x

0) and f(y; y0)) on the respective spatial

axes are equal.

Equation (4.24) can be written as

pz;M(xf )=Z~i(xi; x

0i)

0@qM2

11+M212

0@ xfq

M211+M

212

xi cosx0i sin1A1A dxi dx0i (4.27)

and after simplifying the argument of the -function

pz;M(xf )=1q

M211+M

212

Z~i(xi; x

0i)

0@ xfq

M211+M

212

xi cosx0i sin1A dxi dx0i : (4.28)

We arrive at the Cartesian rotation transform by introducing

~pz;(u) =q

M211 +M2

12

pz;M(xf ) with u =

xfqM2

11 +M212

: (4.29)

In other words, the measured intensity proles are scaled byp

M2

11+M2

12, while the axis of

projection ~x is scaled by 1=p

M2

11+M2

12to compensate for the distortions introduced by the

Phase spacedistribution ρ (x ,x’ )

x’ x

x

x’

# ch

arge

s

φ x~

# ch

arge

s

uφ

# ch

arge

s

~0 0 0

Beam profileRadon transformp (x )

z,M 1

Cartesian Radontransform p (u)

z,M

~

Figure 4.13: A Gaussian phase space distribution is shown as a function of x and x0 ina three dimensional view and as a contour plot (upper graph). Middle graph: Radontransform according to relation (4.24). The transport matrix elements are converted torotation angles. The quadrupole magnet strength has been chosen such that the angularintervals of consecutive projections are equal. Bottom graph: the associated CartesianRadon transform according to Eq. (4.29).

beam transfer matrix. A similar derivation holds for the vertical plane, where M11 andM12 are substituted by M33 and M34 [43, 44].

Equation (4.26) denes the Cartesian rotation angle in terms of the transport matrixelements. The strength of the quadrupole magnets in the scanning section has to be setsuch that the Radon transform spans an angular interval of 180 and that the individualproles are taken at a constant angular spacing.

The previous steps are illustrated in Fig. 4.13. The upper graph shows a Gaussian phasespace distribution at the position zi. The middle graph shows the Radon transformof the Gaussian distribution as obtained from a simulation using the beam line shownby Fig. 4.11. Equation (4.24) has been applied. Notice the defocusing of the beam asa function of the phase space rotation angle. The magnetic elds of the quadrupolemagnets have been chosen such that the angular spacing between consecutive projectionsis equidistant. The lower plot shows the associated Cartesian Radon transform accordingto Eq. (4.29). Each prole is corrected in height and width as described in Eq. (4.29).

4.1.6 Determination of the Beam Parameters

The transverse beam parameters ij, 1 i; j 2 and 3 i; j 4, are evaluatedstatistically as outlined in appendix A. Starting from the reconstructed phase spacedistribution gR we obtain

ij =Zi j gR (i; j) di dj (4.30)

where = (x; x0; y; y0). The beam centroid is assumed to be located at the origin of thecoordinate system. The beam emittance is evaluated by

x = 1122 212 and y = 3344 234 : (4.31)

4.1.7 Incorporation of Space Charge Eects

The high bunch charges in the TTF linac have a signicant in uence on the focusing ofthe beam, in particular in the low energy section. The space charge forces have to beincluded in the beam transfer matrixM of the accelerator section used for the quadrupolescan. It is sucient to consider linearized space charge forces by dividing the lattice inintervals of about 1mm length. At each point, defocusing lenses with a strength

kx =2Nrep2

1

x (x + y) z 3(4.32)

ky =2Nrep2

1

y (x + y)z 3(4.33)

are added to the magnet lattice. Here, denotes the Lorentz factor, re is the classicalelectron radius, x, y and z the rms bunch dimensions. Gaussian shaped bunches areassumed. The transfer matrix M with space charge defocusing is evaluated in severaliterations [20]:

1. Assume reasonable transverse beam parameters 11, 12, 33 and 34 at the begin-ning of the accelerator section used for the quadrupole scan.

2. Calculate the beam transfer matrix without space charges and record the beamdimensions x, y and z at every subdivision.

3. Introduce space charge kicks according to formulae (4.32) and (4.33) where therequired beam dimensions are obtained from the previous calculation. Record theobtained beam dimensions at every subdivision.

4. Repeat step 2 until the result of the beam transfer matrix converges. The calculationneeds approximately 10 iterations.

The beam transfer matrix M is used to calculate the setting of the quadrupole currentsto record beam proles spaced by equal phase space rotation angles. The ltered back-projection algorithm is applied to reconstruct the phase space distribution and a set ofreconstructed beam parameters at the beginning of the accelerator section is evaluated.If the initially assumed and the reconstructed beam parameters do not coincide, the ini-tial parameters in 1) have to be varied and the procedure has to be iterated until thereconstructed and the initial beam parameters are self-consistent.

4.2 Parabola Fit Analysis

The transverse emittance and the transverse beam parameters can also be obtained by aparabola t analysis. The beam size xrms at the measurement point zf is related to thebeam parameters at the reconstruction point zi by

x2rms =M211 11jzi 2M11M12 12jzi +M2

12 22jzi (4.34)

−4 0 4 8

5

M / M [1/m]11 12

0

1

2

3

4

x 10−4

σ /

M122

2

-6ε = 2.47⋅10 m

11 σ = 5.8⋅10 m-6

12 σ = 1.3⋅10 m-6

0

1

2

3

4

x 10−4

σ /

M122

2

5

−4 0 4 8M / M [1/m]11 12

-6ε = 1.06⋅10 m

11 σ = 1.0⋅10 m-6

12 σ = 7⋅10 m-10

Figure 4.14: Quality of the parabola t technique for a Gaussian charge distribution (left)and the asymmetric charge distribution (right). The reconstructed beam parameters areshown in the graphs. The input parameters are: = 1:0 106m, 11 = 1:0 106m,12 = 0m (left graph) and = 1:8 106m, 11 = 2:0 106m, 12 = 1:0 106m (rightgraph).

where M11 and M12 denote the cosine- and sine-like elements of the transfer matrix.Dividing Eq. (4.34) by M2

12 yields

x2rms

M212

=M2

11

M212

11jzi 2M11

M1212jzi + 22jzi (4.35)

Hence x2rms=M212 depends parabolically on the ratio of M11=M12. A parabolic t to the

data yields the beam parameters at the reconstruction point zi. The beam emittance isthen derived by Eq. (4.31). Figure 4.14 shows the quality of the parabola t technique. AGaussian charge distribution yields a perfect parabolic shape and the second-order poly-nomial t recovers the initial beam parameters of = 1 106m, 11 = 1 106m2 and12 = 0m. The parabola t for an asymmetric charge distribution consisting of threeGaussians of dierent height and width does not work well. The initial beam parameters = 1:8 106m, 11 = 2 106m2 and 12 = 1 106m cannot be recovered by the ttingprocedure.

The conclusion is that the parabola t technique is suitable only for nearly Gaussiancharge distributions but fails for more complicated transverse charge distributions.

Chapter 5

Determination of the Transverse

Beam Parameters

Quadrupole scans using the beam produced by the rf photo injector and the thermionicinjector will be presented in this chapter. The experimental setup, the processing of thebeam intensity proles, the steps of the tomographic reconstruction and a discussion ofthe results will be given.

5.1 Experimental Setup

The experimental setup consists of the optical diagnostics system which is used to recordthe transverse beam intensity distributions and the quadrupole scan accelerator section.

Radiation Screens

Precise beam size measurements are performed with transition radiation which is producedat a screen consisting of a 40 nm thick layer of aluminum evaporated onto a 25m thickKapton foil. The foil (34:5mm 28:5mm) is stretched in a metallic frame (50mm 40mm). The screen is mounted at an angle of 45 with respect to the axis of the acceleratoryielding a 90 re ection of the radiation. A transition radiation, a scintillating (Al2O3)and a calibration screen are mounted on a vertical mover and can be inserted into thebeam pipe.

Camera and Readout

The light is collected by an achromatic lens (f = 100mm) and focused directly onto theCCD array of a camera. The light intensity is regulated by gray density lters. Themagnication is V = 0:25 and the resolution is 60m [45]. Standard, interlaced CCDcameras with 512 768 pixels of 20 20m size are used. The images are digitized by aMacintosh frame grabber system and made available for data processing.

Calibration Screen

The calibration screen [46] is an aluminum sheet with six markers arranged in a rectan-gular pattern (2 3). The distance of neighboring markers is 10mm. The calibration

screen is used to convert the pixels of the CCD image to the transverse dimension of thebeam intensity distribution.

Magnetic lattice

The quadrupole scan section consists of two quadrupole magnets which can be poweredseparately. The 102mm long magnets are separated by 64mm. The gradient g is givenas a function of the quadrupole current I by

g [T/m] = 0:0039281 + 0:71659 I [A] : (5.1)

The distance between the second quadrupole and the location of the transition radiationscreen is 1873mm.

5.2 Measurements with the Photo Injector Beam

The rf gun photo injector produced 20 bunches of 1 nC charge at a repetition rate of1MHz. The beam energy in the quadrupole scan section was 16MeV. Space chargeeects have to be taken into account for the evaluation of the beam transfer and duringthe tomographic reconstruction by the application of the self-consistency method outlinedin chapter 4. Tomographic measurements using the photo injector beam will be used topresent the steps of the analysis.

4080

0

40

80

0

40

80

120

x [bin]y [bin]

char

ge d

ensi

ty [a

.u.]

4080

0

40

80

0

40

80

x [bin]y [bin]

char

ge d

ensi

ty [a

.u.]

4080

0

40

80

0

40

80

x [bin]y [bin]

char

ge d

ensi

ty [a

.u.] 0

40

80

−20

20

60

x [bin]y [bin]

char

ge d

ensi

ty [a

.u.]

4080

Figure 5.1: Image processing of the acquired beam intensity proles. Upper left: rawimage. Upper right: background subtracted and X-rays removed. Lower left: noisereduction by a two dimensional Fourier transform lter. Lower right: Smoothing of thefringe of the distribution.

0.8 1.6 2.4

0

200

400

600

800

x [mm]

p(x)

[a.u

.]

0

Raw data

Fourier filter

Averaged

Figure 5.2: Projections belonging to the upper left, lower left and lower right graph ofFigure 5.1.

5.2.1 Data Processing

The post-processing of the beam intensity distributions is shown by Fig. 5.1. The raw im-age taken by the CCD has a non-zero oset and noise is superimposed (upper left graph).The subtraction of a background image, taken without beam at the same quadrupolesetting, removes the oset and corrects for damaged pixels on the CCD. The dark current(electrons accelerated by the strong elds of the rf gun which do not originate from thephoto eect at the cathode) is subtracted by the background picture as well. X-rays, cre-ated by the beam passing the screen, cause excessive signals of single CCD pixels. Thesespikes are removed and replaced by the average of the signals of the neighboring pixels(upper right graph). A two dimensional Fourier transformation is performed to removenoise from the image. The contour line including 86% (2f ) of the charges is evaluated.The charges included by the contour line are used for the inverse Fourier transformationshown by the lower left graph in Fig. 5.1. The high frequency noise is suppressed yieldinga smooth central maximum. The low frequency components, however, become visible inthe fringe of the image. Simulation calculations show that these curvy structures are ar-ticial and need to be suppressed. An averaging procedure is applied for the pixel signalswith less than 5% of the maximum signal. These signals are replaced with the average ofthe 20 next neighbors. The resulting image shows a clear peak rising smoothly from thebackground. Figure 5.2 shows the intensity proles belonging to the upper left, lower leftand lower right graph of Fig. 5.1. The overall shape of the prole does not change, thenoise is reduced signicantly.

The beam intensity distributions recorded during the quadrupole scans using the photoinjector beam are shown by Fig. 5.3. The horizontal and the vertical beam proles aresuperimposed on the beam intensity distribution. The second beam spot appearing inthe images of the third row originate from an inhomogeneity of the Pockels cell selectingthe laser pulses sent to the photo cathode. The resulting inhomogeneity of the laser beam

2 4 6 8

2

4

6

82 4 6 8

2

4

6

82 4 6 8

2

4

6

82 4 6 8

2

4

6

8

2 4 6 8

2

4

6

82 4 6 8

2

4

6

82 4 6 8

2

4

6

82 4 6 8

2

4

6

8

2 4 6 8

2

4

6

82 4 6 8

2

4

6

82 4 6 8

2

4

6

82 4 6 8

2

4

6

8

x [mm] x [mm] x [mm] x [mm]

y [m

m]

y [m

m]

y [m

m]

Figure 5.3: Representative set of beam spots and beam proles during a quadrupole scanperformed at the TTF photo injector.

is transferred directly to the electron beam yielding a vertical phase space distributionwhich diers signicantly from a simple Gaussian. The problem of the laser turned out tobe a valuable check for the quality of the tomographic reconstruction as will be presentedin the following section.

−10

0

10

100

200

0

0.04

0.08

y [mm]

inte

nsity

[a.u

.]

Φ [ ]o

Figure 5.4: Radon transformation of the vertical phase space distribution of the rf gunbeam in the injector section. The corrected beam intensity proles see Equation (4.29) are plotted as a function of the phase space rotation angle .

−50

5

−50

50

1

x [mm]x’ [mrad]

char

ge d

ensi

ty

−4 −2 0 2 4

−1

0

1

x [mm]

x’ [m

rad]

Figure 5.5: Reconstruction of the horizontal phase space distribution from a quadruplescan performed at the TTF photo-injector. Upper graph: three dimensional view. Lowergraph: contour plot.

5.2.2 Tomographic Reconstruction

To apply the ltered backprojection algorithm, the beam proles are corrected in heightand width according to Equation 4.29 and then plotted as a function of the phase spacerotation angle in Fig. 5.4. For graphical clearness only eight vertical beam proles arepresented. The double peak is clearly visible in several proles. For the Radon transformthirteen proles covering an angular interval of 130 are recorded. It is not possible tocover a larger angular range with the two quadrupole magnets available. The beam isdefocused such that the intensity on the transition radiation screen does not suce for aclear beam image. The data does therefore not cover an angular interval of 180o and themissing projections are approximated by a linear extrapolation.

Figures 5.5 and 5.6 show a three-dimensional view and a contour plot of the reconstructedhorizontal and vertical phase space distributions of the photo-injector beam (E = 16MeV,Q = 1nC). The reconstruction point, shown at the beginning of the beam line in Fig. 2.8,is located in front of the quadrupole doublet Q1INJ1 behind the capture cavity. A Radontransformation containing 120 (interpolated) proles has been generated to obtain thereconstructed distributions. The horizontal phase space distribution consists of a singlepeak and the contour lines are of nearly elliptical shape. A Gaussian charge distribution,yielding ellipses in phase space, is therefore a good approximation. The vertical phasespace distribution shows the double peak already seen in the direct image in Fig. 5.3.This result emphasizes the quality of the tomographic reconstruction and shows that thediagnostic technique is capable to reconstruct complex phase space distributions.

−4 0 4−5

05

0

1

y [mm]y’ [mrad]ch

arge

den

sity

−4 −2 0 2 4

0

2

y [mm]

y’ [m

rad]

Figure 5.6: Reconstruction of the vertical phase space distribution from a quadruple scanperformed at the TTF photo-injector. Upper graph: three dimensional view. Lowergraph: contour plot.

5.2.3 Transverse Beam Parameters and Emittance

The transverse beam parameters 11, 22, 12, 33, 44, and 34 are evaluated by applyingEq. (4.30)

ij =Zi j k (i; j) di dj where = (x; x0; y; y0)

to the reconstructed phase space distribution 1(x; x0) and 2(y; y

0). Charge densitieslower than a threshold value are not considered to prevent reconstruction artifacts fromaecting the evaluation of the beam parameters. The threshold is adjusted dynamicallyfor every measurement, generally about 95% of the integral charge density distribution isused to evaluate the beam parameters. The beam emittance x and y is then determinedusing Eq. (4.31)

x = 1122 212 and y = 3344 234 :

Figure 5.7 shows the phase space contour enclosing 39% (1 emittance) and 86% (2emittance) of the charge density. The measured normalized rms emittances are

x = 5:5 106m and x = 9:5 106m :

The transverse beam parameters are

11 = 3:3 107m2 and 12 = 5:2 108m33 = 3:9 107m2 and 34 = 1:3 107m :

−5 0 5

x [mm]

−5

0

5

x’ [m

rad]

ε = 5.5mm mradσ = 3.3⋅10 m

x

11-7 2

σ = 5.2⋅10 m12-8

−5 0 5y [mm]

−4

0

4

y’ [m

rad]

ε = 9.5 mm mradσ = 3.9⋅10 m

y

33-7 2

σ = 1.3⋅10 m34-7

1σ emittance2σ emittance

1σ emittance2σ emittance

Figure 5.7: Contour plot of the reconstructed horizontal phase space density. The contourlines enclosing the area of the 1 and 2 emittance are shown.

5.2.4 Discussion of the Systematic Errors

The accuracy of the emittance measurement is determined by the systematic error ofthe reconstruction code due to the resolution of the optical imaging system, beam jitter,longitudinal misalignment and eld errors of the quadrupole magnets. The interpolationof missing proles and the uncertainty of the space charge dominated beam transfer addto the systematic uncertainty.

Simulations

The quality of the tomographic reconstruction is estimated by simulations using a sym-metric and an asymmetric charge distribution. The symmetric distribution is a Gaussian,the asymmetric distribution consists of three Gaussians of dierent rms width and heightarranged asymmetrically in phase space (see Fig. 4.8). The initial distribution is trackedto the OTR screen for dierent quadrupole settings and projected onto the x and x0 axis.During tracking, a beam jitter and an uncertainty of the transfer matrix M can be intro-duced by allowing a random variation of the tracked phase space distribution and of thetransfer matrix elements. The obtained Radon transform is then used for the tomographicreconstruction. The quality of the reconstruction is judged by the relative dierence ofthe input and output emittances.

Beam Jitter

The beam stability was investigated in the injector by analyzing the beam intensity dis-tributions over a time interval of 40 minutes, which is longer than the time needed for aquadrupole scan. Figure 5.8 shows the horizontal and the vertical beam position and thebeam width (determined by the rms of the beam proles). The horizontal beam centroidis gently drifting towards positive values while the vertical centroid remains stable. Thecorrelated beam movement is corrected for during the reconstruction and should there-fore have no in uence on the result of the measurement. Notice the appearance of suddenhorizontal beam movements towards smaller values. At the time being this eect is notwell-understood and needs further investigation. The variation of the beam width overthe measurement period of 40 minutes is less than 40 microns (5%).

Distrib

ution

=

11 =

11[%]

12 =

12[%]

Symmetric

4:00:1

<1

<1

Asymmetric

11:51:1

3:80:4

5:10:3

Table

5.1:System

aticerror

onthebeam

param

etersdeterm

ined

bysim

ulation

ofthe

tomograp

hicrecon

struction

.18

projection

sat

acon

stantangular

stepsof

10 are

com-

puted

.TheRadon

transform

isinterp

olatedto

120pro

les.Nosm

ooth

ingisapplied

intheltered

back

projection

.Asymmetric

andaasy

mmetric

phase

space

distrib

ution

has

been

used

.

Table5.1

show

stherelative

system

aticerror

oftherecon

structed

emittan

cesandbeam

param

eterssim

ulated

with

themeasu

redbeam

instab

ility.Thesymmetric

phase

space

distrib

ution

canberecovered

with

=

=(4:0

0:1)%,theasy

mmetric

distrib

ution

with

=

=(11:5

1:1)%.Thebeam

dimension

inthesim

ulation

isadapted

tothe

param

etersofthemeasu

redcharge

densities

(seeFig.

5.7).18

projection

swith

anangular

spacin

gof10

have

been

evaluated

andtheRadon

transform

isinterp

olatedto120

pro

les.Nosm

ooth

ingbytheltered

back

projection

isapplied

.Thecom

putation

assumes

alon

gitudinalposition

ingofthequadrupole

magn

etsandthetran

sitionrad

iationscreen

ofbetter

than

1mm.Therelative

errorof

thequadrupole

eld

strength

g=g

should

not

exceed

1%.

In uenceofSpace

ChargeDefocusin

g

Space

charge

forceshave

tobeinclu

ded

tocalcu

latethebeam

transfer

intheinjector

sec-tion

.Theaddition

alspace

charge

defo

cusin

gred

uces

theavailab

leran

geofrotation

angles

becau

setheinten

sityon

thetran

sitionrad

iationscreen

does

not

suce

forabeam

image

atcertain

quadrupole

settings.

Itistherefore

not

possib

leto

obtain

aRadon

transform

covering180 .

Themissin

gbeam

pro

lesin

theRadon

transform

have

tobeextrap

o-lated

.Simulation

sshow

that

theresu

ltof

thetom

ographicrecon

struction

dependson

010

2030

40

0.4

0.8

1.2

Average centroid [mm]

time [m

in]

vertical centroidhorizontal centroid

010

2030

40

0.80

0.88

0.96

1.04

RMS [mm]

time [m

in]

horizontal RM

S

vertical RM

S

Figu

re5.8:

Left

graph:horizon

talandvertical

beam

position

foraperio

dof

40minutes.

Righ

tgrap

h:horizon

talandvertical

beam

sizefor

thesam

etim

eperio

d.

10910

10

0.6

0.8

1

1.2

1.4

# electrons per bunch

Rel

ativ

e ε,

σ

, σ11

12 11

Emittance

σ

σ 12

Figure 5.9: Relative emittance and transverse beam parameters as a function of thenumber of electrons per bunch. The transfer matrix fullls the self-consistency conditionfor a simulated bunch charge of 6:25 109 electrons. The solid, dashed and dotted curvesindicate the systematic error of the reconstructed emittance and beam parameters if thetrue bunch charge is dierent than assumed for the self-consistency condition.

which beam proles are missing. If the waist of the beam is scanned symmetrically, thenthe systematic error of the reconstructed emittance due to the missing beam proles willbe smaller than = = 10%.

Figure 5.9 shows the relative systematic change of the emittance and the beam parametersas a function of the bunch charge. The description of the space charge dominated beamtransfer depends on the accuracy of the bunch charge measurement. The transfer matrixM has been adjusted to fulll the condition of self-consistency for 6:25 1010 electrons perbunch. The solid, dashed and dotted curve in Fig. 5.9 show the variation of the emittanceand the beam parameters if the true bunch charge diers from this value. Then thebeam transfer is not modeled correctly by the program and wrong beam parameters aredetermined although the self-consistency condition is fullled. A bunch charge uncertaintyof 10% yields a systematic uncertainty of = = 4%, 11=11 < 1% and 12=12 = 8%.

Error Discussion

The transverse emittance of the 16MeV beam is

x = (5:5 +0:20:6 0:5 0:3) 106m (5.2)

y = (9:5 +0:41:1 1:3 0:5) 106m : (5.3)

The rst error represents the systematic error of the tomographic image reconstruction(beam jitter, longitudinal misalignment, quadrupole eld error and interpolation), thesecond error the uncertainty due to space charge forces and the third error the statisticalerror of three consecutive measurements. The uncertainty of the space charge term isderived from an estimated uncertainty of the bunch charge measurement of 0:1 nC andfrom the result of simulations containing only 13 projections in the Radon transform. The

f(x,x’)f(x,y)

−5 0 5

1

0

φ = 45o

−5 0 5

φ = 58o1

0 0−5 0 5

φ = 92o1

−5 0 50

1z

[a.u

.] φ = 118o

x [mm]−5 0 5

0

1 φ = 152o

x [mm]−5 0 5

0

1 φ = 167o

x [mm]

z [a

.u.]

Figure 5.10: Comparison of measured beam proles and projections of the reconstructedphase space distribution tracked to the OTR screen.

study of systematic eects indicates that there is a tendency to overestimate the emittance.Hence the error bars in Eq. (5.2) and (5.3) are asymmetric. The beam parameters 11,12, 33 and 34 evaluated from the reconstructed horizontal and vertical phase spacedistribution are

11 = (3:3 0:1 0:5 0:2) 107m2 (5.4)

12 = (5:2 0:1 1:2 0:2) 108m (5.5)

33 = (3:9 0:1 0:5 0:2) 107m2 (5.6)

34 = (1:3 0:1 0:6 0:1) 107m (5.7)

where the errors represent the systematic errors due to the tomographic image reconstruc-tion, due to space charge forces and the statistical error of the measurement.

The measured horizontal emittance of 5:5 106m at the end of the injector represents thebest beam quality expectable for the operating parameters used. Smaller emittances canbe realized with a higher-gradient rf-accelerating eld of about 50MV/m and a smallerlaser pulse length of t < 1:5ps.

Self-consistency Check of the Reconstruction

The reconstructed phase space distribution is tracked forward to the transition radiationscreen using the quadrupole settings as in the tomographic scan. Figure 5.10 shows acomparison of the phase space distribution projected onto the spatial coordinate axis andthe measured beam intensity distribution. The agreement of the measured and recon-structed beam proles is excellent. Only slight dierences are visible at the tails of theproles. The reconstruction of the transverse phase space distribution therefore yieldsreliable numbers for the beam parameters. This result demonstrates the applicability oftomographic image reconstruction techniques for transverse phase space diagnostics.

160 170 1804

6

8

10

12

Nor

mal

ized

ver

tical

em

ittan

ce [m

m m

rad]

Main Solenoid Current [A]

ε

x(tomography)

εy(tomography)

y(slit)

ε

0

2

4

6

8 x 10−7

160 170 180Main Solenoid Current [A]

σ [

m ].

σ

[m],

σ [

m ],

σ

[m]

1112

3334

22

σ11

σ12

σ33

σ34

Figure 5.11: Left graph: horizontal and vertical emittance measured at the TTF photoinjector by phase space tomography and the slit technique (only vertical emittance de-termined). Right graph: beam parameters determined by phase space tomography. Theemittance and the beam parameters are plotted for dierent gun solenoid magnetic eldsand solenoid currents respectively. Both the primary and secondary solenoid have beenvaried while keeping a constant ration of Ip=Is = 1:8.

5.2.5 Further Emittance Studies using the Photo Injector Beam

Slit Emittance Measurement

The emittance determination by transverse phase space tomography has been comparedto a slit emittance measurement. For this purpose a screen with horizontal slits of 100mwidth and separation is moved into a beam of 1 nC charge and 16MeV energy. The imageof the beam has been recorded using optical transition radiation and a CCD camera.The idea of the method resembles the hole-camera principle of geometrical optics. Theobserved distribution and the width of the images of the individual slits can be usedto determine the beam width and angular divergence [47]. Figure 5.11 (left) shows acomparison of the vertical beam emittances measured by the tomographic and the slittechnique using the photo injector beam. The emittances agree within errors.

Optimization of Solenoid Currents

The elds of the gun solenoids are varied to minimize the emittance of the rf gun beam.Two solenoid current scans with dierent laser spot diameters on the photo cathode havebeen performed to determine the optimum solenoid setting. Figure 5.11 shows the beamemittance and the beam parameters measured for three dierent settings of the solenoidcoils, i. e. 158A & 86A, 165A & 90A, 176A & 96A for the primary and secondarysolenoids respectively. The normalized horizontal and vertical emittance (left) and thebeam parameters (right) are plotted versus the primary solenoid current. The laser spotdiameter on the photo cathode is d = 3mm. The horizontal emittance is increasingwith increasing solenoid current. The variations of the vertical emittance are not signif-icant. Currents of 165A and 90A yield a normalized horizontal and vertical emittance

of x = 5:5 106m and y = 9:5 106m respectively and have been used as standardsolenoid settings.

Figure 5.12 shows the normalized horizontal and vertical emittance as a function of thethe variation of the primary and secondary solenoid currents for a laser spot diameter ofd = 10mm. A xed ratio between the solenoid currents Ip=Is = 1:8 is maintained. Theminimum emittance is obtained at Ip = 166A as shown in the left column. In a secondstep, the primary solenoid is xed at Ip = 166A and the current of the secondary solenoidis varied. The optimum emittance of x = 9 106m and y = 17 106m is found atIp = 166A and Is = 82A. The larger emittances are mainly caused by the larger laserspot on the photo cathode.

120 160 200 240 280

10

12

14

16

18

Primary Solenoid Current [A]

γ ⋅ ε

x [m

m m

rad]

Secondary Solenoid / Primary Solenoid = 1 / 1.8

0 120804010

20

30

Secondary Solenoid Current [A]

Primary Solenoid = 166 A

10

20

30

40

120 160 200 240 280Primary Solenoid Current [A]

10

15

20

25

0 1208040Secondary Solenoid Current [A]

γ ⋅ ε

y [m

m m

rad]

γ ⋅ ε

y [m

m m

rad]

γ ⋅ ε

x [m

m m

rad]

Figure 5.12: Normalized horizontal and vertical emittance measured at the TTF photoinjector using a quadrupole scan with tomographic image reconstruction techniques. Leftcolumn: variation of the primary and secondary gun solenoid current at constant ratio ofIp=Is = 1:8. Right column: variation of the secondary solenoid with the primary xed atIp = 166A. Top row: horizontal emittance. Bottom row: vertical emittance. The laserspot diameter on the photo cathode is d = 10mm

−4 −2 0 2 4

−4−2024

0

1

x [mm]x’ [mrad]ch

arge

den

sity

−2.5 0 2.5−1

0

1

x [mm]

x’ [m

rad]

Figure 5.13: Horizontal phase space distribution measured at the TTF thermionic injector.Upper graph: three-dimensional view. Lower graph: contour plot of the phase space area.

5.3 Measurements with Thermionic Injector Beam

The TTF thermionic injector was operated with beam currents between 1 and 5mA, with2Hz macro-pulses of 30s length and with a bunch repetition rate of 216MHz. The totalnumber of electrons per bunch is 2.3108. The beam energy at the quadrupole scan sectionwas 9MeV. Space charge eects in the scanning section can be neglected because of thelower bunch charge.

Figure 5.13 shows the reconstructed horizontal phase space distribution from a scan cov-ering an angular interval of 160. The charge distribution shows a single peak whosecontour lines are almost elliptic close to the center. The elliptic contour lines hint at aGaussian charge distribution. A close inspection of the upper graph reveals still somestar-like blurring which is a remainder of the missing angular interval of 20o. Figure 5.14shows the contour lines corresponding to the and the 2 emittances. The transversephase space distribution in the vertical plane is of similar shape.

The normalized beam emittance and the beam parameters obtained for an average beamcurrent of I = 3mA are

x = (3:8 0:5 + 0:2 0:4) 106m y = (3:3 0:4 + 0:2 0:4) 106m11 = (1:1 0:1 0:1) 107m2

−2 −1 1 2x [mm]

−1

0

1

x’ [m

rad]

2σ emittance

1σ emittance

Figure 5.14: Horizontal phase space distribution measured at the TTF thermionic injector.The contour lines represent the and the 2 emittances.

12 = (4:6 0:3 0:2) 108m33 = (1:3 0:1 0:1) 107m2

34 = (9:8 0:5 0:3) 108m :

The horizontal emittance and the beam parameters 11 and 12 have been measured asa function of the beam current shown in Fig. 5.15. The transverse emittance is increas-ing with beam current while the beam parameters remain constant. The increase of thetransverse emittance can be explained by stronger space charge forces in the thermionicgun. The design emittance for an average beam current of 8mA is = 5 106m.The design current could not be achieved because of eciency problems of the cathode.However, it can be seen that an extrapolation of the normalized emittance will is largerthan the design value of the thermionic injector.

3

4

5

6

7

8

2 3 4 5Beam Current [mA]

γ ⋅ ε

[m

m m

rad]

x

σ [m

], σ

[m

]11

122

Beam Current [mA]2 3 4 5

−1

0

1

2

x 10−7

σ [m ]11 2

σ [m]12

Figure 5.15: Normalized horizontal emittance (left) and the horizontal beam parameters11 and 12 measured at the TTF thermionic injector versus the average beam current.

5.3.1 Parabola Fit Analysis

The beam proles recorded during a quadrupole scan measurement are analyzed using theparabola t analysis. The beam size x2rms divided by the beam transfer matrix elementM2

12 is plotted versus the quotient of M11=M12. The data obtained from a quadrupole

−6 −4 −2 0 20

1

2

3

4 x 10−6

M12

xrms2

M11 M12[1/m]

ε = 4.0⋅10 mx

-6

σ = 1.9⋅10 m11

-7 2

σ = 1.1⋅10 m12

-8

Figure 5.16: Parabola t analysis of the beam proles taken during a quadrupole scan.The error bars indicate the statistical error of the measurement.

scan performed with the thermionic injector beam are shown in Figure 5.16. The beamcurrent is 3mA. The parabolic t to the data yields the normalized emittance and thebeam parameters

x = (4:0 0:5) 106m (5.8)

11 = (1:9 0:2) 107m2 (5.9)

12 = (1:1 0:2) 108m : (5.10)

The error bars indicate the quadratic sum of the systematic (image processing, beamstability uctuations, beam transfer) and the statistical error of three consecutive mea-surements. The emittance and the beam parameters, as indicated in Figure 5.16, yieldsimilar results as obtained by phase space tomography. The error of the emittance andthe beam parameters can be obtained directly from the t procedure. The quality of theparabola t could be improved by recording additional data points at larger M11=M12.

Chapter 6

Time Domain Techniques for Bunch

Length Measurements

6.1 Streak Camera

6.1.1 Experimental Setup

The streak camera is a device for the detection and the time-resolved investigation oflight pulses in the picosecond regime. Figure 6.1 shows the working principle. The light

Slit

Photo-cathode

Sweep

Phosphorscreen

Micro-channelplate

CCD

RF

Figure 6.1: Principle of the streak camera.

pulse travels through a dispersion-free optical system and a slit before hitting the photocathode of the streak camera. The slit reduces the transverse dimension of the image onthe photo cathode. The light pulse is converted to an electron pulse, which is acceleratedand swept transversely by a fast rf electric eld. The resulting transverse distribution isprojected onto a phosphor screen. The image is amplied by a multi-channel plate andthen detected by a CCD camera.

SpectrometerDipole

Quadrupole

Synchrotron light mirror

shielding

Al mirror Al mirror

Al mirror

Al mirrorin vacuum

Streakcamera

Top View

Side View

Figure 6.2: The TTF spectrometer dipole magnet with the synchrotron light mirror. Thelight is re ected by a polished aluminum block through a quartz glass window verti-cally out of the vacuum chamber. The light pulses are guided through a 11m long lightbeam line out of the tunnel. Three aluminum mirrors re ect the synchrotron light to anexperimental area where the streak camera is installed.

The streak camera1 manufactured by the ARP company has an intrinsic time resolutionof res = 2ps. A degradation of the temporal resolution may be due to the transversedimension of the photo-electron pulse, which is convoluted with the longitudinal electrondistribution during the sweep. To fully exploit the time resolution of the streak cameratube a slit width of slit < res has to be used. In the present experiment, the bunchlength was expected to be in the order of 10 picoseconds, so a slit of 1:9mm width (t =(6:2 0:1) ps) was inserted in front of the photo cathode. The slit is oriented suchthat the narrow dimension is parallel to the orientation of the sweep. During the bunchlength measurement, the residual transverse dimension of the electron pulse slit has tobe deconvoluted from the measured length of the phosphor screen l by

z =q2l 2slit (6.1)

yielding the bunch length z. The timing of the streak camera was adjusted by slowlyincreasing the sweeping speed of the tube. The delay of the phase stabilized timingsignal, a 216MHz sub-harmonic of the master oscillator, is re-adjusted at each step. The

1I thank Dr. Terry Garvey, Michel Bernard and Bernard Leblond from LAL, Orsay, France formaking the LAL streak camera available.

camera is operable when stable light pulses are detected by the CCD camera at maximumsweeping speed.

Synchrotron Light Delivery Synchrotron light generated in the spectrometer dipoleat E = 170MeV is extracted by a synchrotron light mirror as shown in Fig. 6.2. Themirror is a 50 50mm polished aluminum block. The light is re ected through a quartzwindow vertically out of the beam pipe and is guided through an underground beam lineof 11m length to the optical diagnostics experimental area. The light beam line consistsof three aluminum mirrors of 76mm diameter. No focusing lenses are used. The streakcamera and the synchrotron light beam are aligned by adjusting the last two mirrors.

6.1.2 Bunch Length Measurements

The electron bunches produced by the TTF photo injector (Q = 3nC, UV laser pulselength t = 15 ps) have been transfered to the spectrometer magnet with the magneticcompressor being switched o. During the acceleration from 16 to 160MeV, the longi-tudinal charge distribution can be considered as invariant. Synchrotron radiation lightpulses, produced in the spectrometer dipole, are detected with the streak camera. Figure6.3 shows the measurements yielding an rms bunch length of z = (1:95 0:08)mm andz = (3:27 0:06)mm. The left graph has been obtained for an rf phase in the guncavity of 20 yielding an optimum bunch compression without a signicant reduction ofthe bunch charge. The right graph is obtained at a rf phase of 45 yielding a reducedbunch compression.

The streak camera measurements of the injector bunch length are summarized in Fig. 6.4.The left graph shows the rms bunch length as a function of the rf gun phase at a xedbunch charge ofQ = 3nC. The bunch length decreases with decreasing rf phase. The eectis explainable by longitudinal bunching due to the head-tail velocity modulation imposedon the bunch by the rf eld. The phase = 0 indicates the phase of the rf eld where the

2 6 10 14 18

100

200

300

400

500

Inte

nsity

[a.u

.]

σ = (1.95 ± 0.08)mmz

σ [mm]z

0 5 10 15 20 25

200

400

600

800

1000

Inte

nsity

[a.u

.]

σ = (3.27 ± 0.06)mmz

σ [mm]z

Figure 6.3: Longitudinal bunch charge proles produced by the TTF photo injector atrf gun phase settings of 20 (left) and 45 (right). Data is presented as circles, the solidcurve indicates a Gaussian t.

5020 40 60

2

3

4

5

σ [m

m]

z

RF Gun Phase [ ]o

PARMELA

2

3

4

5

σ [m

m]

z

Bunch Charge [nC]84 6 10 12

Figure 6.4: Streak camera bunch length measurements. Left: rms bunch length versus rfgun phase (Q = 3nC). Right: rms bunch length versus bunch charge (rf gun phase setto 30). The solid line indicates the prediction by PARMELA. The error of the bunchlength measurement is within the marker size.

electric eld changes sign and thus from acceleration to deceleration. The solid line is theprediction by PARMELA [48]. The right graph shows the rms bunch length as a functionof the bunch charge at a xed gun rf phase of 30. The bunch length does not changesignicantly for bunch charges smaller than 8 nC. Larger bunch charges cause a graduallengthening of the longitudinal charge distribution. This lengthening can be explained bylongitudinal space charge forces whose strength is rising proportional to the bunch charge.

The laser pulse length has been reduced by a factor of 2 during the run in spring 1999.The reduction of the laser pulse length reduces the electron bunch length by the samefactor since the longitudinal space charge forces are of equal magnitude (Q = 3nC, 16 pslaser pulse length compared to Q = 1nC, 8 ps laser pulse length). Unfortunately, thestreak camera was no longer available to investigate the electron bunches produced withthe short laser pulses.

6.2 Longitudinal Phase Space Rotations

6.2.1 Theory and Simulation

The longitudinal dispersion of a magnetic chicane followed by an o-crest acceleration inan rf cavity can be used to translate the longitudinal charge prole into an energy proleof the bunch. Time slices are transfered to slices of constant energy behind the rf cavity.An energy prole measurement in the spectrometer behind the rf cavity yields then animage of the longitudinal bunch charge distribution in front of the chicane.

Figure 6.5 shows a sketch of the TTF linac including the rf module #1, the magneticchicane, rf module #2 and the dispersive section at the experimental area. To demonstratethe method, an asymmetric charge distribution is used in the simulation. The evolution of

−5 0 5−0.05

0

0.05

Length [mm]

dE/E

Dipole

Dipole Dipole

Dipole rf cavity Dipole

Screen

rf cavity

−1.5 0 1.5

−0.03

0

0.03

Length [mm]

dE/E

−3 0 3

−0.03

0

0.03

Length [mm]

dE/E

−1.5 0 1.5

−0.03

0

0.03

Length [mm]

dE/E

−4 −2 0 2 40

10

20

Length [mm]

Cha

rge

[a.u

.]

0 20 40−0.04

0

0.04

∆E /

E

Charge [a.u.]

−2 −1 0 1 20

20

40

Length [mm]

Cha

rge

[a.u

.]

0 10 20−0.02

0

0.02

∆E /

E

Charge [a.u.]−4 −2 0 2 4

0

10

20

Long

itudi

nal C

harg

e D

istr

ibut

ion

[a.u

.]

Length [mm]

Injector bunch chargedistributionReconstructed bunchcharge distribution from ∆E/E

Figure 6.5: The longitudinal charge distribution of an electron bunch can be measuredusing the longitudinal dispersion of the magnetic chicane compressor followed by an o-crest acceleration in an rf cavity. The longitudinal phase space distribution produced bythe injector, behind rf module #1, behind the magnetic chicane and behind rf module #2is shown. The parameters of the rf module #1 and the magnetic chicane are adjusted foroptimum bunch compression. The parameters of the rf module #2 are adjusted to rotatethe longitudinal phase space distribution between the beginning of the chicane and theexperimental area by =2. The center graph shows a comparison of the original longitu-dinal charge distribution with the reconstructed distribution using the energy prole inthe spectrometer section. The agreement is perfect.

this distribution along the linac is shown in Fig. 6.5. The parameters of the rf module #1and the chicane are adjusted to obtain optimum bunch compression. The gradient andthe phase of rf module #2 are adjusted to rotate the longitudinal phase space distributionbetween the beginning of the chicane and the dispersive section by =2. The simulationparameters are listed in Tab. 6.1.

Equation (2.23) describes the bunch energy spread (66) as a function of the injector bunchlength (55), the injector energy spread (66), the parameters of rf module #1 (M65, M66

Uacc1 10MV Uacc2 10MV 1 14:4 2 15:9R65 3:24m1 R66 0:58 UA 275:0m2 M56 0:18m

M65 5:76m1 M66 0:18 SA 543:3m2

iz 1:8mm fz 0:27mm iE 8:7 103 fE 5:7 103

Table 6.1: Parameters of the simulation.

and UA) and #2 (R65, R66 and SA) and the longitudinal dispersion M56 of the bunchcompressor. Time slices in front of the chicane are transformed into energy slices behindrf module #2 if one chooses the transfer matrix elements such that M56R65 + R66 ! 0.Equation (2.23) then reduces to

66 = 55R265 + U2

A

n55(1 +M56M65)

2 + 66M256M

266 + S2

AM256

255

o2 55R

265 :

The non-linear term of the second rf acceleration stage can be neglected because of theshort bunch length behind the chicane.

The longitudinal charge distribution in front of rf module #1 and the energy distributionbehind rf module #2 are shown in Fig. 6.5. From the projections one can see that theenergy distribution behind module #2 is a direct image of the longitudinal charge distri-bution in front of module #1. The center graph in Fig. 6.5 shows in a quantitative waythat the reconstructed and the original charge prole agree perfectly.

The phase space rotation method can also be used to observe the longitudinal compressionwithin the chicane by varying the parameters of the second rf module. The position ziwhere the longitudinal charge prole is to be determined is again given by the conditionof the form M56R65 +R66 ! 0. Here, M56 denotes the longitudinal dispersion between ziand the end of the chicane.

Resolution The resolution of the method is limited by the resolution of the spectrom-eter and the validity of linear beam transfer including the applied quadratic corrections.The energy resolution of the TTF spectrometer is E=E = 2 104. Hence, the spatialresolution is given, in linear approximation, by

z =E=E

R65

: (6.2)

z can be minimized by choosing R65 (at a given eld gradient) as large as possible,

RM65 = lim

!=2

dE sin

Ei + dE cos

2

=dE

E

2

2

= 27:2

1

m

(6.3)

since both rf modules operate at the same eld gradient. For maximum resolution thesecond module has to be operated with a phase = =2. The TTF spectrometer thenyields a spatial resolution of

zjres = 7:5m (6.4)

reconstructed at M56(z0 ! z1) = 3:7 cm corresponding to a chicane position at the endof the third dipole magnet.

The spatial resolution at the beginning of the chicane (M56 = 0:18m) is

zjres =M56E=E

R66

= 36m (6.5)

where R66 is chosen unity (rf phase 0 = =2). In the present conguration, the resolu-tion is sucient to determine electron pulse length produced by the chicane compressor.Non-linear eects like cavity wake-elds and longitudinal space charge forces are smalland can be neglected.

The bunch length determination by phase space rotations is, in a sense, a single bunchtime-domain measurement with a resolution of about 40m. Intensied CCD cameraswith a gating of faster than 100 ns are commercially available and capable to resolve thebunch spacing at the TTF linac. An excellent application of this method is the directobservation of electron beam micro-bunching at the Stanford FEL center [49].

6.2.2 Bunch Length and Energy Distribution Measurements

For the measurements the TTF linac was operated with macropulses consisting of 20bunches of 1 nC charge with a repetition rate of 1MHz. The gun cavity rf phase was 20

to produce the shortest bunches possible. The laser pulse length was t = 8ps.

Energy Spread at the Injector

The energy spread at a beam energy of 16MeV was measured in the dispersive sectionof the injector by means of optical transition radiation. The minimum energy spread wasE=E = 1:9 103.

Energy Distribution at the Experimental Area

RF module #1 is operated at a gradient 10MV/m and a de-phasing angle of 14 tooptimize the bunch compression in the magnetic chicane (E = 90MeV). The beam isaccelerated in the second rf module to the nal beam energy of 170MeV. The beamenergy distribution is recorded in the spectrometer located in the experimental area usingoptical transition radiation.

Figure 6.6 shows the measured energy proles at the experimental area. The measure-ments were taken at dierent machine settings as shown by Tab. 6.2. The energy proleon the left corresponds to a relative energy deviation of 9:2 103 (FWHM), the proleon the right to 3:3 103 (FWHM). The lower plots show the compression of the longitu-dinal charge distribution through the chicane with the vertical dashed line indicating thechicane position where the condition M56R65 + R66 ! 0 is fullled. The energy proleon the left corresponds to a FWHM bunch length of 2:6mm at the rst dipole magnet(z = 1:1mm, t = 3:8 ps), the prole on the right to 280m in the third dipole magnet

0 1 20

1

∆E/E [%]

Inte

nsity

3.3 %(FWHM)

o

∆E/E [%]0 1 2

Inte

nsity

0

1

9.2 %(FWHM)

o

Position in compressor [m]1 2 3 4

Leng

th (

FW

HM

) [m

m]

2

0.5

0

2.5

1.5

1

Position in compressor[m]1 2 3 4

Leng

th (

FW

HM

) [m

m]

2

0.5

0

2.5

1.5

1

σ = 120µm z

σ = 1.1mmz

3

Figure 6.6: Energy prole measurements performed behind the TTF spectrometer dipolemagnet at dierent rf parameter settings. The energy deviation yields 6 103 (left) and3:3 103 (right) FWHM. The lower plots show the evaluated rms condence intervalsof the bunch length throughout the chicane. The dashed line indicates the longitudinalposition where the measured energy prole matches the longitudinal charge distribution.Left: optimum compression. Right: over-compression.

(z = 120m, t = 400 fs).

The step-wise reduction of the rms bunch length in the magnetic chicane is shown inFig. 6.6. The condence interval is obtained from a comparison of the measurements andsimulation. The rms energy spread at the spectrometer dipole magnet is computed for aset of the following parameters: bunch length and energy spread in front of module #1,gradient and phase of both rf modules and longitudinal dispersion of the chicane compres-sor. Gradient and phase of the rf modules and the nal rms energy spread are measuredwithin their error intervals. A comparison of the measured data and the results of thesimulation yields a set of possible solutions describing the compression of the longitudinalcharge distribution for the accelerator setting used. The set of possible bunch lengths isindicated as the condence interval (shaded area) in Fig. 6.6.

E=E [103]p55 [mm] E1 [MV/m] E2 [MV/m] 1 [

o] 2 [o]

Left 9:2 0:2 1:1 0:1 76:5 0:3 65:2 0:3 13:5 5 14 5

Right 3:3 0:2 0:14 0:03 76:5 0:3 65:2 0:3 20:5 5 34 5

Table 6.2: TTF machine settings yielding the energy proles shown in Fig. 6.6. Thecolumns indicate from left to right: relative energy spread measured in the dispersivesection, reconstructed rms bunch length, eld gradient of cavity module #1 and #2, rfphase of module #1 and #2. The injection energy is 16:5MeV. The errors are statisticalerrors of the measurements.

The average rms bunch length at E = 16MeV is z = (990 180)m. The result is inaccord with the streak camera measurement (z = (1:95 0:08)mm) if a correction ismade for the dierent laser pulse lengths.

The lower left graph of Figure 6.6 shows the optimum bunch compression of the longitu-dinal charge distribution. Here, the parameters of the rst acceleration module are setto obtain a nal bunch length of z = (350 130)m, t = (1:1 0:4) fs, and energyspread of (1:5 0:1)MeV behind the magnetic chicane. The lower right graph showsan example for a longitudinal over-compression, where the minimum bunch length is ob-tained at the third dipole magnet. The nal bunch length at the end of the chicane isz = (500 120)m, t = (1:7 0:4) ps. The nal energy spread of (550 30) keV issmaller than for optimum compression. The correlation of the longitudinal phase space isopposite with higher energy leading lower energy electrons. This opposite correlation isdecreased, eventually removed and inverted, by the o-crest acceleration in the second rfmodule increasing the energies of the trailing with respect to the leading electrons. Theresult is an energy prole of reduced width.

Chapter 7

Fourier Transform Spectroscopy

A Martin-Puplett interferometer is used to determine the longitudinal pulse shape byautocorrelation of the incoming radiation pulse. The coherent power spectrum and thebunch form factor is derived using a Fourier transformation. This section outlines theworking principle and the spectral acceptance function of the Martin-Puplett interferom-eter. Several data analysis techniques will be presented. Simulation results showing theerrors and limitations of the method and measurements will be presented.

7.1 Description of the Martin-Puplett interferometer

A Martin-Puplett interferometer is a Michelson-type interferometer used in the millime-ter and sub-millimeter wavelength range. A schematic view is shown in Fig. 7.1. Theinterferometer used at TTF consists of two parabolic mirrors, three wire grids, two roofmirrors and two pyroelectric detectors. These components are mounted on a stable alu-minum plate which is supported by an adjustable table. The interferometer componentsare designed to have high precision and optical quality [50].

Wire Grids

The wire grids with an aperture of 90mm are wound from 20m diameter gold-platedtungsten wire with a spacing of 100m. The grids have been built on a winding machinein the workshops of the University of Aachen. The wire grids are mounted in a metallicframe which is placed into the interferometer. The use of a microscope is essential duringthe assembly of the grid to straighten out the wires and ensure a regular spacing.

Parabolic Mirrors

Aluminum parabolic mirrors with optical quality have been built in the workshops of theFraunhofer Institute fur Produktionstechnik (IPT) in Aachen and of the University ofAachen. The mirrors have a focal length of 200mm and an aperture of 100 100mm.The parabolic mirrors are mounted on a three-point suspension and are equipped withmicrometer screws to vary the position of the re ective plane and the re ection angle.

Polariser

Analyser

BeamDivider

Roof mirror

movableRoof mirror

PyroelectricDetector 1

PyroelectricDetector 2

Quartz window

Al foil

e

AnalyserPolariser

x

y

BeamDivider

ξη

Figure 7.1: Schematic view of the Martin Puplett interferometer consisting of parabolicmirrors, wire grids, roof mirrors and two detectors.

Roof Mirrors

The roof mirrors consist of two polished aluminum mirrors which are glued under anangle of 90 into an aluminum girder. The aperture of the roof mirrors matches the clearaperture of the wire grids. The movable roof mirror is mounted on top of an optical moverdriven by a stepping motor. The roof mirrors are equipped with a single micrometer screwto vary the vertical re ection angle of the mirror.

Pyroelectric Detectors

Two pyroelectric detectors Molectron PI-45 with integrated FET operational amplierare used in the interferometer. The specication of the PI-45 detector can be taken fromTable 7.1. A resistor of 100M is connected to the FET to achieve an signal of 30V/Wat a bandwidth of 4 kHz. A second variable amplier is available. The detector and theelectronic circuit is housed in an aluminum box which is mounted in the interferometer. Apolished aluminum horn antenna of 83mm length and 35:8 full opening angle is mountedin front of the pyroelectric detector to increase its active diameter. The analogue signalproduced by the pyroelectric detector is transfered out of the TTF tunnel using an ad-ditional current driving amplier. A 12 bit, 1 MHz bandwidth ADC digitizes the signal

Figure 7.2: Dierent views on the TTF Martin-Puplett interferometer. The radiationis collected by a parabolic mirror located in front of the vacuum window, and re ectedinto the interferometer by the polarizing wire grid. The beam splitter transmits / re ectsradiation into both interferometer arms and sends the recombined beam towards thesecond parabolic mirror. The radiation pulse is passing the smaller aperture analyzinggrid and is focused onto two pyroelectric detectors equipped with horn antennas.

produced during the 800s long macropulse.

Assembly of the Interferometer

The rst parabolic mirror of the interferometer is placed in front of the vacuum window.The height of the paraboloid is xed by the height of the transition radiation screen andthe vacuum window. The transverse position of the mirror is xed by the condition thatthe transition radiation target has to be in the focal plane of the paraboloid. The mirrorproduces a parallel wavefront radiation beam which is re ected vertically up towards thepolarizing wire grid. The grid re ects the radiation horizontally into the interferometer.The radiation is horizontally polarized because of the wire orientation. The beam dividergrid is re ecting and transmitting polarization components parallel and orthogonal tothe wire orientation towards the roof mirrors. The beam divider grid is oriented at anangle of 54:73 with respect to the vertical to obtain an \apparent" angle of 45 of thedividing grid (see Figure 7.6)1. The position of one roof mirror is xed while the other

1Notice that: tan ( 6 BAC) = tan= cos.

Material LiTaO3

Critical temperature 610C

Temperature stability 0.2%=C

Crystal thickness 100m

Crystal diameter 5mm

Capacity 75 pF

Voltage Supply 15Vmax. DC current 6mA

max. output power 50mW

Table 7.1: Specication of the pyroelectric detector PI-45 [51]

is mounted on an optical mover equipped with a remotely controllable stepping motor.The recombined radiation from both arms is focused onto the pyroelectric detectors bya second parabolic mirror. The analyzing grid which is located between the mirror andthe detectors, transmits the radiation polarized orthogonal to the wires to detector #1and re ects the radiation polarized parallel to the wires to detector #2. The pyroelectriccrystals are located in the focal plane of the parabolic mirror.

Compared to a Michelson, the Martin-Puplett interferometer is a polarizing interferometerand has the advantage of two radiation output ports. The ratio of the dierence and thesum of the two detector signals yields an interference pattern which is less sensitive tobeam intensity uctuations than the individual detector signals [52].

7.2 Working Principle of the Interferometer

The incoming radiation pulse is polarized horizontally by the rst wire grid (polarizer) andsplit by the beam divider to both interferometer arms. The beam divider grid is orientedat an apparent angle of 45 with respect to the polarizer to deliver equal eld amplitudesto both arms. The orientation of the polarization is ipped by two roof mirrors so thatthe radiation pulse transmitted rst, is now re ected by the beam divider and vice versa.Generally, the linear polarization in front of the beam divider is transformed into ellipti-cal polarization after the recombination, depending on the position of the movable roofmirror. The analyzing grid transmits one principal axis of the elliptical polarization intothe pyroelectric detector #1 while the other is re ected into the pyroelectric detector #2.Hence the two detectors measure anti-correlated signals when the roof mirror is moved.The sum signal measures the total radiation power passing the interferometer.

The amplitude and the state of polarization of the millimeter wavelength radiation beamcan be described by a complex two-dimensional vector, each of the components represent-ing the eld amplitude along the axis perpendicular to the direction of propagation. Anoptical device can therefore be represented by a 22 transfer matrix S relating the inputand output of the eld amplitude and polarization state [29]. We choose a coordinate

system where the x and y axis are oriented parallel and orthogonal to the wires of thepolarizing grid respectively. The optical elements of the Martin-Puplett interferometercan be represented by the following transfer matrices.

Polarizing grid The polarizing grid polarizes the millimeter wavelength radiation.With p1 and p2 denoting the spectral re ectivity of the horizontal and vertical polar-ization state, we obtain the transfer matrix

P =p1 00 p2

(7.1)

The coordinate system is chosen to be aligned with the principal axes of the polarizinggrid.

Analyzing grid The analyzing grid is oriented at an angle of = 90 with respectto the polarizing grid, hence

A =

0 11 0

a1 00 a2

0 11 0

=a2 00 a1

(7.2)

where cos sin sin cos

=0 11 0

(7.3)

has been used. The coecients a1 and a2 denote either the re ectivity or the transmissionof the analyzing grid, depending on which detector is used.

Beam Splitter and Roof Mirrors: The transfer matrix of the beam splitter and roofmirrors is evaluated in a coordinate system where the axis is oriented parallel to thewires of the beam splitter and the axis orthogonal to the wires. In a second step, thetransfer matrices are rotated by 45 to transform ~e and ~e to ~ex and ~ey.

The radiation incident on the beam splitter is transfered to either arm of the interferome-ter according to the grid's spectral re ectivity and and spectral transmission and for the horizontal () and vertical () eld component. An incident polarization stateof 1=

p2(~e+~e) yields the re ected eld component 1=

p2 (~e+~e) or the transmitted

eld component 1=p2 (~e + ~e).

The transfer matrix of the roof mirror (horizontal de ection) is

R = 1 0

0 1

(7.4)

as can be seen from Fig. 7.3. The matrix R has to be rotated by 45 because of theorientation of the roof mirror in the (,) coordinate system. We obtain

~R =

p22

p22

p22

p22

! 1 0

0 1

p

22

p22p

22

p22

!=0 11 0

(7.5)

After the re ection by the roof mirrors, the radiation pulses have the polarization states1=p2 ( exp i~e + exp i~e) (re ected by the beam splitter) and 1=

p2 (~e + ~e)

E

EE

axis

Figure 7.3: Polarization components during re ection at a vertical roof mirror. Theincident light is polarized under 45 with respect to the roof mirror axis. The electric eldcomponent parallel to the orientation of the roof mirror (Ek) is invariant, the orthogonalcomponent (E?) is inverted.

(transmitted by the beam splitter) respectively. denotes the path dierence of bothinterferometer arms.

The beamlets are recombined after passing the beam splitter a second time yielding thestate vector 1=

p2 (( exp i + )~e + ( exp i + )~e). The transfer matrix of

the entire beam splitter and roof mirror complex therefore yields

~B =

0 exp i + exp i + 0

=

0 ED 0

(7.6)

The matrix ~B has to be rotated by 45 to transform ~e and ~e to ~ex and ~ey. Hence

B =1

2

1 11 1

0 DE 0

1 11 1

=

1

2

(D + E) D E(D E) D + E

(7.7)

Notice that a perfect beam splitter yields = 1, = 0, = 0, = 1, = 0, hence

B = 1 0

0 1

(7.8)

Incident horizontally polarized radiation will therefore remain horizontally polarized (since = 0), but acquires a phase shift of . As stated above, elliptical polarization will beobtained for 6= 0. The optical transfer matrix of the entire interferometer is given byS = A B P .

Intensity at the Detector The detector is not sensitive to the polarization state ofthe incident radiation. Initially horizontally and vertically polarized radiation yield theradiation intensity jS11j2+ jS21j2 and jS12j2+ jS22j2 respectively. The normalized responsefunction can therefore be expressed as

R =1

2

Xi;j

jSijj2 = 1

2TrS+S

: (7.9)

where the Sij denote the elements of the optical transfer matrix. For S = A B P weobtain

R =1

2Tr(ABP )+(ABP )

=

1

2TrP+B+A+ABP

=

1

2TrPP+B+A+AB

(7.10)

yielding

R =1

8

n(D + E)(D + E)

p21a

21 + p22a

22

+(E D)(E D)

p21a

22 p22a

21

o(7.11)

where D and E are dened by Eq. (7.6). The expansion of R into R = R0 +Q exp(i) +Q exp(i) yields a term Q exp(i) varying with the phase dierence imposed by themovement of the roof mirror and a dc oset R0. The signal enhancement factor of theinterferometer is dened by Q=R0. The expansion yields

R0 =1

4

np21a

22 + p22a

21

( + )

2 +p21a

21 + p22a

22

( )

2o

(7.12)

Q =1

8

np21a

22 + p22a

21

( + )

2 p21a

21 + p22a

22

( )

2o

(7.13)

and Q = Q [53]. The optimum performance of the interferometer is obtained by usingideal wire grids, hence setting p1 = a1 = 1, p2 = a2 = 0, = = 1 and = = 0,yielding Q = Q = 1=8 and R0 = 1=4, hence

R =1

8(2 exp(i) exp(i)) = 1

4(1 cos ) : (7.14)

a1 and a2 can either denote the re ectivity or the transmission of the analyzing grid. Leta1 and a2 in Eq. (7.14) denote the re ectivity, then R describes the detector responseof the re ected radiation (pyro #2). The detector response caused by the transmittedradiation (pyro #1) can be described by choosing a1 = 0 and a2 = 1 yielding

R =1

8(2 + exp(i) + exp(i)) = 1

4(1 + cos ) : (7.15)

The recorded re ected and transmitted radiation intensity R will be anti-correlated. Forexample, for = 0, the entire radiation power is transmitted and only one detectorrecords a signal. As the mirror moves, the re ected radiation intensity increases, whilethe transmitted intensity decreases and so forth. This feature has been utilized in thepresent interferometer design by the installation of two detectors recording the re ectedand the transmitted electric eld component of the analyzing grid (installed at an angleof 45). The total radiation power reaching the analyzing grid is recorded by the sum ofthe two signals. The ratio of the dierence and the sum of the two detector signals yieldsan interference pattern which is insensitive to beam instabilities.

7.3 Frequency-dependent Acceptance Function

7.3.1 Spectral Transmission and Re ectivity of Wire Grids

A theoretical description of the scattering of electromagnetic waves by a regular grid ofparallel cylindrical wires with circular cross-section and an experimental conrmation of

*

1 THz

Figure 7.4: Computed power transmission of a wire grid as a function of d= (ratio ofwire spacing and the radiation wavelength). The curves denote dierent ratios of wirediameter and wire spacing. Upper graph: polarization of the electric eld vector parallelto the wire grid. Lower graph: polarization of the electric eld vector perpendicular tothe wire grid [54]. The star marks the curve predicted for the interferometer grids.

the theory is well-established in literature [55, 56, 57]. Figure 7.4 shows the power trans-mission of a wire grid versus the quotient of the wire spacing and the radiation wavelengthd=. The resistance of the gold plated tungsten wire is = 5:5 109m. The radiationis approaching the grid under at incidence angle of 45. The upper and the lower graphshow the power transmission of the polarization components parallel and perpendicularto the orientation of the wires.

The wire grid acts as an ideal polarizer (k = 0 and ? = 1) only in the limiting caseof !1. At shorter wavelength the transmission of the \forbidden" polarization com-ponent is increasing while the transmission of the \permitted" component is slightlydecreasing. At a critical wavelength c = 0:59d the grid becomes transparent to incidentradiation of both polarization states. The same applies for = c=n, where n = 1; 2; 3; : : :.The factor 0.59 is caused by the orientation of the wires and is derived in the next section.Wire grid serve as a proper polarizers, beam dividers and analyzers only for wavelengthssignicantly longer than c. Unwanted diraction patterns will occur at shorter wave-lengths. The critical wavelength for 100m wire spacing is c = 170m (correspondingto fc = 1:76THz), which is beyond the range covered in the experiments.

0 0.5 1 1.5

0.6

1

1.4T

rans

mis

sion

Frequency [THz]

Parametrization

Data

Mok &Chambers

0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

Tra

nsm

issi

on

Frequency [THz]

Theory byMok & Chambers

Paramtrization

Data

Figure 7.5: Transmission of the wire grids used for the Martin-Puplett interferometeras a function of radiation frequency. Left: transmission of the polarization componentparallel to the wires. Right: transmission of the polarization component orthogonal tothe wires. Data obtained by THz spectroscopy are indicated as circles. The dashed curvedenotes the prediction of [55]. The solid curve is the parameterization used to describethe spectral transmission of the wire grids.

Transmission Measurement of the Wire Grids The spectral transmission of thewire grids has been measured [58] by I. Wilke and M. Khazan using time-domain THzspectroscopy2. Figure 7.5 shows the measured amplitude transmission of the wire gridsused in the Martin-Puplett interferometer. The left graph shows the spectral transmissionof the polarization component parallel to the wires, the right graph shows the transmission

2I thank Dr. Ingrid Wilke and Mr. Maxim Khazan for performing spectral transmission measure-ments using time-domain Fourier-transform spectroscopy.

A

B

C

D

E

F

G

α

β

θ

Figure 7.6: Beam divider geometry.

of the orthogonal polarization component. There is good agreement between the mea-surement and the theoretical prediction by [55] (dashed line). The solid line indicates anexponential parameterization of the data and is used to describe the spectral transmissionof the wire grids.

7.3.2 Spectral Acceptance of the Wire Grid Beam Divider

The spectral acceptance of a wire grid depends not only on the wire parameters (diameter,wire spacing, wire material), but also on the orientation of the wire grid relative to theincident radiation. Figure 7.6 shows the geometry of the beam divider grid. The wire gridis located in the ABCD plane with the wires oriented along AC. The radiation is incidentalong CE. The plane ABEF represents the projected plane of the wire grid perpendicularto the propagation of the electromagnetic radiation. The projected wire orientation isalong AE.

The following coordinate system is introduced: the z-axis points along the direction ofthe wires along AC, the x-axis is oriented perpendicular to the wires in the plane of thegrid ABCD. The y-axis is oriented perpendicular to the plane of the grid, along GE. Inpolar coordinates, the unit vector ~e is dened as

~e = (sin cos; sin sin; cos ) (7.16)

where is the angle 6 ACE in Fig. 7.6. is oriented in the x; y plane.

An incident electromagnetic wave with momentum ~k and magnitude jkj = !=c has amomentum component perpendicular to the orientation of the wire grid (in the x,y plane),

which is = j~e ~kj = jkj sin . is called the eective wave number of the radiationinteracting with the grid. The wire grid is operated in zeroth diraction order, for wavenumbers smaller than jkj0 = 2=d (normal incidence). For arbitrary incidence, the Braggdiraction condition is (kx+2n=d)

2 = 2, where n denotes an integer. Using kx = cos,the lowest critical wave number is given by

jkjc = jkj0sin (1 + cos)

: (7.17)

Equation (7.17) can be more conveniently expressed by the angle = 6 CBE, denotinghalf of the re ection angle of the wire grid, and = 6 BAE, denoting the apparent angleof the wire grid in the projection plane with respect to the vertical. The colatitude canbe expressed in terms of and by

sin =AE

AC=

AEqCD2 +AD2

=AEq

AB2 + BC2=

1qcos2 + sin2

cos2

(7.18)

where BC2=AE2 = (BC2BE2)=(BE2AE2) = sin2 = cos2 has been used. We obtain nally

sin =cosq

1 sin2 cos2 : (7.19)

The azimuthal angle can be found by equating the y component of the unit vectorEq. (7.16, sin sin, with

sin( ) =EG

CE= cos (7.20)

(see Fig. 7.6), hencecos = sin cos : (7.21)

Equation (7.19) and (7.21) introduced into Eq. (7.17) yields for the lowest critical wavenumber

jkjcjkj0 =

1

cos

s1 sin cos

1 + sin cos : (7.22)

The quotient jkjcjkj0 yields for = 45 (90 de ection at the wire grid): jkjc

jkj0 = 0:59 for

= 0, jkjcjkj0 = 0:82 for = 45 and jkjcjkj0 = 1:41 for = 90.

Equation (7.22) describes the variation of the critical wavenumber for non-perpendicularincidence on a wire grid. Fig. 7.4 shows the transmitted radiation power at a wire gridwhere = 45 and = 0. Notice that the critical wave number, hence the performanceof the grid, can be improved by a factor of 2.4, if is changed to 90. The abscissa ofFig. 7.4 will be scaled by 2.4 while the characteristic of the data is invariant. The opti-mum spectral acceptance of the beam divider can be achieved for = 90. More generallystated, the orientation of the beam divider wires should be as parallel as possible to there ection direction. To optimize the present interferometer assembly, the beam dividershould be oriented horizontally. As a direct consequence, the roof mirrors, the polarizingand the analyzing grid then have to rotated by 45.

Figure 7.7 shows the quantity Q (Eq. (7.13)) of the beam dividerroof mirror congura-tion of the Martin-Puplett interferometer as a function of the radiation frequency. Thespectral transmission and re ectivity functions , , and (see Fig. 7.4) are insertedinto Eq. (7.13). The polarizing and analyzing grids are assumed to be ideal. The beamdivider angles are = 45 and = 45.

The spectral acceptance of the beam dividerroof mirror conguration decreases grad-ually from Q = 1:0 to Q = 0:8 for frequencies up to 1THz (a=d = 0:2) which is thefrequency range covered by the experiments. The acceptance is decreasing linearly atlarger frequencies. Ideally, a ratio a=d = 0:3 yields a at response up to 1:2THz.

Q

0.6

Frequency [THz]

a/d = 0.2

a/d = 0.3

a/d = 0.4

1.2 1.8 2.4

Figure 7.7: The factor Q (Eq. (7.13)) of the beam divider - roof mirror conguration [55].The curves correspond to dierent quotients of wire diameter a and wire spacing d.

7.3.3 Diraction in the Interferometer

Diraction phenomena of freely propagating electromagnetic elds can be described in theslowly varying envelope (SVE) approximation [59]. The time dependent wave equation

1

c2@2 ~E

@t2+r2 ~E = 0 : (7.23)

can be solved by the general ansatz for an electromagnetic wave traveling in z direction

E(r; z; t) = E0(r; z) exp(i (kz !t)) : (7.24)

We obtain in cylindrical coordinates

@2E0(r; z)

@r2+1

r

@E0(r; z)

@r+

@

@z

(@E0(r; z)

@z ikE0(r; z)

) ik

@E0(r; z)

@z= 0 : (7.25)

The SVE approximation implies that the variation of the eld amplitude E0 within awavelength is small compared to the eld amplitude, hence @E0=@z kE0. The equationthen yields

@E0(r; z)

@r2+1

r

@E0(r; z)

@r 2ik

@E0(r; z)

@z= 0 : (7.26)

Equation (7.26) is suitable to describe diraction phenomena as the decrease of the eldamplitude along the direction of propagation z is coupled to an increase of the radialboundary of the eld distribution.

Equation (7.26 is solved by

Ep(r; ; z)

E0= L0

p

2r2

w2

!w0

wexp

r

w

2!exp(i(!t L T )) (7.27)

r

z = 0 z = zR

r

w(z)

w0

R(z)

θ0

Figure 7.8: Field intensity distribution of the I0(r; z)-mode of waist w0 and Rayleighlength z0. R(z) denotes the radius of the surfaces of constant phase, 0 denotes thefar-eld divergence angle.

where

T =kr2

2R, L = kz (2p+ 1) arctan

z

z0

: (7.28)

L0p(r) denotes the Laguerre polynomials of order p. The azimuthal order is set to zero

to describe the radial symmetry of transition radiation. p indicates the number of zerosin the radial eld distribution. A Gaussian eld distribution (\Gaussian beam") is thesolution for p = 0.

The angular distribution of transition radiation can in principle be expressed by a su-perposition of the Ep(r; ; z), p = 0; 1; 2; : : :. For the discussion of diraction eects inthe Martin-Puplett interferometer it is sucient to consider a Gaussian eld distribution(p = 0) of appropriate dimension because diraction aects the eld amplitudes at largeradii only. The intensity distribution of a Gaussian eld distribution I0(r; z) = E

0E0 isshown in Fig. 7.8. The shape of the radial eld amplitude distribution is invariant whilethe eld propagates along the z-axis. Starting from z = 0, the beam size increases as

w = w0

s1 +

z

z0

2

(7.29)

PG BS RM PMAG

HA

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

70

80

90 BS

Path Length in Interferometer [m]

w(z

) [m

m]

λ = 10mm

λ = 1mm

Figure 7.9: The diracted electromagnetic eld envelope in the Martin-Puplett interfer-ometer. The aperture radius of the following optical elements is shown. PG: PolarizingGrid, BS: Beam Splitter, RM: Roof Mirror, PM: Parabolic Mirror, AG: Analyzing Grid,HA: Horn Antenna. The eld envelope is drawn for = 1 cm and = 1mm.

where w0 = 2 denotes the waist dimension of the light beam and

z0 =w2

0

(7.30)

is called the Rayleigh-length. The divergence angle 0 is

=

w0: (7.31)

The beam size w(z) at a given axial position z is, by convention, dened as E0;0(w) := E0=eor I0;0(w) := I0=e

2 respectively. The transverse phase factor T describes the wave frontof the propagating light wave. The bending radius of these surfaces of constant phase is

R(z) = z

1 +

z0z

2!

(7.32)

yielding R ! 1 for z ! 0 (plane waves at the waist), R = 2z0 for z = z0 (maximumbending) and R ! z for z ! 1 (spherical waves with origin at z = 0). The longitu-dinal phase term L indicates the phase of the eld distribution along the direction ofpropagation.

Propagation through an optical system A complex beam parameter

q = z + iz0 or1

q=

1

R i

w2(7.33)

is introduced to transform the eld amplitude along the optical beam line. Transfermatrices

Ld =1 d0 1

(drift d) or Lf =

1 0 1

f1

!(thin lens, focal length f) (7.34)

describe the beam transfer [29] and can be used to transform the complex beam parameterq [59] according to

qf =L11qi + L12

L21qi + L22: (7.35)

Diraction Figure 7.9 shows the envelope of the electromagnetic elds traveling throughthe Martin-Puplett interferometer. The envelope covers the rst parabolic mirror (z = 0)entirely (see chapter 3), so that w0 is equal to the eective mirror radius. The opticaltransfer matrix L of the Martin-Puplett interferometer consists of a drift, a parabolicmirror with 200mm focal length, a drift of 1200mm including the passage through thepolarizer, beam splitter, roof mirrors, a second parabolic mirror with 200mm focal lengthand the nal drift of 110mm through the analyzing grid to the detector. The transitionradiator is in the focal plane of the rst parabolic mirror, hence the initial complex beamparameter is q0 = w0=(i) (the slope of the light envelope after the rst parabolic mir-ror vanishes). The aperture of the optical elements is suciently large that the evaluatedlight beam envelope passes the elements without losses. However, a limiting eect occursat the horn antenna of the detector. The detector dimension (see appendix F) thereforelimits the spectral acceptance of the interferometer. The acceptance becomes wavelengthdependent only for > 2mm as is indicated for the = 10mm envelope.

7.3.4 Detector Acceptance

The detection of far-infrared radiation is based on energy absorption by the pyroelectriccrystal leading to a heating which is accompanied by the creation of surface charges.The spectral dependence of the detection process is in uenced by the dimension (5mmdiameter and 100m thickness) and the permittivity of the pyroelectric crystal. Thecrystal diameter causes a long-wavelength cut-o because diraction eects reduce thecoupling of electromagnetic elds whose wavelength exceeds the detector diameter. Thethickness and the permittivity of the crystal lead to an interference eect at the front andthe back surface of the crystal. The interference eect was discovered in the course of themeasurements and not known before. The eect was investigated also by other groups[34, 60] and improved the understanding of the measured coherent power spectra. Inthe following we let t be the amplitude transmission of electromagnetic elds entering thecrystal and r be the amplitude re ection at the back and the front surface. E0 denotes theincident electric eld amplitude, then the nal electric eld amplitude inside the detectoris

Ein = fE0t exp(i!t) + E0tr exp(i!t) exp(i)g1Xn=0

r2 exp(2i)

2(7.36)

= E0t fexp(i!t) + r exp(i!t) exp(i)g 1

1 r2 exp(2i) (7.37)

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

Frequency [THz]D

etec

tor

acce

ptan

ce 1

0

Figure 7.10: The acceptance function of the pyroelectric detector is dominated by aninterference eect at the front and the back face of the crystal. Minima of the acceptancefunction repeat every 250GHz.

where = 2nd= is the phase of a re ected beamlet with respect to the incident waveE0. n denotes the refractive index of the crystal. The detected intensity Iin = Ein(Ein)

is

Iin =E20t

2 (1 + r2)

1 + r4 2r2 cos 2(7.38)

Figure 7.10 shows the expected detector response Iin=I0 as a function of radiation fre-quency. The interference eect yields an oscillatory acceptance function. Minima andmaxima of the acceptance function repeat every 250GHz which is determined by theoptical path length of nd = 600m.

Calibration Measurement of the Detector Acceptance The pyroelectric detectoracceptance is shown in Fig. 7.11. The data originate from a calibration measurement3 [61]performed with output power calibrated GaAs Read-type IMPATT diodes. The detectoracceptance has been measured at 10 dedicated frequencies (see appendix F). The resultis a slight increase of the acceptance between 72 and 128GHz, a maximum at 156GHzand a decrease toward larger frequencies. The increase below 156GHz is explainable bythe reduced radiation coupling due to the detector diameter, the decrease towards largerfrequencies by the interference eect. The oscillatory characteristic of the latter eect isshown as solid curve, representing the optimum t of Eq. (7.38) to the data. Only fourdata points (f > 150GHz) of the calibration measurement are available to perform theadjustment. Besides, the coherent power spectra measured with the shortest bunches atthe TTF linac have been investigated to determine the frequency interval between themaxima and the minima of the acceptance function. The rapid decrease of the detectoracceptance at low radiation frequencies is described by a vertical line at f = 135GHzin Fig. 7.11 followed by smooth interpolation of the data towards smaller frequencies.The gray curve is nally used to model the acceptance of the pyroelectric detectors.The quality of the calibration data is not yet satisfactory and it is advisable to repeatthe calibration for an increased frequency range. For the time being, the uncertainty of

3thanks to Dr. Jurgen Freyer, Lehrstuhl fur Allgemeine und Angewandte Elektronik, TU Munchen,for the use of the millimeter-wave radiation sources to perform the calibration measurement ofthe pyroelectric detectors.

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

Frequency [THz]

Det

ecto

r A

ccep

tanc

e

Data

Interference characteristic

input to themodel

Figure 7.11: Spectral acceptance of the pyroelectric detector. The data of the calibrationmeasurement are shown as circles. The solid line indicates the interference characteristicof the detector crystal matched best to the result of the calibration measurement. Thedetector acceptance is reduced below 130GHz due to diraction eects. The gray curveis included into the interferometer acceptance to account for the destructive interferencein the crystal and the small frequency cut-o of the pyroelectric detectors.

the detector acceptance function limits the precision of the Martin-Puplett interferometer.

A second result of the calibration measurements is the eective diameter of the hornantenna mounted on the pyroelectric detector (see appendix F). The eective diameterof the horn antenna is d = 32:1mm, while the true diameter is 55:9mm.

7.3.5 Transmission of the Quartz Window

The spectral transmission of the quartz Suprasil II vacuum window has been measured [58]using time-domain THz spectroscopy. Figure 7.12 shows the amplitude transmission whichis decreasing continuously from T = 0:8 at f = 200GHz to T = 0:2 at f = 1:5THz. Alinear t to the data yields T = 0:860:52f [THz] and is used to model the interferometeracceptance function.

7.3.6 Aluminum Foils of Finite Thickness

The electric characteristics of the aluminum transition radiation lms can be describedby macroscopic quantities because the thickness of 40 nm corresponds to about 80100atomic layers. This thickness is sucient for high re ectivity in the optical regime butthe question is whether the incident sub-millimeter radiation may penetrate the metalliclayer since the wavelength is much larger than the thickness of the aluminum lm. Ifpenetration occurs, it will be frequency selective and will lead to an additional acceptancefunction.

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

Tra

nsm

issi

on

Frequency [THz]

T [ ] = 0.86 - 0.52 ⋅ f [THz]

Figure 7.12: Transmission of the Suprasil II quartz glass window as a function of radiationfrequency.

For simplicity the bunch elds are replaced by a free-propagating sub-millimeter elec-tromagnetic wave incident perpendicularly onto the metallic boundary. The electric andmagnetic elds inside the conductor can be described by

~E = ~E0 expr0!

2zexp

i

r0!

2z i!t

(7.39)

~H = (1 + i)

s

20!~n ~E (7.40)

where ~n points along the direction of propagation of the electromagnetic wave. denotesthe conductivity and the permeability of the metal. The conductivity of aluminum at atemperature of 20C is = 3:54 107 1=m. Both eld amplitudes drop exponentially inthe metal. The penetration depth , where the elds have dropped to 1=e of their surface

z

E

B

Figure 7.13: A free-propagating electromagnetic wave incident perpendicularly onto ametallic boundary.

z = 0 z = a

k→

k→

II

k→

IV

zk→III

→k I

Figure 7.14: A free-propagating electromagnetic wave with momentum k is incident per-pendicularly onto an aluminum sheet of thickness a. The wave is partially re ected (kII)and transmitted (kI) at the rst boundary. At the second boundary the transmitted partis again partially re ected (kIII) and transmitted (kIV ).

values, is called the skin depth

=

s2

0!: (7.41)

For aluminum, = 1, we obtain a skin depth of 280 nm at a frequency of f = 100GHz,which is clearly larger than the thickness of the aluminum layer. Obviously the secondboundary of the aluminum layer has to be taken into account to calculate a re ection andtransmission coecient for the incident wave. Assume a metallic layer of thickness a anda coordinate system arranged as in Fig. 7.14. The electromagnetic wave with momentum(k) is incident from the left and is split into a re ected (kII) and transmitted wave (kI) atthe rst boundary at z = 0. At the second boundary (z = a) the wave with momentum(kI) is split again into a re ected and transmitted component with momentum (kIII) and(kIV ) respectively. In steady state, the electric elds in vacuum are given by

~Em = ~Em0 exp(ikmz i!t) (7.42)

~Hm =~km ~Em

0!(7.43)

and in the metal by

~En = ~En0 exp(Az) exp(iAz i!t) (7.44)

~Hn =(1 + i)p

2

s

0!~n ~En (7.45)

where A =q!=2 and ~k = k~ez has been used. The superscripts m and n denote the

dierent eld components either in vacuum or in the material, so Eq. (7.42) describes E,EII and EIV and Eq. (7.44) EI and EIII respectively.

The tangential component of the electric eld (~n ~E) and of the magnetic eld (~n ~H)

have to be continuous at z = 0. Using ~n~k ~E

= ~E

~n ~k

, we obtain a set of four

equations

E0 + EII0 = EI

0 + EIII0 (7.46)

E0k

0! EII

0 kII

0!=

(1 + i)p2

s

0!

EI0 EIII

0

(7.47)

EIV0 exp

ikIV a

= EI

0 exp(Aa + iAa) + EIII0 exp(Aa iAa) (7.48)

k

0!EIV0 exp ikIV a =

(1 + i)p2

s

0!

EIII

0 exp(Aa iAa) + (7.49)

EI0 exp(Aa + iAa)

:

Equations (7.46) (7.50) can be solved for EI0 , E

II0 , EIII

0 and EIV0 as a function of the

amplitude of the incoming wave E0. We obtain for the re ected amplitude EII0

EII0 =E0

0B@

1

20c2 i

0!

sinh(Aa iAa)

120c2+i

0!

sinh(AaiaA)+ 2(1+i)

0c

q

20!cosh(AaiAa)

1CA: (7.50)

Figure 7.15 shows the amplitude re ectivity EII0 =E0 as a function of the thickness of the

aluminum layer. The frequency of the electromagnetic wave is f = 100GHz ( = 3:1mm).We get the important result that a 40 nm aluminum lm is not at all penetrated by theelectromagnetic wave.

An experimental test was made to verify this surprising result. The transmission ofthe transition radiation screen was measured using time-domain THz-spectroscopy [58].No signal was observable. From the signal to noise ration of better than 6000 we canconclude that the transmission in the frequency band between 200GHz and 1:5THz issmaller than 1:5 103. The important conclusion is that the re ection of far-infraredtransition radiation by thin 40 nm aluminum foil amounts to 100% without noticeablefrequency dependence.

7.3.7 Spectral Acceptance of the Martin-Puplett Interferometer

The suppression of high-frequency and low-frequency components of the far-infrared tran-sition radiation eld distribution inside the Martin-Puplett interferometer has been dis-cussed in this section. To summarize:

0 5 10 15 20 25 30 35 40 45 500

0.2

0.4

0.6

0.8

1

Thickness [nm]

Ref

lect

ivity

Scr

een

Thi

ckne

ss

Figure 7.15: Amplitude re ectivity EII0 =E0 as a function of the thickness of the aluminum

lm. The radiation frequency is f = 100GHz.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

0

0.2

0.4

0.6

0.8

1

Frequency [THz]

Inte

rfer

omet

er A

ccep

tanc

e

Figure 7.16: The spectral acceptance function of the TTF Martin-Puplett interferometer.

1. Suppression of low frequencies:

(a) At low frequencies the source dimension of transition radiation exceeds thediameter of the aluminum foil. (chapter 3).

(b) The enlarged opening angle due to Fresnel diraction (chapter 3).

(c) Diraction in the interferometer.

(d) Eciency reduction of the pyroelectric detector at frequencies lower than 130GHz.

2. Attenuation of high frequencies:

(a) The transmission of the quartz window (chapter 3).

(b) The re ectivity of the polarizing grid.

(c) The spectral acceptance of the beam splitter and the roof mirrors.

(d) The transmission and re ectivity of the analyzing grid.

3. Attenuation of intermediate frequencies:

(a) The interference eect in the pyroelectric detector.

(1a), (1b) and the in uence of the transverse beam size on the form factor (3) depend onbeam energy and the transverse beam emittance. Figure 7.16 shows the overall interfer-ometer acceptance for E = 170MeV, = 5 106m.

7.3.8 Systematic Error of the Acceptance Function

The systematic error of the spectral acceptance function has been evaluated using theerrors of the transmission measurements performed with the optical elements of the in-terferometer and by varying the parameters of the diraction calculation.

Quartz Window and Wire Grids: A linear t of the quartz window transmission(Fig. 7.12) has been included in the interferometer model. The transmission data ofthe wire grid have been tted by an exponential curve (Fig. 7.5). The rms scatter of thewindow and grid transmission data around the t yield (wT )rms = 0:022 and (gT )rms = 0:01respectively. The errors of the window, polarizer, beam splitter and analyzer transmissionand re ectivity coecients are uncorrelated and add quadratically.

Diraction: The in uence of diraction on the spectral transfer function can be es-timated by varying the dimension of the transition radiation screen, the size of rstparabolic mirror of the interferometer, the active detector dimension and the beam en-ergy. The transition radiation screen and the parabolic mirror have rectangular shapebut are approximated by circles of equal area in the model. To obtain an error estimateof the approximation, the diameter is varied to either enclose or inscribe the rectangularscreen and mirror. The active detector diameter is varied by 10% and the beam energy by5%. A small size of the transition radiation screen, parabolic mirror and active detectordiameter in combination with a higher beam energy causes the strongest suppression oflong radiation wavelengths. On the other hand, a large transition radiation screen, mirrorand detector diameter and a lower beam energy reduces the suppression. The dierenceof both computations yields an error band for the diractive terms in the interferometeracceptance function. The diractive error is added quadratically to the errors obtainedpreviously.

Transverse Beam Size: The assumed transverse beam size is varied by a factor be-tween 0.5 and 2. The dierence of the frequency domain expressions is added quadraticallyto the previously obtained errors. The error caused by the transverse beam size is smallcompared to the other errors because of the small transverse emittance.

Detector: The thickness and the permittivity of the pyroelectric crystal, which deter-mine the separation of the maxima and minima of the acceptance function, is adapted tothe calibration measurements and the data taken with the shortest bunch length measuredat TTF. A variation of the optical path length within 30m was possible to describe themeasurements. The amplitude re ectivity in Eq. (7.38) is a free parameter to t the in-terference characteristic to the calibration measurement. Unfortunately only four pointsof the calibration measurement, f > 130GHz, are suitable to determine r yielding a largeuncertainty of this parameter.

The error interval of the interferometer acceptance function is shown in Fig. 7.17.

7.4 The Autocorrelation Function

Autocorrelation and Power Spectrum

The radiation intensity at the pyroelectric detectors (Eq. (7.14) and Eq. (7.15)) can berewritten in terms of radiation intensities I

I =1

4I0 (1 cos(!)) : (7.51)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.2

0.4

0.6

0.8

1

Frequency [THz]

Inte

rfer

omet

er A

ccep

tanc

e

Figure 7.17: The spectral acceptance of the Martin-Puplett interferometer. Superimposedis the error interval of the acceptance function.

The phase dierence has been substituted by the angular frequency ! and the timedierence corresponding to the path dierence in the two arms of the interferometer.In case of a continuous spectrum

I() =1

4

Z 1

0

~I(!) (1 cos(!)) d! : (7.52)

The dierence of both detector signals is evaluated to subtract correlated signal uctua-tions originating from beam instabilities, hence

I+() I() =1

4

Z 1

0

~I(!) cos(!) d! : (7.53)

The radiation spectrum can now be evaluated by inverting the cosine Fourier transforma-tion of Eq. (7.53)

~I(!) =8

Z 1

0(I+() I()) cos(!) d : (7.54)

The Fourier transformation of the dierence interferogram I+() I() yields the fre-quency spectrum of the incident radiation pulse. The dierence interferogram is normal-ized by the sum of the two signals to correct for radiation intensity uctuations duringthe measurement.

Simulation

Three longitudinal charge distributions are considered: a Gaussian and two asymmetricdistribution composed of two Gaussians. The beam energy is 170MeV and the normal-ized emittance is = 5 106m. The transverse charge distribution is assumed to beGaussian. The coherent transition radiation spectrum and the autocorrelation responseof the Martin-Puplett interferometer are simulated as follows:

1. Determination of the longitudinal bunch form factor ~q;! by Fourier transformationof the longitudinal charge distribution.

2. Convolution of the form factor with the spectral acceptance function of the inter-ferometer.

3. Computation of the coherent power spectrum j~q;!j2 and application of an inverseFourier transformation to obtain the autocorrelation function (Eq. 7.54).

Figure 7.18 shows the longitudinal charge distribution of the Gaussian (left) and theasymmetric (middle, right) bunches. The second row shows the coherent power spectrumj~q;!j2. The power spectrum is obtained by Fourier transformation of the longitudinalcharge distribution, by a multiplication with the acceptance function and by squaringthe result. The autocorrelation function (third row, solid line) is obtained by an inverse

−5 0 50

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Time [ps]−10 0 10

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Time [ps]

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Angular frequency [1/ps]0 2 4

0

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0.1

0.15

Angular Frequency [1/ps]0 2 4

0

0.05

0.1

0.15

Angular Frequency [1/ps]

ρ q,ω

2

Figure 7.18: Top: Three longitudinal charge distributions versus time. Middle: Theexpected longitudinal from factor versus angular frequency !. Bottom: The expectedautocorrelation in the Martin-Puplett interferometer (solid curve). Superimposed is theautocorrelation of the longitudinal charge distribution disregarding the spectral accep-tance (dashed curve).

0 2 40

0.1

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0 2 40

0.1

0.2

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0.1

0.2

−10 0 10−1

0

1

−10 0 10−1

0

1

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0

1

t [ps] t [ps] t [ps]

ω [1/ps] ω [1/ps] ω [1/ps]

Aut

ocor

rela

tion

ρ q,ω

2

Figure 7.19: Top: the longitudinal from factor versus angular frequency. Bottom: the au-tocorrelation function. The shaded area denotes the uncertainty interval of the conversionfrom the charge distribution to the form factor and autocorrelation function.

Fourier transformation. Superimposed is the autocorrelation function without any spec-tral corrections (dashed line).

The spectral corrections cause signicant oscillations of the autocorrelation function due tomissing frequency components. The main maximum is narrower with spectral correctionsthan without. The FWHM of the autocorrelation function is therefore not a reliablemeasure of the bunch length. To learn about the shape of the longitudinal distribution,more sophisticated analysis techniques have to be applied to the entire autocorrelationand the coherent power spectrum.

Error Investigation The uncertainty of the bunch form factor and the autocorrela-tion function are dominated by the imprecise knowledge of the spectral acceptance ofthe interferometer. Figure 7.19 shows the error intervals of the form factors and theautocorrelation functions for the three charge distributions.

7.5 Reconstruction of the Longitudinal Charge Dis-

tribution

The variety of eects in uencing the spectral acceptance of the interferometer make adirect determination of the longitudinal bunch charge distribution dicult. The analysispresented in this section therefore follows the opposite way: a \reasonable" longitudinalbunch charge distribution is assumed and Fourier transformed. After the application of theinterferometer acceptance function, a prediction of the coherent power spectrum and the

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[a.u

.]

σ = 300fst

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(b)

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(c)

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[a.u

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[a.u

.]

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ω [1/ps]

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er S

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[a.u

.]

(a)

Figure 7.20: Reconstruction of a symmetric longitudinal bunch charge distribution. Leftcolumn: bunch charge distribution. Middle column: coherent power spectrum. Rightcolumn: autocorrelation. Row (a): simulated bunch charge distribution, coherent powerspectrum and autocorrelation. Row (b) shows a Gaussian distribution of amplitude A1 =1:2 and rms width of z = 300 fs tting best to the highest order maximum of thesimulated power spectrum. The power spectrum and the autocorrelation are shown assolid curves. Row (c) shows a second Gaussian distribution superimposed contributingthe missing lower frequency contributions (A2 = 1 and 2 = 1ps). The correspondingpower spectrum and the autocorrelation are presented as solid curves.

autocorrelation function is made. The parameters of the initial bunch charge distributionare varied until the predictions are in good agreement with the measured autocorrelationand power spectrum. This procedure does of course not yield a unique description butappears to be the most useful compromise in view of the many limiting eects. In general,the measured data cannot be described by a single Gaussian distribution. A sum ofseveral Gaussian distributions of dierent amplitudes Ai and widths i, which can evenbe displaced, have to be considered. The reconstruction of a symmetric and an asymmetricbunch charge distribution will be discussed in the following.

Symmetric Charge Distribution

Consider a charge distribution composed of two Gaussians of amplitudes A1 = 1:2, A2 = 1and variance 1 = 300 fs and 2 = 1ps respectively. Row (a) of Fig. 7.20 shows the bunchcharge distribution, the coherent power spectrum and the autocorrelation of the chargedistribution as circles. To reconstruct the initial charge distribution, a single Gaussian isassumed rst. The amplitude and the width of the Gaussian is varied until its Fouriertransformation ts the highest order maximum of the coherent power spectrum (b). Asecond Gaussian distribution is superimposed to match the lower frequency components.In the present case, the rms width of the second distribution may not be shorter than 1 psto prevent contributions at frequencies larger than 500GHz in the power spectrum. Theamplitude and the rms width is varied until an agreement is reached.

Asymmetric Charge Distribution

The question arises whether asymmetric charge distributions are analyzable with thepresented method. See Fig. 7.21, row (a), as an example and consider the asymmetriccharge distribution introduced in Fig. 7.18. The simulated coherent power spectrum andthe autocorrelation are shown as circles. Row (b) shows a Gaussian charge distributionwhose frequency contribution ts to the highest order maximum of the coherent powerspectrum. In contrast to Fig. 7.20, we obtain too large low frequency components whichcannot be compensated by just adding a Gaussian curve of smaller rms width. Hence amore complicated bunch charge distribution must be considered.

The Gaussian charge distribution of amplitude A1 is separated into two Gaussians ofamplitude A1=2 and equal rms width. In a second step the two Gaussians are separatedin time. The adaption of the highest order maximum will degrade but while continuingthe separation, the adaption will eventually improve. The optimum separation of the twopeaks is found for the best re-adaption of the highest order maximum of the coherentpower spectrum (c). The lower frequency components are also reduced. Thirdly, we su-perimpose a Gaussian distribution of larger rms width. The amplitude and the width ofthe charge distributions are varied until the coherent power spectrum and the autocorre-lation function agree with the simulation (d).

It can be very time-consuming to work out arbitrary longitudinal charge distributionsusing the presented method. The use of a superposition of Gaussian charge distributionsis sucient for the present data quality which is heavily impeded by the interferometeracceptance.

Error Investigation The three longitudinal charge distributions of Fig. 7.19 have beenused to investigate the systematic error which is due to the interferometer acceptance.The autocorrelation and the coherent power spectrum are evaluated for the upper andlower boundary curves of the interferometer acceptance function. The charge distributionis then reconstructed using the method outlined above. The error range of the longitu-dinal charge distributions is shown as the shaded area in Fig. 7.22. The error is largestat the maxima and local minima of the distributions. Allowing for a variation of the lineshape within the full range of the error band, it is still possible to distinguish between a

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(a)

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σ = 0.5pst A = 1.56

(b)

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[a.u

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∆ = 1.7ps

(c)

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[a.u

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rge

Den

sity

[a.u

.]

σ = 0.5pst

A = 1.16σ = 0.6pst

A = 0.7

∆ = 1.7ps

(d)

Figure 7.21: Reconstruction of an asymmetric longitudinal bunch charge distribution.Left column: bunch charge distribution. Middle column: coherent power spectrum. Rightcolumn: autocorrelation. Row (a): simulated bunch charge distribution, coherent powerspectrum and autocorrelation. Row (b) shows a Gaussian distribution of amplitude A1 =1:56 and rms width of 1 = 500 fs tting best to the highest order maximum of thesimulated power spectrum. The corresponding power spectrum and the autocorrelationare shown as the solid curves. The lower frequency contributions exceeding the simulatedpower spectrum indicate a more complicated charge distribution. Row (c) shows twoGaussian distributions of amplitude A1=2 and rms width 1 separated by = 1:7 ps. Ata separation of , the highest order maximum of the simulated (circles) and predictedpower spectrum (solid curve) are, again, in agreement. The lower frequency componentsare reduced. The amplitude and the rms width of both Gaussians has been optimized, asshown in (d), to obtain the simulated charge distribution.

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ity [a

.u.]

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rge

dens

ity [a

.u.]

−5 0 50

0.5

1

1.5

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Cha

rge

dens

ity [a

.u.]

Figure 7.22: Estimation of the systematic error of the time- and frequency-domain ttingtechnique to determine the longitudinal bunch charge distribution. The charge distribu-tions introduced in Fig. 7.19 are reconstructed. The shaded area denotes the uncertaintyof the reconstructed bunch shape. The error is mostly due to the uncertainty of thespectral acceptance function of the interferometer.

single peak, shoulder and double peak distribution. The method is, on the other hand, notable to predict the forward-backward symmetry of the longitudinal charge distribution,because this information is contained neither in the power spectrum nor in the autocor-relation.

More sophisticated data analysis techniques, as the application of a Kramers Kroniganalysis on the bunch form factor, make sense only if the form factor is known preciselyover a large spectral interval. A Kramers Kronig analysis [62], which has been successfullyapplied at the linac of the Cornell synchrotron [63], can be used to reconstruct the phaseinformation of the longitudinal form factor. Once the phase is known, a direct inverseFourier transformation can be used to derive the bunch charge distribution. In the presentcase, a deconvolution of the longitudinal form factor with the interferometer acceptancewould lead to large uncertainties in frequency intervals where the acceptance function issmall. The technique would become applicable when a dierent detection device with a at spectral acceptance, for example a photo acoustic detector, is used.

7.6 Determination of the Bunch Length

7.6.1 Analysis of the Autocorrelation

Under the assumption of a Gaussian charge distribution, the bunch length and the widthof the autocorrelation function are closely related. A Gaussian of variance z yields anautocorrelation with a variance of 2z. The idea is to predict the length of the longitudinalcharge distribution directly from the width of the autocorrelation function. The in uenceof the interferometer acceptance function on this simple method will be investigated inthis section.

A simulation is used to investigate the error of the bunch length determination using the

−2 −1 0 1 2

0.7

0.8

0.9

1

FW

HM

(A

utoc

orr)

/ F

WH

M (

Cha

rge)

Translation [ps]

Figure 7.23: FWHM of the autocorrelation used for the determination of the bunchlength. The simulated charge distribution consists of two Gaussians of equal rms width but dierent height Ai ( = 1ps, A1 = 1, A2 = 0:6). The ratio of the width of theautocorrelation and the width of the charge distribution is plotted versus the separationof the two Gaussians. The error bars are due to the uncertainty of the spectral acceptanceof the Martin-Puplett interferometer.

FWHM of the autocorrelation function. The longitudinal charge distribution is describedby two Gaussians of equal rms width ( = 1ps) but of dierent height Ai (A1 = 1,A2 = 0:6). To vary the FWHM of the initial charge distribution, the two Gaussians areseparated in small steps relative to each other. Figure 7.23 shows the ratio of the FWHMevaluated from the autocorrelation function and the FWHM of the initial charge distri-bution as a function of the translation of the two Gaussians. The error bars are due tothe uncertainty of the spectral acceptance of the interferometer.

For a negligible separation of the two Gaussians, the ratio of the two is close to unity.The ratio is decreasing for a larger separation of the charge contributions. The initialcharge distribution becomes longer, at large variations a shoulder or a double peak isformed. The width of the autocorrelation, on the other hand, does not change, which isexplainable by the suppression of the low frequencies by the acceptance function. Thebunch length is therefore under-estimated.

To conclude, the FWHM of the interferogram measured by the TTF Martin-Puplettinterferometer is a good measure of the FWHM bunch length only under the assumptionof Gaussian longitudinal bunch shapes. More complicated bunch shapes lead to resultswhich deviate from the correct FWHM value easily by a factor of two.

7.6.2 On-line Monitoring of the Bunch Length

It is possible to monitor the bunch length on-line by observing the detector signal ampli-tude. Figure 7.24 shows the coherent power spectrum of Gaussian charge distributions oft = 1ps (left) and t = 2:5 ps (right). The spectral acceptance function of the Martin-

0 1 2 30

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0.15

Angular frequency [1/ps]0 1 2 3

0

0.05

0.1

0.15

Angular Frequency [1/ps]

ρ q,ω

2

ρ q,ω

2

σ = 1pst σ = 2.5pst

Figure 7.24: Coherent power spectrum of a Gaussian charge distribution. Left: t = 1ps.Right: t = 2:5 ps. The detector response is proportional to the integral coherent powerspectrum marked by the shaded area. Shorter bunches yield larger signals.

Puplett interferometer is included in the computation. The detector signal is proportionalto the integral of the power spectrum shown in Fig. 7.24 (shaded area). For a constantbunch charge and average beam current, shorter bunches will cause larger detector sig-nals. Figure 7.25 shows the expected relative detector signal versus bunch length underthe assumption of the charge distributions in Fig. 7.19. This method has been extensivelyused in optimizing the bunch compression.

7.7 Bunch Length Measurements

Bunch length measurements performed with the Martin-Puplett interferometer are pre-sented in this section. The instrument has been used to determine the longitudinal chargedistribution of the beam produced by the photo injector and the thermionic gun.

1 2 30

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σ [ps]t σ [ps]t σ [ps]t

Figure 7.25: Pyroelectric detector signal as a function of the bunch length. Bunch chargeand average beam current are invariant. The error bars are due to the uncertainty ofthe acceptance function. The left, middle and right graph corresponds to the respectivelongitudinal charge distribution introduced in Fig. 7.19.

7.7.1 Measurements with the Photo Injector Beam

The rf gun photo injector produced 20 bunches of 1 nC charge with a repetition rateof 1MHz. To measure the bunch length, the beam was transfered to the high energyexperimental area (E = 170MeV) with and without the use of the magnetic chicane.Bunch compression using the magnetic chicane was investigated.

Alignment of the Interferometer

The interferometer is aligned using the light of a diode laser which is guided into the beampipe about 3m upstream of the interferometer and re ected by the transition radiationscreen. The laser beam is aligned on the accelerator axis by centering the spot on therst mirror and on the transition radiation screen.

The polarizing grid of the interferometer is covered with a stretched aluminum foil tore ect the laser beam. The beam divider is replaced by a stretched transparent foil act-ing as an optical beam divider. The plane of the foil is identical to the wire grid. Thepyroelectric detectors and the analyzing grid are replaced by two small apertures of 2mmdiameter distant by 130mm to dene the position and the angle of the laser beam in theinterferometer.

Figure 7.26: The signal of the pyroelectric detectors sampled by an ADC. The signal is theintegrated response over a macropulse. The duration of the detector signal is explainedby the 4 kHz bandwidth of the amplier. The dierence between the main maximum andthe oset prior to the maximum is proportional to the incident radiation intensity.

0 5 10 15 20 250

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o S

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]

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Bunch Charge [nC]

Pyr

o S

igna

l [V

]

Pyro 1Pyro 2

Figure 7.27: Left graph: pyroelectric detector signal as a function of the number ofbunches (Q = 1:2 nC). Right graph: pyroelectric detector signal as a function of bunchcharge for three bunches.

The interferometer table is arranged such that the table is leveled out and that the laserbeam hits the center of the rst parabolic mirror. The parabolic mirror is adjusted thatthe re ected laser beam hits the beam splitter at the correct height and that it hits thecenter of the two roof mirrors. The tilt of the roof mirrors is adjusted to produce a singlelight spot on the transparent beam splitter foil. The second parabolic mirror is thenadjusted to transfer the laser beam through the two apertures.

Data Acquisition

Figure 7.26 shows the signal shape of the pyroelectric detectors. The incident radiationintensity is proportional to the signal peak. The peak is determined by evaluating themaximum of a parabolic t applied to a data sample surrounding the peak. The detectorsignals are sampled every macropulse by the ADC. The detector signals, typically vesignals per position of the roof mirror, are stored for o-line data processing.

Coherence of Transition Radiation

The size of the pyroelectric detector signal depends on the number of bunches and thebunch charge. A linear increase of the detector signal is expected when the number ofbunches is increased and the bunch charge is kept constant. This linear increase is indeedobserved as shown in the left graph of Fig. 7.27. In the contrary, according to Eq. (3.23),a quadratic increase of the detector signal is expected for a variation of the bunch charge.The non-linear dependence of the detector signal versus bunch charge is presented in theright graph of Fig. 7.27. The conclusion is that the transition radiation generated by theelectron bunches is in fact coherent.

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Figure 7.28: Upper left: raw signals detected by the pyroelectric detectors during aninterferometer scan. Upper right: the two signals are scaled, and plotted with an arbitraryoset of 3 and 6V. Signal #2 is inverted. Lower left: normalized dierence interferogram.Lower right: coherent power spectrum obtained by Fourier transformation.

Autocorrelation and Coherent Power Spectrum

Figure 7.28 shows the raw signals of the pyroelectric detectors during an interferometerscan. The anti-correlated response of the two detectors is caused by observing either thetransmitted or the re ected radiation component behind the analyzing grid. The sum ofboth signals is proportional to the radiation power. A scaling correction is needed forthe slightly dierent amplication of the two detectors. The scaling factor is obtained bycomparing the main maxima (obtained by a parabolic t of the data around the peak)of both signals and the signal average. The scaled signals are depicted in the upperright graph of Fig. 7.28 with an oset of 3 and 6V. The detector signals are subtractedfrom each other and normalized to their sum yielding an improved signal-to-noise ratio.Correlated intensity uctuations which are caused by bunch charge variations and orbitinstabilities cancel in he normalized dierence interferogram shown in the lower left graphof Fig. 7.28. The coherent power spectrum (lower right graph) is evaluated by Fouriertransformation.

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.u.]

MeasurementError

Figure 7.29: Shortest bunch measured at the TTF linac. Upper left: autocorrelation.Upper right: coherent power spectrum. Data is represented by circles. Lower left: lon-gitudinal charge distribution matching the measured autocorrelation and the coherentpower spectrum. The solid curve shown in the upper graphs depicts the predicted auto-correlation and coherent power spectrum of the longitudinal pulse shape. Lower right:error estimate of the longitudinal charge distribution.

Bunch Length and Bunch Shape Determination

The time- and frequency-domain tting analysis can be applied to the data shown inFig. 7.28. A superposition of three Gaussian distributions of dierent amplitude and vari-ance is required to reproduce the measured autocorrelation and coherent power spectrum.Figure 7.29 (lower left) shows the longitudinal bunch charge distribution

L(t) = A1 exp

t2

221

!+ A2 exp

t2

222

!+ A3 exp

t2

223

!(7.55)

with

A1 = 0:62 ; 1 = 270 fs ; A2 = 0:20 ; 2 = 1ps ; A3 = 0:18 ; 2 = 2ps (7.56)

matching the measured autocorrelation and coherent power spectrum. The autocorrela-tion and the coherent power spectrum computed from the longitudinal charge distribution

−15

−5

515

−0.1 0

0.1

0.2

0.3

Tim

e [ps]

Autocorrelation [a.u.]

Φ =

25o

−15

−5

515

−0.1 0

0.1

0.2

0.3

Tim

e [ps]

Autocorrelation [a.u.]

Φ =

15o

−15

−5

515

−1 0 1 2

Tim

e [ps]

Autocorrelation [a.u.]

Φ =

45o

−15

−5

515

−0.2 0

0.2

0.4

Tim

e [ps]

Autocorrelation [a.u.]

Φ =

35o

Figu

re7.30:

Autocorrelation

measu

redbytheMartin

-Puplett

interferom

eterfor

dieren

trf

gunphases.

Itisnot

possib

leto

reproduce

theinterferogram

susin

gthetim

e-and

frequency-dom

ainttin

gtech

nique.

ofEq.(7.55)

arepresen

tedin

Fig.

7.29(top

row).Themeasu

rementsare

describ

edvery

well.

Theerror

ofthelon

gitudinalcharge

distrib

ution

isdeterm

ined

bycon

volutin

gthe

charge

distrib

ution

Eq.(7.55)

with

theupper

andlow

erboundof

theinterferom

eterac-

ceptan

cecurve.

Thelow

errigh

tgrap

hofFig.

7.29show

stheresu

ltingcon

dence

interval

asshaded

area.Thesolid

linedepicts

themean

bunch

charge

distrib

ution

.Thenarrow

peak

ofthebunch

charge

distrib

ution

isdeterm

ined

more

precisely

than

thetails

ofthe

distrib

ution

which

iscau

sedbytheuncertain

tiesof

theaccep

tance

curve

atlow

frequen-

cies.

TheFWHM

bunch

length

ofthecharge

distrib

ution

ofEq.(7.55)

is(1:33

0:45)ps,cor-

respondingtoz=

(20070)

m

ort=

(66025)

fsresp

ectively.Themeasu

rement

presen

tedbyFig.

7.29depicts

theshortest

bunch

studied

with

theMartin

-Puplett

inter-

ferometer

attheTTFlinac.

Theresu

ltisin

agreementwith

theshortest

bunch

length

determ

ined

bylon

gitudinalphase

space

rotations.

Also

thepulse

shape,anarrow

peak

superim

posed

onto

awider

basis,

iscon

sistent.

Measurementswith

outCompressio

n

Thelon

gitudinalbunch

charge

distrib

ution

has

been

measu

redwith

outtheuseofthechi-

canecom

pressor

toinvestigate

theapplicab

ilityof

themeth

odin

thelim

itof

longbunch

−20 −10 0 10 20

−0.5

0

0.5

1

time [ps]

Aut

ocor

rela

tion

[a.u

.] DifferenceInterferogram

0 0.1 0.2 0.3 0.4 0.5 0.60

0.2

0.4

0.6

0.8

1

1.2

1.4

frequency [THz]

Pow

er S

pect

rum

[a.u

.]

CoherentPower Spectrum

Figure 7.31: Interferogram and coherent power spectrum measured with the beam of thethermionic injector. Data are represented by circles. The solid line represents the predic-tion of the longitudinal bunch charge distribution of Eq. (7.55) using the interferometermodel.

lengths. The injector parameters used for the streak camera measurements in chapter 6have been applied. Figure 7.30 shows four interferograms measured by the Martin-Puplettinterferometer for dierent phases of the gun rf eld . It is conspicuous that the shapeof the interferogram is dierent from previous measurements. The shape can, in fact,not be predicted by the interferometer model. The model predicts, independent of theshape of the charge distribution, larger oscillations at the tail of the interferogram. Theproblem consists in the Fourier transformation of the measurement. The coherent powerspectrum reaches merely a frequency of 200GHz which is below the rst minimum of theinterferometer acceptance curve. The coherent power spectrum is suppressed in the entirefrequency interval so that the analysis is very much impeded. The investigation of theFWHM of the autocorrelation function yields bunch lengths which are under-estimatingthe result obtained with the streak camera measurement by a factor of 24.

The present analysis is not applicable for coherent power spectra which do not exceedthe frequency of the rst minimum of the interferometer acceptance function. Only withthe rst appearance of frequency contributions larger than 250GHz bunches shorter thant = 2ps can be condently predicted.

7.7.2 Measurements at the TTF Thermionic Injector

The TTF thermionic injector was operated at an average beam current of 5mA and pro-duced 30s long macropulses with a bunch to bunch repetition rate of 216MHz. Thebunch charge is 37 pC. The beam was accelerated to an energy of 80MeV and transferedto the experimental area.

Figure 7.31 shows an interferogram and the coherent power spectrum. The small sec-ondary maxima in the coherent power spectrum indicate a bunch length shorter than2 ps. A Gaussian bunch charge distribution tted to the data yields an rms bunch length

of z = 0:6mm (t = 1:9 ps). Further investigations of the thermionic injector bunchlength using the Martin-Puplett interferometer have been performed by [50] yielding aminimum length of z = 0:5mm (t = 1:6 ps).

7.8 Possible Improvements of the Interferometer

The Martin-Puplett interferometer is limited by the acceptance of the pyroelectric detec-tors suppressing frequencies due to an interference eect on the front and back side of thecrystal. It is advisable to replace the detector by a Golay cell (photo-acoustic detector)having a at acceptance in the in the frequency range between 30GHz and 3THz [64].Golay-cells have a larger sensitive detector dimension which improves the low frequencyacceptance.

The overall acceptance of the interferometer can be improved by wire grids wound withwires of 10m diameter and 30m spacing. The ratio of wire diameter and wire spacingof a=d = 0:3 is favorable for a at acceptance of the beam splitter. The wire grid ac-ceptance can further be improved by a rotation of the grids: a vertical orientation of thepolarizer in the present setup improves the acceptance limit of the interferometer alreadyby a factor of 1:4.

The transmission out of the beam pipe can be improved by the use of a crystalline quartzwindow. These windows are transparent for radiation up to 3THz. A further improve-ment can be achieved only by the use of a diamond window, which is transparent in theentire spectrum of infrared radiation.

The acceptance of the pyroelectric detector can be improved by increasing the distance tothe second parabolic mirror. Diraction losses inside the beam pipe cannot be signicantlyreduced, because of the limited size of the pipe and the vacuum window. An interestingoption is the use of a diraction radiation screen, which is an aluminum foil with a circularhole at the center. The electron bunch is traveling through the hole whereas the coherentradiation is re ected from the foil yielding an almost \non-destructive" bunch lengthmeasurement [65].

Chapter 8

Hilbert Transform Spectroscopy

High-Tc Josephson junctions oer a new spectroscopic method to determine the longitu-dinal bunch form factor. The principle is to investigate the modication of the current-voltage characteristic of a Josephson junction due to incident radiation. Applying aHilbert transformation to this modication the frequency spectrum of the radiation canbe derived. Since the principle of Hilbert transform spectroscopy is based on a purelyelectric measurement, it oers the possibility of high-speed spectroscopy in the millimeter-and sub-millimeter wavelength regime.

8.1 Electric Properties of a Josephson Junction

8.1.1 The Josephson Eect

A Josephson junction consists of two superconductors separated by a thin insulating layer.The electrons in a superconductor form Cooper-pairs which can be described by a singlemacroscopic quantum mechanical wave function. If the insulating layer is sucientlythin, Cooper-pairs will penetrate from one superconductor to the other by the quantummechanical tunnel eect. The Josephson equations are

IJ = Ic sin(t) (8.1)

d

dt=

2e

hU(t) (8.2)

where Ic is the critical current of the junction, (t) is the phase dierence of the Cooperpair wave functions on both sides of the insulating layer and U(t) is the voltage acrossthe junction. For currents jIj < Ic the dc Josephson eect occurs. According to (8.2) Uvanishes in this case. The detection of microwave radiation is based on the ac Josephsoneect, hence the junction is operated with a dc current I = I0 > Ic, leading to a non-vanishing voltage U . Incident radiation modies the current-voltage characteristic of thejunction, and this change is used to determine the spectral distribution of the radiation.

Josephson junctions used for a spectral analysis in the range 100-1000 GHz have to bedesigned to have a negligible capacitance. Their electrical performance is then well de-scribed by the so-called Resistively Shunted Junction (RSJ) model. An equivalent circuit

I RIJ

I

J RU

Figure 8.1: Equivalent circuit of a resistively shunted Josephson junction (RSJ model).

is shown in Fig. 8.1. Both Cooper pairs and unpaired electrons are able to tunnel throughthe insulating layer of the Josephson junction. An applied current I splits up into aCooper-pair (or Josephson) current (IJ) and a single-electron \resistive" current (IR).The Cooper pair current obeys the Josephson equations.

8.1.2 DC Characteristic of the Josephson Junction

The equivalent circuit for an \autonomous" Josephson junction (without incident mi-crowave radiation) is shown in Fig. 8.1. The bias current of the junction is a pure dccurrent I0 > Ic. According to Kirchho's law, the current I0 splits into both branches ofthe circuit and the following equation holds:

I0 = IJ + IR = Ic sin+h

2eR

d

dt: (8.3)

Solving the dierential equation for (t) and using Eq. (8.2), an expression for the time-dependent voltage U(t) across the junction is derived. As shown in appendix E.1 theresult is [66, 67]

U(t) = RIc (I0=Ic)2 1

I0=Ic cos (!0 t)for I0 > Ic (8.4)

where !0 =2ehRIc

q(I0=Ic)

2 1 is the Josephson frequency. An arbitrary constant phasehas been omitted. This equation shows that for I0 > Ic an oscillating voltage U(t) ariseswhich derives from an alternating Cooper-pair current as well as a resistive single-electroncurrent. The voltage U(t) has a dc component and is not purely sinusoidal as can be seenfrom Fig. 8.2. For a bias current just above the critical current a strongly distorted sinewave is observed while with increasing bias current the oscillation becomes more and moresinusoidal. Taking the time-average of Eq. (8.4) one obtains (see appendix A)

U =

(0 for I0 IcRqI20 I2c for I0 > Ic

): (8.5)

The current-voltage characteristic of the junction, obtained by plotting I0 as a functionof U , is shown as the dashed-dotted curve in Fig. 8.3.

0 5 10 15 20 25 30 350

0.5

1

1.5

2

2.5

t [ps]U

(t)

[mV

]

0 0.5 1 1.5 2 2.5 3 3.53

3.5

4

4.5

5

t [ps]

U(t

) [m

V]

Figure 8.2: Time dependence of voltage U(t) across the Josephson junction for the au-tonomous case (no incident radiation). Top graph: I0=Ic = 1:02, bottom graph: I0=Ic = 4.The resistance R is assumed to be 1 , the critical current is 1 mA.

8.1.3 Josephson Junction with Incident Radiation

The incident microwave radiation is modeled by a small sinusoidal ac current which issuperimposed with the dc bias current. We consider rst the case of monochromaticradiation, the generalization to a continuous spectrum is straight forward. The total biascurrent is now the sum of a dc component I and an ac component with amplitude Is andfrequency !s=2

I(t) = I + Is sin!st : (8.6)

The generalization of Eq. (8.3) reads

I + Is sin!st = Ic sin+h

2eR

d

dt: (8.7)

The dierential equation (8.7) has no analytic solution. It can be solved approximately ifthe amplitude Is of the ac current, representing the incident radiation, is small comparedto the dc bias current. A second-order perturbation approach [66, 67, 68], outlined inappendix E, yields a correction to the dc current

I = 1

4I

2eRIch

2 I2s!2s !2

0

for (!s 6= !0) (8.8)

where

!0 =2eU

h(8.9)

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

U [mV]

I [m

A]

1.2 1.3 1.4 1.5 1.6 1.71.6

1.8

2

0

Figure 8.3: Dashed-dotted curve: time-averaged voltage across the junction as a func-tion of the dc bias current. Solid curve: modication of dc characteristic curve due tomonochromatic incident radiation. The dc current-voltage characteristic acquires a stepat the voltage U = h=2e!s. The height of the step is proportional to the incident radiationpower. The RSJ model is assumed.

denotes the frequency of the unperturbed Josephson oscillation at the given dc bias. Thefrequency !0 is often called the working point of the junction, since it gives the frequencyof the unperturbed Josephson oscillation. The working point is changed by varying thedc bias current.

Equation (8.8) diverges for !0 = !s, but this divergence disappears if the internal noisein the junction is taken into account. With noise included the equation can be written as

I = 1

8I!0

2eRIch

2

!s !0(!s !0)2 + 2

!s + !0(!s + !0)2 + 2

!(8.10)

where is a damping term introduced by the noise. Hence the junction responds to theincident monochromatic radiation in such a way that the dc current-voltage characteris-tic acquires a step I in current at the voltage U = h=2e!s. The height of the step isproportional to the incident radiation power. This is shown schematically in Fig. 8.3.

8.1.4 Hilbert Transform Spectrometry

Incident radiation consisting of a number of discrete spectral lines will lead to corre-sponding discrete current steps in the I-U curve. Now we want to demonstrate that the

Josephson junction, as described by the RSJ model, may equally well serve as a detectorfor electromagnetic radiation with a continuous spectral distribution Si(!s) [69]. The ba-sic point is the observation that the voltage response of the junction depends quadraticallyon the amplitude Is of the ac current (see Eq. (8.10)), hence intensities can be added. Forthis reason the voltage response to a continuous radiation spectrum is simply obtainedby integrating Eq. (8.8) over the frequency range of the radiation, taking the spectralintensity Si(!s) as a weight factor.

I = 2e

h

2 R2I2c4I

Z 1

0

S(!s) d!s!2s !2

0

: (8.11)

The pole in the integrand requires a careful mathematical treatment. The equation canbe written in the following way

I = 1

8I!0

2eRIch

2PZ 1

1S(!s) d!s!s !0

=

8I!0

2eRIch

2H[S(!s)] (8.12)

where P denotes the principal value of the integral and

H[S(!s)] =1

PZ 1

1S(!s) d!s!s !0

(8.13)

is the so-called Hilbert transform (see appendix E) of the radiation spectrum S(!s). Notethat by evaluating the principal value of the integral one can use the noise-free Eq. (8.8)instead of (8.10). The spectral intensity of the incident radiation, computed by the inverseHilbert transform of (8.12), is

S(!s) = H1(g(!0)) (8.14)

with the function

g(!0) =8

h

2e

!2I(!0)I(!0)!0

R2I2c: (8.15)

Equation (8.15) shows the quantities that have to be measured to determine the radiationspectrum. The current-voltage characteristic of the Josephson junction has to be scannedboth with and without incident radiation. During the scan the bias current is increasedin small steps. At each step the Josephson frequency !0 = 2eU=h and the modicationU(!0) due to the radiation are measured. The current modication I is derived fromU via the dierential resistance Rd = dU=dI of the unperturbed I-U curve. Thesequantities are used to compute the function g(!0) which is Hilbert-transformed to obtainthe spectral intensity S(!s) of the incident radiation. The square root of S(!s) then yieldsthe longitudinal bunch form factor.

8.2 The Josephson Junction

High-Tc Josephson junctions have been fabricated by epitaxial growth of YBa2Cu3O7xon NdGaO3 bicrystal substrates [70]. A schematic view of the detector which incorpo-rates the antennas for millimeter and sub-millimeter wave detection is shown in Fig. 8.4.The size of the antenna varies from 1.8 mm at the outer bound to several micrometers

YBa Cu O2 3 7

m20

3NdGaO

µ

I

U

1.8 mm

Grainboundary

Figure 8.4: A schematic view of the Josephson junction used as a detector for millimeterand sub-millimeter wave radiation.

at the center. Here, a grain boundary in the substrate cuts the thin lm of the high-Tcsuperconductor, which then works as a Josephson junction. The electrical properties ofgrain boundary high-Tc junctions are very close to those predicted by the RSJ model [71].An electrical circuit to bias the junction with a dc current and to measure the potentialdierence across the contact is connected to the antennas. The junction features a largedynamic range of approximately 105 and a high sensitivity of 1014 W=Hz1=2 NoiseEquivalent Power (NEP) to millimeter- and sub-millimeter radiation [72]. The resolutionis around 1 GHz in the temperature range from 4 to 78 K [70].

The Josephson spectrometer was mounted at the same diagnostic station of the linacwhich had been used to determine the bunch length with the Martin-Puplett interfer-ometer. Coherent transition radiation generated at the front face of an aluminum foilwas guided through a quartz window to the antenna of the Josephson junction, using anarrangement of mirrors and wire grid polarizers. A grid served as a low pass lter formillimeter radiation. The junction was installed inside a pumped aluminum pipe placedin a liquid nitrogen bath (T = 78K). The millimeter wavelength radiation was transferedto the junction through an over-sized cylindrical wave-guide equipped with a copper hornantenna for better radiation coupling1. The Josephson junction was current-biased. Thevoltage response signal of the junction was amplied by a factor of 104, time-averaged bya digital oscilloscope and recorded as a function of the unperturbed Josephson frequency!0, derived from the time-averaged junction voltage U . The operation of the spectrometerwas controlled by a personal computer through an IEEE-488 interface.

1see also [73] for a more complete description of the experimental setup.

0 0.2 0.40

0.1

0.2

0.3

I [m

A]

0 0.2 0.40

1

2

3

R [Ω

]

U [mV] U [mV]0.6

Data

RSJ Model

Data

RSJ Model

Figure 8.5: Left: the current-voltage characteristic of the Josephson junction withoutincident radiation. Measured data points are marked by circles. Dashed-dotted line: thetheoretical I-U curve (8.5) with tted values for Ic and R. Solid curve: convolution of thetheoretical I-U curve with Gaussian-distributed noise. Right: the dierential resistanceRd as determined from dierentiation of the I-U characteristic (solid curve), comparedwith point-to-point dierence quotients (circles).

8.3 Bunch Length Measurements

The TTF thermionic gun was operated with the same parameters as for the interferometermeasurements. The gun produced bunches of 2:3 108 electrons at a repetition rate of216MHz. Te macropulse length was 30s at a repetition rate of 2Hz. The buncher cavityrf phase was adjusted for optimum bunching.

8.3.1 The Intrinsic Parameters of the Junction

Figure 8.5 shows the characteristic I-U curve of the Josephson junction used. The mea-sured data points are presented as circles. A non-linear characteristic caused by the acJosephson eect is observed at low bias currents while an Ohmic characteristic (linearrelation between potential dierence and bias current) is observed for large bias currents.The intrinsic parameters Ic and R of the junction are measured by tting the theoreticalI-U curve (8.5) to the data. The result is

Ic = 0:11 mA R = 1:62 : (8.16)

The t is shown as the dashed-dotted line in the left graph of Fig. 8.5. The agreementwith the data is quite satisfactory except in the vicinity of Ic where a gradual increaseof the voltage is measured instead of the steep rise predicted by theory. The dierenceis due to noise in the junction. A better representation of the experimental I-U curve is

0 0.1 0.2 0.3 0.4 0.50

0.01

0.02

0.03

0.04 Detector Response

U [mV]∆

U [m

V]

0 0.1 0.2 0.3 0.4 0.5−0.5

0

0.5

1

1.5

U [mV]

∆ U

I ω

0 [a.u

.] Characteristic function g

Figure 8.6: The analysis of the spectroscopic measurement to obtain the coherent tran-sition radiation spectrum. Upper graph: The detector response U as a function of U .Lower graph: The characteristic function g(!0), as dened by Eq. (8.15), plotted versusU = h=2e!0. The solid curve shows the extrapolation of g(!0) for small U . The measuredcharacteristic I-U curve of the Josephson junction without radiation is shown in Fig. 8.5.

obtained if the theoretical curve is convoluted with a Gaussian noise distribution.

U(I) =1q

(22)

ZRqI 02 I2c exp

(I

0 I)2

22

!dI 0 : (8.17)

The variance is a t parameter. The resulting solid curve in Fig. 8.5a, correspondingto = 0.02 mA, provides a perfect description of the data.

The dierential resistance of the junction Rd = d U=dI is computed by dierentiation ofthe I-U curve with noise (8.17) or by taking the slopes between adjacent data points. Theright graph of Fig. 8.5 shows that both methods are in good agreement. In the furtheranalysis, the dierential resistance obtained from the I-U t curve is used to avoid thepoint-to-point uctuations in the data.

8.3.2 The Spectroscopic Measurement

The spectroscopic information is contained in the dierence U between the characteristiccurves with and without radiation, which is shown in the upper graph in Fig. 8.6. U istransformed to I by I = U=Rd, where Rd is the dierential resistance shown in theright graph of Fig. 8.5. The detector current response I, the dc bias current I and thetime-averaged voltage across the junction U are multiplied to compute the characteristicfunction g(!0) as dened by Eq. (8.15). g(!0) is shown in the lower plot in Fig. 8.6. The

100 150 200 250

0

0.4

0.8

1.2

frequency [GHz]

Pow

er S

pect

rum

[a.u

.]

σPower Spectrum

f = (102 ± 16) GHz

Figure 8.7: The coherent radiation spectrum as obtained from a discrete Hilbert transformof the characteristic function g. Corrections must be applied for data points marked withcrosses. The decrease below f = 100 GHz is due to the WR-10 waveguide cut-o. Thesedata points are not used in the analysis. The solid curve indicates a Gaussian t to thepower spectrum.

determination of I is problematic for small frequencies (i. e. dc bias currents close toIc). The dierential resistance Rd drops to zero when I approaches Ic. The data at smallvalues of U are quite sensitive to measurement errors and noise in the junction, whichhas a signicant in uence on the dierential resistance in the vicinity of Ic. Thereforethe frequency range 0 < f0 < 50 GHz has been omitted from the present analysis. Since,however, g(!0) is needed at all frequencies to perform the Hilbert transform, a smoothextrapolation function of the form

g(!0) / !0!20 + const

for f0 = !0=(2) < 50 GHz : (8.18)

is used in the range 0 < f0 < 50 GHz, shown as a solid curve in Fig. 8.6. The extrapola-tion function has to vanish at U = 0 to generate an antisymmetric characteristic functiong(!0). The Hilbert transformation of g(!0) then leads to a symmetric power spectrum.

The intensity spectrum was calculated using an algorithm of discrete Hilbert-transform[42]. Figure 8.7 shows the evaluated coherent radiation spectrum. The spectrum is plottedin the frequency range between 60 and 260 GHz. The power spectrum has a maximumat a frequency close to 100 GHz. The decrease towards smaller frequencies is explainedby transmission losses of the radiation for frequencies close to cut-o frequency (60 GHz)of the WR-10 waveguide, diraction of the radiation delivery line, transmission losses inthe quartz window and diraction of the bunch electromagnetic elds at the nite sizetransition radiator. These eects have been discussed in detail in chapter 7.

8.3.3 Data Analysis

The main uncertainty of the experiment originates from the wavelength-dependent ac-ceptance of the detector system. Read-out errors of the voltage response U , which wasdetermined from a digital oscilloscope after averaging over 15 seconds, cause additionaluncertainties. The precision of the read-out of the Josephson voltage is estimated to bebetter than 1 V, the bias current power supply stability to 1 A. The analysis of thisproof-of-principle experiment has been carried out without a detailed consideration ofthe spectral acceptance function. The main interest was to show the applicability of aJosephson junction as detector for coherent transition radiation.A Gaussian-shaped frequency spectrum of the radiation is assumed for the determinationof the rms bunch length. A Gaussian t applied to the data, shown as a solid curve inFig. 8.7, results in

f = (102 16) GHz : (8.19)

f refers to the intensity spectrum which is proportional to the square of the form factor.The rms bunch length is therefore given by

t =1

2p2f

= 1:2 ps : (8.20)

The resulting rms bunch length is smaller than the bunch lengths determined by theMartin-Puplett interferometer. The experiment, however, shows that a Josephson junc-tion can be applied to record the form factor of picosecond electron bunches and that apulse length of correct magnitude is derived. Improvements can be expected from a quasi-optical coupling of the radiation to the junction and an improved electronic read-out. Aspecial cryostat equipped with a quartz window is in preparation. The bandwidth of theread-out electronics will be adapted to a bandwidth sucient to measure selected buncheswithin TTF macropulses [74]. The experience collected during the proof-of-principle ex-periment and the improvements proposed show great promise that Hilbert transform spec-troscopy will become a quick and reliable technique to determine sub-picosecond electronbunch lengths.

Chapter 9

Conclusion and Outlook

Transverse phase space tomography has been successfully applied to determine the beamemittance and beam parameters at the TESLA Test Facility linac. The beam emittanceproduced by the photo injector was optimized by adjusting the gun solenoids to providea compensation of the correlated emittance growth. The solenoid currents leading to anormalized beam emittance of x = (5:5 0:7

0:9) 106m and y = (9:5 1:71:8) 106m are

Iprimary = 165A and Isecondary = 90A. The vertical emittance agrees with an emittancemeasurement performed with a screen of narrow horizontal slits. In the tomographicmethod it turned out to be essential to include linearized space charge forces in the beamtransfer computations and to demand self-consistency between the reconstructed beamparameters and the parameters assumed for the beam transfer computation. The re-sulting beam parameters have been successfully used as input values for beam transfercomputations, yielding a magnetic lattice with 100% beam transmission to the end of thelinac.

Phase space tomography has also been used to determine the transverse phase space dis-tribution of the beam produced by the thermionic injector. At a beam current of Ibeam =3mA, a normalized emittance of x = (3:8 0:4

0:7) 106m and y = (3:3 0:40:7) 106m has

been measured. Due to limited optical resolution it was not possible to obtain reliablenumbers for the transverse emittance and the beam parameters behind the rst rf module.Measurements with an improved optical system are needed.

The longitudinal charge distribution has been investigated using a streak camera and ro-tations of the longitudinal phase space density. The measurements of the streak camera(t = 2ps resolution) reliably determined the rms bunch length and the shape of thelongitudinal charge distribution of the photo injector beam. The bunch length varied be-tween z = (1:95 0:08)mm and z = (4:72 0:09)mm depending on the rf phase of thegun cavity. The longitudinal bunch shape agrees well with a Gaussian distribution. Lon-gitudinal phase space rotations yield an rms injector bunch length of z = (990 90)m.The injector bunch length determined by the two methods is compatible, because thelaser pulse length was reduced by a factor of 2 between the two measurements.

The proper operation of the magnetic chicane compressor has been demonstrated by longi-tudinal phase space rotations and the Martin-Puplett interferometer yielding a minimumrms bunch length of z = (350 130)m and z = (200 70)m respectively. The pulse

shape reconstructed by both techniques is similar: a narrow peak superimposed onto awider basis. The shape of the charge distribution can be explained by an uncorrelatedbunch energy distribution in front of the chicane compressor.

A Martin-Puplett interferometer and a Josephson junction have been applied as far-infrared coherent transition radiation spectrometers. Considerable eort has been madeto evaluate the spectral acceptance of the Martin-Puplett interferometer. The measure-ment of the coherent transition radiation power spectrum is aected by diraction eectsat the nite size transition radiator, the transmission of the quartz window, the transmis-sion and the re ectivity of the wire grids, the radiation diraction in the interferometerand the acceptance of the pyroelectric detectors. In spite of the fairly large corrections,it was possible to model the autocorrelation and the coherent power spectrum of thecompressed pulses and to successfully determine the rms bunch length and the bunchshape. However, a comparison of bunch length measurements of the uncompressed beamof the photo injector showed a discrepancy between the interferometer and streak cam-era results. The Martin-Puplett interferometer is not suited for bunch lengths exceedingt = 2ps because of severe acceptance limitations at frequencies below 100GHz.

A Josephson junction has been applied as a frequency-selective far-infrared spectrometerfor the rst time to determine the longitudinal bunch form factor. The Josephson junc-tion oers a possibility for high-speed spectroscopy in the millimeter- and sub-millimeterwavelength range. A proof-of-principle measurement has been performed at the TTFlinac using the thermionic injector beam. It is planned to improve the Josephson junctiondetector by using a cryostat with direct optical coupling of the radiation. The read-outelectronics will be adjusted to a bandwidth sucient to measure selected bunches withinTTF macropulses. The Josephson spectrometer will be set up in parallel with a Martin-Puplett interferometer to have a direct comparison of both methods. The measurement ofa particular electron bunch within a macropulse will be possible by triggering the appara-tus on the same bunch at every beam crossing. The other option is a measurement of anaverage bunch length within a macropulse by sampling the current-voltage characteristicof the junction within a single macropulse.

Appendix A

Beam Dynamics and Parameters

The beam transfer, the phase space description of charged particle beams and the rstorder beam-matrix formalism for the denition of the beam parameters will be outlinedin this section. A beam line is a set of magnetic elements and accelerating structuresplaced along a reference path. The reference particle follows the reference trajectory, thepath through the center of the magnetic elements, and has the design momentum. Thetransverse coordinates used in this chapter are labeled x for the horizontal plane and yfor the vertical plane. The longitudinal coordinate is labeled l.

A.1 The Transfer Matrix Formalism

A charged particle in a beam line can be represented by a six-dimensional vector ~X

~X = (x; x0; y; y0; l; ) (A.1)

where x and y denote the horizontal and vertical displacement of the particle, x0 and y0

the horizontal and vertical angle of the particle's trajectory with respect to the referencetrajectory. l and denote the longitudinal displacement and the fractional momentumdeviation ( = p=p) with respect to the reference particle. The knowledge of ~X at aninitial position i of the beam line determines the particle coordinates at a nal position f .If the magnetic elds depend linearly on the deviation of the particle from the referencepath, beam transfer can be reduced to the process of matrix multiplication

(Xf )k =6X

k=1

Mkl (Xi)l (A.2)

where M is a 6 6 square matrix, called the beam transfer matrix.

The entire beam line can be modeled by multiplying transfer matrices sequentially fromthe end to the beginning of the line, hence

Mtot =Mn Mn1 ::: M2 M1 (A.3)

and by replacing M in Equation (A.2) by the product matrix Mtot.

A.2 Beam Transfer Matrices

Drift Space A drift space is a region free of electromagnetic elds. The length of thedrift space is denoted by L. The transfer matrix is

MD(L) =

0BBBBB@

1 L 0 0 0 00 1 0 0 0 00 0 1 L 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

1CCCCCA (A.4)

Quadrupole Magnet A quadrupole magnet focuses the beam in one plane, defocusesit in the other. The transfer matrix for a horizontally focusing quadrupole magnet ofstrength k and length L is

MQ(k; L) =

0BBBBBBBBBBB@

cos ()1pksin () 0 0 0 0

pk sin () cos () 0 0 0 0

0 0 cosh ()1pksinh () 0 0

0 0pk sinh () cosh () 0 0

0 0 0 0 1 00 0 0 0 0 1

1CCCCCCCCCCCA

(A.5)

where =pk L. For a vertically focusing quadrupole magnet, the upper left block

matrix Mq;ij, 1 i; j 2, and the middle block matrix Mq;ij, 3 i; j 4, have to beinterchanged.

Dipole Sector Magnet In a dipole sector magnet the central trajectory has a radius ofcurvature R and exits perpendicular to the pole-face boundaries. The de ection angle ofthe beam is . The transfer matrix is

MB(R; )=

0BBBBBBBB@

cos () R sin () 0 0 0 R (1 cos ())

1

Rsin () cos () 0 0 0 sin ()

0 0 1 L 0 00 0 0 1 0 0

sin () R (1 cos ()) 0 0 1 R ( sin ())0 0 0 0 0 1

1CCCCCCCCA

(A.6)

Rectangular Dipole Magnet The rectangular dipole magnet consists of straight pole-face boundaries which are not perpendicular to the central beam trajectory. The magnetcan be described by a combination of a dipole sector magnet and a \magnetic wedge" ofangle i at the entrance (i = 1) and the exit (i = 2) of the magnet [75]. denotes therotation of the pole-face with respect to the orientation of a sector magnet pole face. Thetransfer matrix yields

MR(R; ; ) = MF ( ) MB(R; ) MF ( ) with (A.7)

MF ( i) =

0BBBBBB@

1 0 0 0 0 0tan ( i)=R 1 0 0 0 0

0 0 1 0 0 00 0 tan ( i)=R 1 0 00 0 0 0 1 00 0 0 0 0 1

1CCCCCCA

The sign convention is such that i > 0 denotes horizontal defocusing and vertical focus-ing.

Standing Wave Cavity The beam gains energy (E) and experiences focusing whiletraveling through a standing wave cavity of length l. The transfer matrix is

MS(E; ; l)=

0BBBBBBBBBB@

cos(a)p2 sin(a)

p8 lE sin(a)

E0 0 0 0

3E sin(a)p8 l (E+E)

EK(E+E)

0 0 0 0

0 0 cos(a)p2 sin(a)

p8 lE sin(a)

E0 0

0 0 3E sin(a)p8 l (E+E)

EK(E+E)

0 00 0 0 0 1 00 0 0 0 0 E

(E+E)

1CCCCCCCCCCA

(A.8)

where a = log1 + E=(

p8E)

, K = cos(a) +

q(2) sin(a) [76]. E denotes the injection

beam energy. Equation (A.8) holds for on-crest acceleration of relativistic beams.

Coordinate Rotation A coordinate rotation of the transverse coordinates is describedby a rotation matrix

MR() =

0BBBBBB@

cos () 0 sin () 0 0 00 cos () 0 sin () 0 0

sin () 0 cos () 0 0 00 sin () 0 cos () 0 00 0 0 0 1 00 0 0 0 0 1

1CCCCCCA

(A.9)

The angle denotes the angle of rotation. It is possible to describe the beam transferthrough a vertical bend or a skew quadrupole by multiplying matrix (A.9) in front andbehind of matrices (A.6) and (A.5) with the respective coordinate rotation angle of =2and =4.

A.3 First Order Beam Matrix Formalism

In an accelerator it is often of more concern to study the dynamics of a bunch of particlesrather than the single particle motion. Every particle in a bunch, in case of the TESLATest Facility there are up to 1010 electrons, appears as a point in a six-dimensional phasespace whose coordinates are x, x0, y, y0, l, and . Assume an arbitrary charge distribution(), where = (1; 2; 3; 4; 5; 6) denotes the phase space variables (x; x0; y; y0; l; ).The characteristics of the charge distribution (transverse beam width, transverse angulardivergence, bunch length and momentum deviation) can be described by a statisticalapproach using the rst moments j and the second moments ij, 1 i; j 6, of the

charge distribution

j =Zj ()d (A.10)

ij =Z(i i) (j j) ()d : (A.11)

The j dene the beam centroid coordinates in phase space. The second moments formthe covariance matrix of the bunch charge distribution . The diagonal elements jjrepresent the variances of the phase space coordinates, the o-diagonal elements ij, i 6= j,the covariances of the respective coordinates. The matrix is called the beam matrix.

The diagonal elements of the beam matrix describe the transverse beam width (11 inthe horizontal and 33 in the vertical plane), the transverse angular divergence (22 in thehorizontal and 44 in the vertical plane), the bunch length (55) and the fractional mo-mentum deviation (66). The o-diagonal elements describe covariances of the transverseplanes (12, 34), coupling between the transverse planes (13, 14, 23 and 24) and thecoupling between the transverse and longitudinal coordinates (15, 16, 25, 26, 35, 36,45 and 46). Since the beam matrix elements describe the characteristics of the bunchcharge distribution in six-dimensional phase space, they are called the beam parameters.

As an example, consider a two-dimensional Gaussian charge distribution

2(x; x0) = exp

x2

211+

121122

xx0 x02

222

!(A.12)

which can be represented by an ellipse in the (x; x0) phase space. A contour line of 2 isdened by 2(x; x

0) = exp (C) with

22x2 212xx

0 + 11x02 = C = constant (A.13)

C denes the phase space area enclosed by the contour line. The ellipse given by Equation(A.13) can be rewritten in terms of a two-dimensional beam matrix 2 yielding

~XT12~X = C (A.14)

where ~XT is the transpose of the coordinate vector ~X. 2 denotes the real, positivedenite and symmetric beam matrix. In particular, it is

2 =11 1212 22

, 12 =

1

det

22 1212 11

, ~X =

xx0

(A.15)

so that Equation (A.13) becomes

22x2 212xx

0 + 11x02 = det 2 : (A.16)

The constant C of Equation (A.13) is evaluated to be the determinant of the 2 matrix.The ellipse described by Equation (A.16) is shown in Figure A.1 along with the physicalinterpretation of the beam matrix elements.

p11 denes the horizontal spatial extent,p

22 the angular extent of phase space occupied by the bunch. The o-axis element 12 isa measure of the covariance of the angular and spatial coordinate, i. e. of the orientation

σ22

A = detσ

σσ

11221

r12

σ22σ11

r12

σ11

x

x

’

Figure A.1: The horizontal phase space ellipse and the physical interpretation of the beammatrix elements.

(tilt) of the ellipse. The area enclosed by the ellipse is A = det.

A six-dimensional ellipsoid can be expressed in terms of Equation (A.14) as well. Thephysical interpretation of the elements of the six-dimensional beam matrix is straightforward: the square roots of the diagonal elements describe the extent of the phase spaceellipsoid on the respective coordinate axis. This is in particular

p11: the width of the beam ellipsoid in the horizontal plane.

p22: the horizontal angular divergence of the beam ellipsoid.

p33: the width of the beam ellipsoid in the vertical plane.

p44: the vertical angular divergence of the beam ellipsoid.

p55: the longitudinal extent of the beam ellipsoid

p66: the fractional momentum deviation of the beam ellipsoid.

The o-diagonal elements describe the covariance of the phase space ellipsoid with respectto a pair of its coordinate axis. A convenient quantity to measure the covariance of thephase space ellipsoid is the correlation dened as

rij =ijpiijj

; for 1 i; j 6 : (A.17)

The correlation ranges between 1 and 1. rij = 0 denotes a circular phase space distri-bution, rij = 1 a linear distribution with slope 1. A correlation between 0 and 1 denotesa tilted phase space distribution as shown in Figure A.1.

A.3.1 Beam Transfer

The beam matrix will undergo a transformation when the bunch of particles passes abeam line. Inserting the identity I =MM1 =M1M into Equation (A.14) we obtain

~XTi

MTMT1 1i

M1M~Xi = 1 (A.18)

M ~Xi

T MiM

T1

M ~Xi

= 1 (A.19)

~XTf

1f~Xf = 1 (A.20)

where M denotes the transfer matrix of the beam line, the indices i and f denote theinitial and nal position of the beam respectively. The beam matrix can be transformedthrough a beam line using the single particle transfer matrix M . The beam parameterscan therefore be obtained at any position of the beam line if they are measured once. Thetransformation law is

f =MiMT : (A.21)

It should be noted that for magnetostatic beam lines the determinant of the transfermatrix is unity, hence the determinants of f and i are identical since

det f = detMiM

T= detM det i detMT = det i (A.22)

A direct consequence of Equation (A.22) is the conservation of the volume enclosed bythe phase space ellipsoid, known as Liouville's theorem.

The transformation laws of the beam matrix elements in a beam line have been derivedfor the special case of Gaussian charge distributions. In contrast to the Gaussian case,an arbitrary charge distribution is not completely described by the covariance matrix.Nevertheless, the transfer of the variances and covariances of an arbitrary distributionthrough a beam line can still be described by the beam matrix formalism.

A.3.2 Beam Emittance

The beam emittance is dened as the phase space volume occupied by a certain fraction,typically 1, of the beam. The emittance is a measure of the beam quality. A lowemittance beam has a small angular divergence after it is tightly focused or when it istransfered with large cross-section over large distances without external focusing. In thebeam matrix formalism, the emittance is equal to the determinant of the beam matrix(see Figure A.1), hence

det = 1122 212 =< x2 >< x02 > < xx0 >2= 2x : (A.23)

Here the two-dimensional horizontal projection of the charge distribution is used. Anidentical relation holds for the vertical transverse (y; y0) and the longitudinal phase spacedistribution (l; ).

Appendix B

Derivation of the Ginzburg-Frank

formula

This chapter presents the derivation of the Ginzburg-Frank formula describing the spec-tral and angular distribution of transition radiation. The radiation originates from thetransition of charged particles in uniform linear motion of velocity ~v through a planeboundary of two media of dierent permittivities 1 and 2. The particles are assumedto travel along the z-axis directed perpendicular to the boundary. The conguration isshown in Fig. B.1.

(1) Electromagnetic elds carried by a particle bunch

Maxwell's equations can be used to derive the wave equation for the scalar and vectorpotential (~r; t) and ~A(~r; t) [77]

c2@2(~r; t)

@t2r2(~r; t) =

(~r; t)

0(B.1)

c2@2 ~A(~r; t)

@t2r2 ~A(~r; t) = 0~j(~r; t) (B.2)

where and ~A are related to the electric and magnetic elds ~E and ~B by

~E = ~r @ ~A

@t; ~B = ~r ~A : (B.3)

(~r; t) denotes the charge density of the bunch and ~j(~r; t) the current density. For parti-cles traveling along a straight line, and ~j are related by ~j = ~v, hence either of Eq. (B.1)and (B.2) gives a complete description of the problem.

The dierential equations (B.1) and (B.2) can be solved in frequency domain, wherethey transform into algebraic equations. The Fourier transformation takes the transversecoordinate ~r? into a transverse momentum ~q and time t into frequency !. The longitudinalcoordinate is not transformed because of the boundary conditions which have to be fullledat z = 0. For the scalar potential we obtain

(~r?; z; t) =1

(2)3

Zd2q d!q;!(z) exp (i~q ~r? i!t) (B.4)

v

ε2ε1

z

medium 2medium 1

z = 0

= 1 = ε

Figure B.1: Conguration used for the derivation of the Ginzburg-Frank Equations.

~A(~r?; z; t) =1

(2)3

Zd2q d! ~Aq;!(z) exp (i~q ~r? i!t) : (B.5)

The algebraic equation for q;! can be derived by inserting Eq. (B.4) into Eq. (B.1)yielding

q;!(z) =q;!(z)

0

"q2 +

!2

v2 !2

c2

#1: (B.6)

Dierential Eq. (B.2) is solved by a similar expression given by

~Aq;! =

c2~vq;! : (B.7)

Using Eq. (B.3), the electric eld yields

~EQq;! = i

!~v

c2 1

v2

q2

q;! : (B.8)

The superscript Q in Eq. (B.8) indicates that the eld is produced by a charged particlebunch. The Fourier transform of a charge distribution q;! has to be derived and insertedinto Eq. (B.8). The Fourier transform of a single particle and, as an example of a particlebunch, a Gaussian charge distribution of transverse variance 2r and longitudinal variance2z will be considered.

Single particle The charge distribution of a single particle in uniform motion withvelocity v is

e(~r; t) = e(~r?)(z vt) (B.9)

yielding the Fourier transformation

eq;!(z) = eZd2r? dt (r?)(zvt) exp (i~q ~r?+i!t) = e

vexp

i!z

v

(B.10)

Radiation Fields

The electric radiation eld has to satisfy the homogeneous wave equation

c2@2 ~E(~r; t)

@t2 ~r2 ~E(~r; t) = 0 : (B.11)

which can be transformed into frequency domain using

~E(~r; t) =1

(2)3

Zd2q dt ~Eq;!(z) exp (i~q ~r? i!t) : (B.12)

The boundary at z = 0 is assumed to be of innite extent, hence the radiation eld can becharacterized by a plane wave propagating in z direction. The ansatz for the z componentis

~n ~ERq;!(z) = i

a+a

exp (ikzz) (B.13)

where a+ and a are yet undetermined coecients and ~n is a unity vector pointing alongthe z axis. Two coecients and two signs are introduced in Eq. (B.13) because theradiation wave can propagate either forward into medium 2 (z > 0, a+, and upper sign)or can be re ected backward into medium 1 (z < 0, a, and lower sign). Equations (B.11)and (B.13) lead to a relation among radiation frequency !, transverse ~q and longitudinalkz momentum

!2

c2 q2 k2z

!~Eq;!(z) = 0 ) kz =

s!2

c2 q2 : (B.14)

The transverse electric eld components can be derived using ~r ~D = 0 yielding

~q

q ~ER

q;! = kzq~n ~ER

q;! : (B.15)

The complete expression for the electric radiation eld is

~ERq;! = i

a+a

24~n ~q

q2

s!2

c2 q2

35 exp

0@i

s!2

c2 q2 z

1A : (B.16)

Boundary Conditions

The coecients a+ for z > 0 (medium 2) and a for z < 0 (medium 1) have to bedetermined from the boundary conditions the sum of the charge and the radiation eldhas to fulll at the interface. The normal component of the ~D-eld and the tangentialcomponent of the ~E-eld have to be continuous at z = 0.

Normal component:

1~n ~EQq;! + ~ER

q;!

medium 1

z=0= 2~n

~EQq;! + ~ER

q;!

medium 2

z=0(B.17)

leads to

1a + 1!v1c2 1

v2

q;!10

"q2 +

!2

v2 1!

2

c2

#1=

2a+ + 2!v2c2 1

v2

q;!20

"q2 +

!2

v2 2!

2

c2

#1:

(B.18)

Tangential component:

~q ~EQq;! + ~ER

q;!

medium 1

z=0= ~q

~EQq;! + ~ER

q;!

medium 2

z=0(B.19)

leads tos1!2

c2 q2aq;!

10

q2q2 + !2

v2 1!2

c2

= s2!2

c2 q2a+q;!

20

q2q2 + !2

v2 2!2

c2

: (B.20)Equation (B.18) and (B.20) can be solved for a+ and a. For the special case of 1 = 1(vacuum) and 2 = (medium), we obtain

a=e

0!

2 ( 1)1 2 +

p 2

(1 2 + 22)

1 +

p 2

p 2 +

p1 2

(B.21)

a+=e

0!

2 ( 1)1 2

p1 2

(1 2 + 22)

1

p1 2

p 2 +

p1 2

(B.22)

where = qc=! has been used [23]. The combination of Eq. (B.21) and (B.22) withEq. (B.16) describes the electric transition radiation eld between vacuum and a mediumof permittivity with the restriction of innite boundaries.

(4) Radiated energy into the backward hemisphere

The energy U contained in a propagating electromagnetic wave is described by

U = 0

Zdx dy

Z 1

1dz

~ER(~r; t)2 : (B.23)

The knowledge of the electric eld is sucient for the evaluation of the radiated energy,since the electric and magnetic elds, in time average, store the same energy. In thefollowing, the radiation into the backward hemisphere, which is used for beam diagnosticsat the TESLA Test Facility, is evaluated.

Let t = 0 be the time of transition from medium 1 to medium 2. To use the plane waveradiation eld, we have to choose the time t large enough that the radiation eld and thecharge eld are well separated. The center of the coordinate system, i. e. z = 0, is shiftedto the center of the radiation pulse. This coordinate transformation causes an unknownphase factor in the coecient a, but justies the use of symmetric boundaries on theintegrals (B.23) [78].

v

z

medium 2medium 1

z = 0

Figure B.2: Translation of the origin of the coordinate system into the middle of theradiation pulse.

The Fourier decomposition of ~ER(~r; t), see Eq. (B.12), is introduced into Eq. (B.23)leading to a frequency domain description of the radiated energy

U = 0

Zdx dy

Z 1

1dzZ d2q d2q0 d! d!0

(2)6

~ERq;!(z) ~E

R

q0;!0(z) expi~r?

~q ~q

0 t (! !0)

:

(B.24)

1. The integration over dx dy yieldsZdx dy exp

i~r?

~q ~q

0

= (2)2 2~q ~q

0

(B.25)

hence

U = 0

Z 1

1dzZ d2q d! d!0

(2)4~ERq;!(z) ~E

R

q;!0(z) exp (i (! !0) t) : (B.26)

2. The radiation eld Eq. (B.16) is introduced into Eq. (B.26) and then integrated overz using

Zdz e

izq

!2

c2q2

q!02

c2q2= 2

0@s!2

c2 q2

s!02

c2 q2

1A (B.27)

We obtain

U = 20

Z d2q d! d!0

(2)4(a)

a

0@s!2

c2 q2

s!02

c2 q2

1A

24~n ~q

q2

s!2

c2 q2

35 24~n ~q

q2

s!02

c2 q2

35 :(B.28)

v

q

k

θ

Figure B.3: Transverse momentum ~q, the total momentum ~k and the polar angle ofemission of the transition radiation.

Note that a and a depend on (q; !) and (q; !0) respectively. The -expression inEq. (B.28) implies ! = !0, hence a a = jaj2. The additional phase factor intro-duced by the coordinate translation cancels out. Equation (B.28) can be simpliedusing the equality of ! and !0 yielding

U = 0

Z d2 d! d!0

(2)4jaj2 !2

c2q2

0@s!2

c2 q2

s!02

c2 q2

1A : (B.29)

3. The integration over !0 can now be carried out using

0@s!2

c2 q2

s!02

c2 q2

1A =

c2

!

s!2

c2 q2 [(! !0) + (! + !0)] (B.30)

hence

U = 20

Z d2q d!

(2)3jaj2 !

2

q2c

s1 q2c2

!2: (B.31)

4. The angular distribution of transition radiation can be evaluated by expressing thetransverse momentum ~q by the total momentum ~k of the radiation and its emissionangle , hence q = k cos . The vectors are shown in Fig. B.3. There is no needto introduce an azimuthal emission angle because of symmetry. The dierential d2qbecomes 2qdq = 2!2=c2 sin cos d and we obtain

U = 2Z 1

0d!Z

0d sin U1 (!; ) (B.32)

with

U1 (!; ) =20

(2)3jaj2 !

2

c

cos2

sin2 (B.33)

being the dierentiated spectral energy density.

Introducing a from Eq. (B.21), we obtain the Ginzburg-Frank [23] formula describingthe opening angle and the spectral power density distribution of transition radiation:

U1 =e2

430c

2 sin2 cos2

(1 2 cos2 )2( 1)2

1 2 +

p sin2

21 +

p sin2

2 cos

p sin2

2 (B.34)

For the special case of !1, that is in case of a metallic boundary for frequencies wellbelow the plasma frequency of the metal, we obtain the simplied expression

U1 =e2

430c

2 sin2

(1 2 cos2 )2: (B.35)

Expression (B.35) denotes the Ginzburg-Frank formula as commonly applied for the com-putation of transition radiation in accelerators.

Appendix C

Electromagnetic Fields of

Relativistic Electron Bunches

The transverse source size of transition radiation is dened by the projection of the elec-tromagnetic elds, carried by a charge distribution traveling in uniform motion througha beam pipe, onto the transition boundary. The charge distribution is assumed to travelalong the axis of a cylindrical vacuum chamber of either perfect or high conductivity.Maxwell's Equations are solved with proper boundary conditions to obtain the electro-magnetic elds inside the pipe.

(1) Fields of the Charge Distribution

The electric and magnetic elds of a charge distribution traveling on the axis of a cylindri-cal beam pipe are described by the vacuum solution of the Maxwell Equations. The chargedistribution is assumed to travel in the center of the beam pipe and to be azimuthallysymmetric. The Maxwell Equations in vacuum are

r ~E =

0, r ~B = 0 (C.1)

r ~E = d~B

dt, r ~B = 0~j +

1

c2d ~E

dt: (C.2)

From symmetry only the radial and longitudinal components of the electric eld (Er andEz) and the azimuthal component of magnetic eld (B) have to be considered. denotesthe charge density. The current j has only an axial component, which originates from themoving charge distribution. The Maxwell Equations for the above eld components readin cylindrical coordinates

r ~E =1

r@r (rEr) + @zEz =

0(C.3)

r ~B =1

r@B = 0 (C.4)

r ~E = @t ~B8<:

1r@Ez = 0

@zEr @rEz = @tB1r@Er = 0

(C.5)

r ~B = 0~j +1

c2@t ~E

(@zB = 1

c2@tEr

1r@r(rB) = 0jz +

1c2@tEz

(C.6)

Here @r, @ and @z denote @=@r, @=@ and @=@z respectively. Equations (C.4), (C.5) and(C.6) show that B, Er and Ez are independent of the azimuthal coordinate (azimuthalsymmetry). The elds depend on the radial coordinate r and the longitudinal coordinate = z ct taking into account that the elds travel together with the charge at avelocity of c. Maxwell's equations are solved in frequency domain where the longitudinalcoordinate is transformed into the wave number k by a Fourier transformation. Thetransverse coordinates are not transformed to retain the transverse component of theelectromagnetic elds at dierent wave numbers k. The eld components are

Er;z(; r) =Z 1

1~Er;z(k; r) exp (ik)dk (C.7)

B(; r) =Z 1

1~B(k; r) exp (ik)dk (C.8)

(; r) =Z 1

1~(k; r) exp (ik)dk (C.9)

The Fourier components obey the equations

1

r

@(r ~Er)

@r+ ik ~Ez =

~

0(C.10)

1

r

@ ~B

@= 0 (C.11)

ik ~Er @ ~Ez

@r= ikc ~B (C.12)

ik ~B = ikc~Er (C.13)

1

r

@r ~B

@r

= ikc

~Ez + 0~jz : (C.14)

Inserting Eq. (C.13) into Eq. (C.12) yields a relation between ~Ez and ~Er

@ ~Ez

@r=ik

2~Er : (C.15)

Equation (C.15) inserted into (C.10) yields a well known second order dierential equationfor ~Ez

@2 ~Ez

@r2+1

r

@ ~Ez

@r k2

2~Ez =

ik

2~

0: (C.16)

The two independent solutions of the homogeneous equation are the modied Besselfunctions I0(kr= ) and K0(kr= ). The inhomogeneous dierential equation will be solvedfor a ring charge distribution of radius a with azimuthal symmetry

~(k; r; a) =(r a)

2aq~(k) (C.17)

where ~(k) denotes the Fourier transform of the longitudinal charge distribution and qthe total charge. ~Ez is written as a superposition of I0 and K0 in the following form

~Ez(k; r) =

8<: c1(k)I0

kr

, r < a

c2(k)K0

kr

, r > a

9=;+ c3(k)I0

kr

!: (C.18)

The terms c1(k)I0 (kr= ) and c2(k)K0 (kr= ) describe the eld ~Ez originating from thering charge distribution ~(k; r). Fields inside the ring (r < a) can only be given byI0 (kr= ), since K0 (kr= ) diverges for r ! 0. Fields outside (r > a) are given byK0 (kr= ), since K0 is decreasing with radius. The term c3(k)I0 (kr= ) describes theeect of the cylindrical boundary. I0 has been chosen because it is not divergent at r! 0.The coecients c1(k), c2(k) and c3(k) have to be determined by boundary conditions atr = a; b.

(2) Vacuum Chamber of Innite Conductivity

The longitudinal component ~Ez(k; r) has to fulll the following boundary conditions

The continuity of ~Ez at r = a.

A vanishing longitudinal eld component at the beam pipe, i. e. ~Ez(k; b) = 0.

Eq. (C.16) has to hold. This introduces the charge term into the ansatz (C.18).

These boundary conditions at r = a; b yield the following equations

c1(k)I0

ka

!= c2(k)K0

ka

!(C.19)

c2(k)K0

kb

!= c3(k)I0

kb

!(C.20)

Equation (C.16) is integrated over an innitesimal radial interval from a r a + to take the charge term into account. The left hand side yields

Z a+

adr r

(1

r@rr@r ~Ez k2

2~Ez

)= r@r ~Ez

a+a + 2

k2

2a ~Ez(k; a) (C.21)

= ak

(c2(k)K1

ka

! c1(k)I1

ka

!)(C.22)

where ! 0 has been used [27]. The rst term of integral (C.21) describes the disconti-nuity of the derivative of the eld at r = a. The second term is of the order of , becauseof the continuity of ~Ez at r = a. The right hand side of Eq. (C.16) integrated over thesame radial interval yields

ik

20

Z a+

adr r

(r a)

2aq~(k) =

ik

2q

20~(k) : (C.23)

where the ring charge distribution Eq. (C.17) has been used. Eq. (C.22) and (C.23) yieldthe third boundary condition

c2(k)K1

ka

! c1(k)I1

ka

!=

iq

20 a~(k) : (C.24)

For the coecients c1(k), c2(k) and c3(k) we obtain

c1(k) = ik

20 2q~(k)K0

ka

!(C.25)

c2(k) = ik

20 2q~(k)I0

ka

!(C.26)

c3(k) =ik

20 2q~(k)I0

ka

!K0

kb

I0kb

(C.27)

where the Wronskian I0(ka= )K1(ka= ) + I1(ka= )K0(ka= ) = =ka has been used [79].These coecients are introduced into ansatz (C.18) yielding an expression for the longi-tudinal component of the electric eld ~Ez(k; r)

~Ez(k; r)=ik

20 2q~(k)

248<:K0

ka

I0kr

, r < a

I0ka

K0

kr

, r > a

9=;+I0

ka

!K0

kb

I0kb

I0 kr

!35 : (C.28)

The eld components ~Er(k; r) and ~B(k; r) can be derived from Eq. (C.15) and (C.13),respectively. The result obtained is

~Er(k; r)=k

20 q~(k)

248<:K0

ka

I1kr

, r < a

I0ka

K1

kr

, r > a

9=;+I0

ka

!K0

kb

I0kb

I1 kr

!35 (C.29)

and ~B = =c ~Er.

(3) Vacuum Chamber of Finite Conductivity

For the vacuum solution of Eq. (C.16) we make the ansatz (C.18)

~Ez(k; r) =

8<: c1(k)I0

kr

, r < a

c2(k)K0

kr

, r > a

9=;+ c3(k)I0

kr

!(C.30)

~Er(k; r) = i 248<: c1(k)I1

kr

, r < a

c2(k)K1

kr

, r > a

9=;+ c3(k)I1

kr

!35 (C.31)

~B(k; r) = i c

248<: c1(k)I1

kr

, r < a

c2(k)K1

kr

, r > a

9=;+ c3(k)I1

kr

!35 (C.32)

where ~Er and ~B are derived from Eq. (C.15) and (C.13) respectively.

The Maxwell Equations have to be solved inside the metal of the beam pipe. The averagecharge density in the metal vanishes in the frequency range considered (frequenciessmaller than the electron plasma frequency in the metal). The time varying magneticeld induces a radial current in the beam pipe obeying Ohm's law ~jr = ~Er, where denotes the conductivity of the metal. The wall currents are described by an additionalterm in Eq. (C.6)

r ~Br= @zB = 0jr +

1

c2@tEr : (C.33)

Hence the Fourier-transformed Maxwell Equations become

1

r

@(r ~Er)

@r+ ik ~Ez = 0 (C.34)

1

r

@ ~B

@= 0 (C.35)

@~Ez

@r+ ik ~Er = ikc ~B (C.36)

1

r

@(r ~B)

@r= ik

c~Ez and (C.37)

~B = ~Er

c 0

ik

!: (C.38)

Inserting Eq. (C.38) into Eq. (C.36) we obtain

@ ~Ez

@r=

ik

2+ 0c

!~Er : (C.39)

A second order dierential equation is obtained for ~Ez if Eq. (C.39) is substituted intoEq. (C.34).

1

r

@

@r

r@ ~Ez

@r

!=

ik

2+ 0c

!1

r

@r ~Er

@r

(C.40)

=

k2

2 ik0c

!~Ez (C.41)

The penetration depth of the high frequency eld inside the metal is small comparedto the thickness of the beam pipe. The radius r is therefore r b = constant. HenceEq. (C.41) reduces to an Helmholtz wave equation

@2r + 2 ~Ez = 0 (C.42)

where 2 = (ik0c k2= 2). Eq. (C.42) is solved by

~Ez(k; r) = d exp (i(r b)) for r b : (C.43)

where the coecient d depends on the wave number k and the radius of the charge dis-tribution a. In the following the positive sign in the exponential is used to describe anexponential drop of the eld for r > b.

Expression (C.43) can be used to evaluate the azimuthal magnetic eld component insidethe metal. The dierentiation of Eq. (C.38) leads to

1

r

@r ~B

@r

=1

r

@r ~Er

@r

c 0

ik

!: (C.44)

and after the substitution of Eq. (C.34) we obtain (while keeping r b = constant insidethe metal)

@ ~B

@r=

0 ik

c

!d exp (i(r b)) : (C.45)

The integration leads to

~B(k; r) = d k

c

1 +

2

k2

!exp (i(r b)) for r b : (C.46)

The radial electric eld ~Er is related to the azimuthal magnetic eld by Eq. (C.38). Thisyields

~Er(k; r) = d kexp (i(r b)) for r b : (C.47)

The longitudinal electric elds ~Ez has to be continuous at r = a, ~Ez and ~B have to becontinuous at r = b and Eq. (C.16) has to hold to fulll the boundary conditions. Thefollowing equations are obtained

c1(k)I0

ka

!= c2(k)K0

ka

!(C.48)

c2(k)K1

ka

! c1(k)I1

ka

!=i

q

20a~(k) (C.49)

c2(k)K0

kb

!+ c3(k)I0

kb

!= d(k) (C.50)

i c

c2(k)K1

kb

!+ c3(k)I1

kb

!!=d(k) k

c

1 +

2

k2

!: (C.51)

For the coecients c1(k), c2(k), c3(k) and d(k) we obtain

c1(k) = ik

20 2q~(k)K0

ka

!(C.52)

c2(k) = ik

20 2q~(k)I0

ka

!(C.53)

c3(k) =

d(k) +

ik

20 2q~(k)I0

ka

!K0

kb

!!=I0

kb

!(C.54)

d(k) =

20bq~(k)

I0ka

I0kb

1i

I1(kb )I0(kb )

1

k+

k

: (C.55)

The elds ~Ez, ~Er and ~B are obtained both in the vacuum and the metal by introducingthese coecients into Eq. (C.30)(C.32). We obtain for the vacuum solution

~Ez(k; r) =ik

20 2q~(k)

8<: K0

ka

I0kr

, r < a

I0ka

K0

kr

, r > a

9=;

+

"d(k) +

ik

2q~(k)I0

ka

!K0

kb

!#I0kr

I0kb

(C.56)

~Er(k; r) =k

20 q~(k)

8<: K0

ka

I1kr

, r < a

I0ka

K1

kr

, r > a

9=;

i "d(k) +

ik

2q~(k)I0

ka

!K0

kb

!#I1kr

I0kb

(C.57)

and ~B = =c ~Er.

(4) The Ultra-relativistic Limit

For high particle energies, the argument of the modied Bessel-functions will tends tozero and an approximate expression for the electromagnetic elds can be derived. Forx 1

I0(x) 1 (C.58)

I1(x) x

2(C.59)

K0(x) lnx

2

+ 0:5772

(C.60)

K1(x) 1

x(C.61)

Vacuum Chamber of Innite Conductivity: Inserting the approximate expressionfor the small argument modied Bessel functions into Eq. (C.27)(C.29) we obtain

~Ez(k; r) = lim !1

ik20 2

q~(k)

(log

kr

2

!+ log

kb

2

!)= 0 (C.62)

~Er(k; r) = lim !1

k

20 q~(k)

(

kr+ log

kb

2

!)=q~(k)

20

1

r(C.63)

~B(k; r) =

c~Er(k; r) : (C.64)

The elds of Eq. (C.62)(C.64) are now identical to the elds evaluated by a steady-statecoasting beam approach [80].

Vacuum Chamber of Finite Conductivity: Inserting the approximations (C.58) -(C.61) into Eq. (C.54) and (C.55) yields the following result for !1

d(k) =q~(k)

20b

1ikb2k+

k

(C.65)

c(k) = d(k) : (C.66)

In the ultra-relativistic limit the coecients c(k; rs) and d(k; rs) become equal. Theelectromagnetic elds simplify to

~Ez = d(k) (C.67)

~Er =q~(k)

20c

1

r ik

2d(k)r (C.68)

~B =q~(k)

20c

1

r ik

2cd(k)r : (C.69)

Notice that the longitudinal eld components do not vanish even for innitely high beamenergies. These elds, called longitudinal wake elds, originating from resistive wall ofthe vacuum chamber in uence the energy spread of short electron bunches. These wakeelds become stronger for a smaller radius of the vacuum chamber b, for shorter bunchesz and for a smaller conductivity of the beam pipe. The bunch length dependenceof Eq. (C.67)(C.69) is hidden in the Fourier transformation of the longitudinal chargedistribution ~(k) (see Eq. (D.24) and (D.25). Longitudinal wake elds are important forthe understanding of the longitudinal charge prole determination as presented in chapter6.

Source dimension of Coherent Transition Radiation from a Uniform ChargeDistribution: The integration of Eq. (C.28)(C.29) weighted with a transverse uniformcharge distribution yields the source dimensions of coherent transition radiation on theradiation boundary. The following calculation considers only the radial component of theelectric eld, because it dominates over the longitudinal component and is related to theazimuthal magnetic eld by =c. The coordinates are chosen such that a denotes theradius of the ring charge distribution and r the radius at which the eld is evaluated.A transverse uniform charge distribution (a) can be composed of a set of ring chargedistributions. The integral over a,

~Er =k

20 q~(k)

(Z r

0d da aK0

ka

!(a)I1

kr

!+

Z 1

rd da aI0

ka

!(a)K1

kr

!+ (C.70)

Z 1

0d da aI0

ka

!(a)

K0

kb

I0kb

I1 kr

!9=; :

is solved for a uniform transverse charge distribution of radius R

(a) =1

R2

1 , a < R0 , a > R

: (C.71)

We obtain

~Er =k

20

q~(k)

2r

(Z r

0d da aK0

ka

!I1

kr

!+

Z R

rd da aI0

ka

!K1

kr

!+ (C.72)

Z R

0d da aI0

ka

!K0

kb

I0kb

I1 kr

!9=; :

The integrals can be evaluated analytically using [79]Z z

0dt tI1(t)dt = zI(z) (C.73)Z z

0dt tK1(t)dt = zK(z) + 21(z) (C.74)

where denotes an integer. We obtain

Z r

0da aK0

ka

!=

r

kK1

kr

! 2

k2(C.75)

Z r

0da aI0

ka

!=

r

kI1

kr

!(C.76)

Z R

0da aI0

ka

!=

R

kI1

kR

!: (C.77)

The radial electric elds of a uniform bunch charge distribution moving with velocity vin straight line uniform motion is

~Er(k; r)=q~(k)

0R2

248<:

kI1kr

, r < R

RI1kR

K1

kr

, r > R

9=;+RI1

kR

!K0

kb

I0kb

I1 kr

!35 : (C.78)

The electric eld of a uniform charge distribution projected onto the transition radiationscreen is rising proportional to I1(kr= ) towards R. At radii larger than R, the eld dropsproportional to K1(kr= ). The vacuum chamber adds to the nal electric eld by a termproportional to I1(kr= ).

Longitudinal Space Charge Forces: The longitudinal space charge forces can beevaluated by Eq. C.28. To derive Eq. (2.3) in chapter 2 we consider a point charge, hencers = 0 and () = ( 0), yielding

~ =1

2exp(ik0) : (C.79)

Equation (C.28) then reduces to

~Ez =ik

420 2qK0

kr

!exp(ik0) : (C.80)

In time-domain we obtain

Ez =q

420 2

Z 1

1dk ikK0

kr

!exp(ik ( 0)) (C.81)

=q

420 2d

d

"Z 1

1dkK0

kr

!exp(ik ( 0))

#(C.82)

=q

420 2d

d

1r

2+ ( 0)

2 1

2

(C.83)

=q

420 2( 0)

r

2+ ( 0)

2 3

2

(C.84)

Equation (2.3) in chapter 2 is obtained by setting r = 0

Ez =q

420 21

( 0)2 (C.85)

Appendix D

Space Charge Forces

A linear expression for the space charge forces experienced by a test particle in a uniformand a Gaussian charge distribution can be evaluated using Eq. (C.29) describing the radialelectric eld of a ring charge distribution of radius a traveling in uniform motion withvelocity v along a straight line. Neglecting the wall eects of the beam pipe we obtain

~Er(k; r) =k

20 q~(k)

8<: K0

ka

I1kr

, r < a

I0ka

K1

kr

, r > a

9=; : (D.1)

where a denotes the radius of the ring charge distribution and r the radius where the eldis evaluated. To describe the eld travelling together with a transverse charge distribution

lambda, Eq. (D.1) has to be wighted with . The integral yields

~Er =k

20 q~(k)

(Z r

0d da aK0

ka

!(a)I1

kr

!+

Z 1

rd da aI0

ka

!(a)K1

kr

!):

(D.2)

Uniform Charge Distribution

For a uniform charge distribution

U(a) =1

R2

1 , a < R0 , a > R

(D.3)

Eq. (D.2) yields

~EUr =

k

20

q~(k)

R2

(Z r

0d da aK0

ka

!I1

kr

!+Z R

rd da aI0

ka

!K1

kr

!)(D.4)

=q~(k)

0R2

8<:

kI1kr

, r < R

RI1kR

K1

kr

, r > R

9=; : (D.5)

A small argument approximation of Eq. (D.5) is obtained by inserting the small argumentapproximation of the modied Bessel functions Eq. (C.58)(C.60)

~EUr (k; r) =

q~(k)

0R2

(r2

, r < RR2

2r, r > R

): (D.6)

The eld is increasing linearly until the edge of the uniform charge distribution. For largerradii it decreases proportional to 1=r.

Gaussian Charge Distribution

To describe a Gaussian charge distribution, Eq. (D.1) has to be weighted with a transverseGaussian distribution

G(a) =1

22rexp

a2

22r

!: (D.7)

Integral (D.2) yields

~EGr =

k

20

q~(k)

22r

(Z r

0d da aK0

ka

!exp

a2

22r

!I1

kr

!+

Z 1

rd da aI0

ka

!exp

a2

22r

!K1

kr

!):

(D.8)

The second integral is split into two components. Applying partial integration we obtain

Z 1

0da a I0

ka

!exp

a2

22r

!=2r+

p2k3r

exp

k2R2

4 2

!I1=2

k22r4 2

!(D.9)

and

Z r

0da a I0

ka

!exp

a2

22r

!=2r

I0

kr

!exp

r2

22r

! 1

!

k2r

Z r

0da I1

ka

!exp

a2

22r

!:(D.10)

where Z 1

0dt exp

a2t2

I(bt) dt =

p

2aexp

b2

8a2

!I=2

b2

8a2

!(D.11)

has been used [79]. The expressions (D.9) and (D.10) have to be evaluated for smalltransverse dimensions r, i. e. close to the center of the Gaussian charge distribution wherewe expect a linear characteristic of the space charge force. Using the small argumentapproximations of the modied Bessel functions and the polynomial expansion of theexponential function we obtain a linear contribution from Eq. (D.10) yielding

~EIr = k

20

q~(k)

22r22r

(I0

kr

!exp

r2

22r

! 1

)K1

kr

!(D.12)

= 1

20q~(k)

(1 r2

22r 1

)1

r(D.13)

=q~(k)

402rr : (D.14)

The two missing integrals bound by 0 and r are evaluated for small radii. We obtain

k2r

Z r

0da I1

ka

!exp

a2

22r

! k2r

Z r

0da

ka

2

1 a2

2 2r

!and (D.15)

Z r

0da aK0

ka

!exp

a2

22r

!

Z r

0da

ln

ka

!0:58

! a a3

22r

!(D.16)

Equation (D.15) yields a term linear in r

~EIIr = k

20

q~(k)

22r

2k22r 2

r2

2K1

kr

!(D.17)

k2

40 2q~(k)r : (D.18)

The evaluation of integral (D.16) yields lowest order terms of O(r). The lowest order ofthe radial electric eld is then of the order of O(r2) because I1(kr= ) kr=2 = O(r).

The electric and magnetic elds close to the center of a transverse Gaussian charge dis-tribution are in linear approximation

~EGr

q~(k)

402rr k2

40 2q~(k)r q~(k)

402rr (D.19)

The second term of Eq. (D.19) is small compared to the rst term because of the factor1= 2 and can therefore be neglected. The linear approximation of the elds of a Gaussianbunch charge distribution is a valid approximation for radii r r.

Forces and Transverse Kicks

The electromagnetic force ~F can be evaluated using

~Fr = e~Er v ~B

= e ~Er

1 2

=e ~Er

2(D.20)

where e indicates a test charge. The linearized space charge force has to be inserted intothe equation of motion. To do so, the force ~Fr has to be transformed into an accelerationd2r=dz2 with respect to the longitudinal position z in an accelerator. For the accelerationwe obtain

d2r

dz2=

1

2c2d2r

dt2=

1

2c2

~Frm

=q ~Er

2c2m 3: (D.21)

The special case of a uniform and a Gaussian charge distribution yields

d2r

dz2

U

=Ne2~(k)

40m2c2 3R2=

2Nre~(k)

2 3R2r (D.22)

d2r

dz2

G

=Ne2~(k)

40m2c2 32rr =

Nre~(k)

2 32rr (D.23)

where q = Ne and N denote the number of particles per bunch. ~(k) denotes the Fouriertransformation of the longitudinal bunch charge distribution. The kick a test charge ex-periences close to the center of a uniform charge distribution is twice the kick close to aGaussian distribution of equal total charge.

A longitudinal Gaussian charge distribution can be expressed in terms of the wave numberk

~(k) = exp

k2

22k

!: (D.24)

The Fourier transformation yields

() =Z 1

1~(k) exp(ik)dk =

1q22z

exp

2

22z

!(D.25)

where z denotes the bunch length. For Gaussian distributions the rms width in time andfrequency are related by z = 1=k. The coordinate = z ct takes into account thatthe elds travel along with the bunch at a velocity v.

The resulting transverse space charge kick is

kr =2Nrep2

1

2 3R2z, kr =

Nrep2

1

2 32rz(D.26)

for a uniform and a Gaussian charge distribution, respectively. Equations (D.26) are writ-ten for the central slice of the bunch ( = 0).

Equations (D.26) describe the linearized space charge forces for a bunch of cylindricalsymmetry. Using an elliptical integration path a more general situation, where the hori-zontal x and vertical y beam dimensions are dierent, can be evaluated. The result ofthis calculation yields for a Gaussian transverse charge distribution [81]

kx =2Nrep2

1

x(x+y) z2 3, ky =

2Nrep2

1

y(x+y)z2 3: (D.27)

Equation (D.27) is used for the evaluation of linearized space charge forces in chapter 4.For the special case of x = y, Eq. (D.26) is obtained.

Appendix E

Hilbert Transform Spectroscopy

This appendix outlines the theory of Hilbert transform spectroscopy for Josephson junc-tions obeying the so-called RSJ-model. The response of the junction without incidentradiation (autonomous case), with incident monochromatic radiation and with an inci-dent radiation spectrum will be treated.

E.1 Current-Voltage Characteristic without Radia-

tion

A Josephson junction obeying the RSJ model is described by Equation (8.3)

I0 = Ic sin+h

2eR

d

dt:

A constant dc current I0 > Ic is applied. In the following we use dimensionless quantities

i0 = I0=Ic; u = U=RIc; = t!c; !c =2eRIch

: (E.1)

Equation (8.3) then transforms to

d

d= i0 sin() : (E.2)

For I0 > Ic, i.e. i0 > 1, one gets d=d = i0 sin() 6= 0 so we can separate variables:

d

i0 sin= d : (E.3)

Integration gives

2qi20 1

arctan

0@ i0 tan(=2) 1q

i20 1

1A = + const (E.4)

or

tan(=2) =1

i0+

qi20 1

i0tan(

qi20 1

2 + ) (E.5)

where is an arbitrary phase. Then is

() = 2 arctan

0@ 1

i0+

qi20 1

i0tan(

qi20 1

2 + )

1A : (E.6)

The normalized voltage is u() = d=d :

u() = i0 (i20 1)1 + tan2

pi201

2 +

i0 +1i0

1 +

q(i20 1) tan

pi201

2 +

2

= (i20 1)1

i0 +1i0

cos

qi20 1 + 2

+q(i20 1) sin

qi20 1 + 2

: (E.7)

The last expression can be simplied by substituting sin = 1=i0 (remember that I0 > Ic,

so i0 > 1). Then cos =p1 sin2 = 1

i0

qi20 1 and

u =(i20 1)

i0 + sinq

i20 1 + 2+ : (E.8)

The constant is determined by the initial conditions. For the following calculations is chosen such that

u =(i20 1)

i0 cosq

i20 1 : (E.9)

Returning to unnormalized quantities we get Equation (8.4). The dc voltage measuredacross the junction is the time-average of u():

u =1

0

Z 0

0

i20 1

i0 cosq

i20 1 d : (E.10)

Here 0 = 2=qi20 1 is the period of oscillation. The integral yields

u =2

0

2664arctan

0BB@(i0 1) tan

qi20 1

2

qi20 1

1CCA3775=0

=0

: (E.11)

The argument of the tangent function is zero at the lower boundary and at the up-per boundary. The arctan function advances by between two successive zeros of itsargument. Therefore we obtain the simple formula

u =2

0 =

qi20 1 : (E.12)

The result can be summarized by

u =

(0 for ji0j 1qi20 1 for ji0j > 1

)(E.13)

and corresponds to Equation (8.5) in unnormalized quantities.

E.2 Current-Voltage Characteristic with Radiation

Incident radiation is modeled by an additional sinusoidal current with amplitude Is andfrequency !s, which has to be superimposed on the dc bias current I. In normalizedquantities, the dierential Equation (8.7) becomes

i+ is sin () = sin() +d

d(E.14)

with

is =IsIc

and =!s!c

: (E.15)

Equation (E.14) has no analytical solution. The radiation-induced current can be regardedas a small perturbation since is 1 [66]. In the following calculation we determine thechange in the dc bias current while keeping the time-averaged voltage u constant (comparegure 8.3). The dc current i and the phase are expanded into powers of the ac-currentamplitude is

i = i0 +1Xk=1

akiks and = 0 +

1Xk=1

bkiks : (E.16)

The sin term in Equation (E.14) is expanded into a second-order Taylor series

sin

0 +

1Xk=1

bkiks

! sin0 + b1 cos0 is +

b2 cos0 b21

2sin0

!i2s : (E.17)

Equations (E.16) and (E.17) are introduced into the dierential Equation (E.14). Sortingfor powers of is we get

d0d

+ sin0 = i0 (E.18)

db1d

+ b1 cos 0 = a1 + sin () for k = 1 (E.19)

db2d

+ b2 cos 0 = a2 +b212sin0 for k = 2 : (E.20)

Equation (E.18) describes the characteristic of the junction without radiation and is solvedby (E.6). Equations(E.19) and (E.20) are of the form

dbkd

+ bk cos0 = fk() : (E.21)

The general solution of this equation is

bk = expZ

cos0 d0 Z

exp

Z 0

cos0 d00!fk(

0) d 0!: (E.22)

An expression for cos 0 can be derived from (E.2). Taking the derivative with respect tothe normalized time

d20d 2

+d0d

cos0 = 0

one obtains

cos0 = 0_0

= d ln_0

d: (E.23)

Inserting Equation (E.23) into (E.22) the latter equation yields

bk = _0

Z fk(0)

_0( 0)d 0 : (E.24)

We are now looking for the change in the dc current at a xed value of the time-averagedvoltage:

u = _0 + _b1 + _b2 + :::::: = _0 : (E.25)

This is fullled if_bk = 0 for all k > 0 : (E.26)

A consequence is that the integrand in Equation (E.24) is not allowed to have a dccomponent. This can be seen as follows. We form the long-term time average of _bk

_bk = limT!1

1

T

Z T

0

_bkd = limT!1

1

T(bk(T ) bk(0)) : (E.27)

Assume now that the integrand in Equation (E.24) had a dc component, say A. Thenbk(T ) bk(0) would be of the form A T plus a bounded term due to the ac components

of the integrand in (E.24). From the condition _bk = 0 follows immediately A = 0, i.e. thedc component is indeed zero.

The condition _b1 = 0 implies

(a1 + sin ()) _10 = 0 (E.28)

or(a1 + sin ()) (i0 cos (u)) = 0 (E.29)

where f1() = a1 + sin () has been inserted. Keeping in mind that a1 describes a puredc current component, Equation (E.29) can be solved for a1:

a1 =1

i0sin () cos (u) ) a1 = 0 for 6= u : (E.30)

From Equation (E.30) follows directly that there is no rst-order dc current componentis introduced by the small ac current1. The rst-order phase correction b1() can now becalculated from Equation (E.24) setting a1 = 0. It follows

b1 = _0

Z sin ( 0)_0

d 0 =1

i0 cos (u)

Z

(i0 cos (u 0)) sin ( 0) d 0 (E.31)

or

b1 = 1

i0 cos (u)

i0 cos ()

cos (( u) )

2 ( u) cos (( + u) )

2 ( + u)

!: (E.32)

1A nite value is obtained for a1 if = u. This special case is excluded by taking the principalvalue of the integral in Equation (E.41)

The next step is the solution of Equation (E.20) with the condition _b2 = 0. The absenceof a dc component in the integrand in (E.24) means for k = 2:

a2 +

b212sin0

!(i0 cos (u)) = 0 (E.33)

An expression for sin0 can be derived from Equations (E.9) and (E.2).

_0 =u2

i0 cos (u)= i0 sin0 (E.34)

and therefore

sin0 =1 i0 cos (u)

i0 cos (u): (E.35)

Inserting the latter expression into Equation (E.33) and solving for the second-ordercorrection a2 of the dc current we nd

a2 = 1

2i0(1 i0 cos (u)) b21 : (E.36)

Now expression (E.32) has to be substituted for b1 and with the identity

1 i0 cosq

i20 1

i0 cos

qi20 1

2 = 21Xk=1

k coskqi20 1

i0 +

qi20 1

k (E.37)

the following expression for the second-order dc current is found

a2 =1

a0

2Xk=1

k coskqi20 1

i0 +

qi20 1

k i0 cos ()

cos (( u) )

2 ( u) cos (( + u) )

2 ( + u)

!2

:

Note that only the rst two terms (k = 1; 2) of the sum are included in taking the timeaverage. Three terms have to be evaluated

T1 =1

Z 1

i0(i0 + u)

a0 cos2 ( 0) cos2 (u 0) ( u)

!d 0

T2 =1

Z 1

i0(i0 + u)

a0 cos2 ( 0) cos2 (u 0) ( + u)

!d 0

T3 =1

Z 2

i0 (i0 + u)20@cos2 (u 0) cos2 ( 0) sin2 (u 0) sin2 ( 0)

cos2 (u 0) sin2 (u 0)

2 (2 u2)

1A d 0 :

The integrations are straight forward and lead to

a2 = T1 + T2 + T3 = 1

4i0 (2 u2): (E.38)

The important result is that in second order a non-zero dc current component remainsafter the integration. This leads to a resonance-type correction of the dc current-voltagecharacteristic of the Josephson junction. In terms of currents, the nal result can bestated as

i = a2i2s =

i2s4i0 (2 u2)

for u 6= : (E.39)

Returning to unnormalized quantities we get Equation (8.8). The current correction insecond order i is proportional to the square of the perturbing radiation amplitude i2sand is sensitive to its frequency !s. The rst property implies that the Josephson junctionacts as a \quadratic detector", i. e. the detector response is proportional to the incidentradiation power.

The generalization of Equation (E.39) to a continuous radiation spectrum is straightforward. The response of the Josephson junction to monochromatic perturbation has tobe convoluted with the normalized continuous power spectrum s(). Equation (E.39)then becomes

i = 1

4i0

Z 1

1i2s s()

2 u2d = 1

4i0

Z 1

1s()

2 u2d (E.40)

where s() = s() i2s. Returning to unnormalized quantities and setting S() = s()=Icwe get Equation (8.11).

Equation (E.40) has to be solved for the power spectrum s(). Consider

Z 1

0

s()2u

2 u2d =

Z 1

0

s()( + u)

2 u2d +

Z 1

0

s()( + u)

2 u2d

=Z 0

1s()( + u)

2 u2d +

Z 1

0

s()( + u)

2 u2d

=Z 1

1s()

ud

where s() = s() is assumed. The latter integral diverges for = u. A correctmathematical treatment, however, demands for a principal value integral, denoted by P,to exclude the pole. Hence

i = 1

8i0uPZ 1

1s()

ud: (E.41)

Equation (E.41) corresponds to Equation (8.12) in the text. The principal value integral(E.41) is related to the Hilbert transform, a well dened integral transform. The Hilberttransformation is derived from Fourier theory in Appendix E.3.

E.3 The Hilbert Transform

In this section the Hilbert transformation will be derived from Fourier theory. The Fourierintegral is written in the following way [82]:

f(x) =Z 1

0(a(y) cosxy + b(y) sinxy) dy (E.42)

where

a(y) =1

Z 1

1f(u) cos yu du (E.43)

b(y) =1

Z 1

1f(u) sin yu du (E.44)

Inserting Equation (E.43) and (E.44) into (E.42) we obtain

f(x) =1

Z 1

0dyZ 1

1f(u) cos (y(u x)) du (E.45)

or written in complex form

f(x) = Re1

Z 1

0dyZ 1

1f(u) exp (iy(u x)) du = Re : (E.46)

Calculating the imaginary part of instead we obtain

g(x) = Im =1

Z 1

0dyZ 1

1f(u) sin (y(u x)) du (E.47)

org(x) =

Z 1

0(b(y) cos xy a(y) sinxy) dy : (E.48)

The integral (E.48) is called the allied integral of Fourier's integral [83]. The allied integralis obtained by replacing a(y) by b(y) and b(y) by a(y) in (E.42). The twice repeatedformation of the allied Fourier integral leads to the negative of the original.

According to Equation (E.48) the Fourier coecient a(y) and b(y) can be written in adierent way:

a(y) = 1

Z 1

1g(u) sinyu du (E.49)

b(y) =1

Z 1

1g(u) cos yu du : (E.50)

Formally we obtain from Equation (E.47)

g(x) =1

lim!1

Z

0dyZ 1

1f(u) sin (y(u x)) du : (E.51)

Integration of the outer integral leads to

g(x) =1

lim!1

Z 1

1f(u)

1 cos ((u x))

u xdu: (E.52)

For !1, however, the term cos ((u x)) averages to zero. Hence

g(x) =1

PZ 1

1f(u)

u xdu (E.53)

where P denotes the principal value at u = x.

The inversion of Equation (E.53) is derived by inserting (E.49) and (E.50) into (E.42)following

f(x) = 1

Z 1

0dyZ 1

1g(u) sin (y(u x)) du: (E.54)

Following the previous steps we obtain

f(x) = 1

lim!1

Z 1

1g(u)

1 cos ((u x))

u xdu: (E.55)

The cos ((u x)) term can be neglected for !1 and the following expressions havebeen derived

g(x) =1

PZ 1

1f(u)

u xdu; f(x) = 1

PZ 1

1g(u)

u xdu (E.56)

The reciprocity expressed by (E.56) was rst discovered by Hilbert. The Hilbert trans-formation can also be derived from Cauchy's integral theorem.

Appendix F

Acceptance Measurement of the

Pyroelectric Detectors

The spectral acceptance of the pyroelectric detectors has been measured at the TUMunichusing GaAs Read-type IMPATT diodes [61, 84]. The output power of the oscillators,shown by Tab. F.1, is calibrated using a calorimetric measurement.

W-band D-band G-band

f [GHz] 78.0 82.3 88.2 101.9 123.4 128.5 156.0 164.8 176.0 203.2

Prf [mW] 46.8 37.1 37.1 15.1 1.2 1.0 5 4 3.6 1.1

Table F.1: Output power and frequency of the radiation sources. An isolator between theoscillator and the horn antenna has been used with the W-band and G-band sources.

Horn antenna

Chopperwheel

Pout Detector

Horn antenna

Distance

EffectiveDetectorArea

Detector Position

Figure F.1: Spectral acceptance measurement of the pyroelectric detectors with powercalibrated dc diodes sources. The diode is connected to an isolator and a horn antenna.The emitted GHz radiation propagates freely to the detector. A chopper in front of thesource is used to impose an harmonic variation on the radiation signal. The pyroelectricdetector equipped with a horn antenna is mounted opposite to the source. The detectoris mounted on a rail to vary the distance between the source and the detector.

171

An isolator between the horn antenna has been used with the W-band and G-band sources.The isolator prevents re ected rf power from traveling back to the oscillator. The atten-uation caused by the use of the isolator is 1:9 dB. The W-band horn antenna has anopening of 36:527:5mm and a length of 61:5mm. The D-band horn antenna opening is24:5 18:5 and the length is 43mm. The attenuation of the horn antenna is determinedby a calorimetric measurement and yields 3:1 dB at 101:9GHz and 1:85 dB at 78:0GHz.The measurement agrees well with numerical simulations performed by [85] yielding

f [GHz] 78.0 82.3 88.2 101.9 123.4 128.5

[dB] 1.97 2.19 2.51 3.33 1.56 1.69

Table F.2: Attenuation evaluated for the source horn antennas.

The calibration measurements of the pyroelectric detectors have been performed usingthe assembly shown in Fig. F.1 (W-band, D-band). The oscillator, the isolator and thehorn antenna are mounted on an optical table. The GHz radiation propagates freelyfrom the horn antenna to the detector. A chopper wheel is used to impose an harmonicvariation on the radiation signal. The pyroelectric detector equipped with a cylindricalhorn antenna is mounted opposite to the source on a rail to vary the distance betweenthe source and the detector.

During the calibration measurement the detector signal is recorded at dierent distancesfrom the source with and without radiation in order to subtract the background fromthe signal. The measurement is repeated for each of the ve pyroelectric detectors. Theleft graph of Figure F.2 shows the resulting signal amplitude for the 78GHz source as

D1 D2 D3 D4 D5 theory

20 40 60 800

0.2

0.4

0.6

Distance [cm]

Inte

nsity

at d

etec

tor

/ Int

enis

ty a

t sou

rce

0 20 40 60 80 1000

0.2

0.4

0.6

0.8Effective Diameter = 32.1 mm

Effective Horn Area [%]

∑ (

I

- I

)m

eas

theo

ry

2

Figure F.2: Left graph: detector response caused by the 78GHz radiation source asa function of the distance between source and detector. D1D5 denote ve dierentpyroelectric detectors. The solid curve shows the theoretical prediction by diractioncalculations. Right graph: theory is tted to the measurement by varying the eectivediameter of the detector horn antenna. The sum of the quadratic dierences betweentheory and measurement (see left graph for best adaption) is plotted versus the eectivediameter of the horn antenna.

80 90 100 110 12020

30

40

50

Frequency [GHz]

Effe

ctiv

e D

iam

eter

[mm

] Pyro Detectors Ideal horn antenna

Figure F.3: Eective diameter of the pyroelectric detector equipped with the horn an-tenna. Data is shown as markers in the frequency range of 78GHz to 128:5GHz. Thesolid line indicates the theoretical expectation of an ideal horn antenna.

a function of the distance between the source and the detector horn antennas. The vepyro detectors are indicated as D1D5 and the theoretical prediction is depicted as solidcurve. The theoretical prediction is evaluated from a diraction calculation of the freelypropagating radiation wave. The electric eld distribution in an rectangular horn antennacan be evaluated by solving Maxwell's equations [86] yielding

E = E0 cosx

2a

exp

ik

x2

2lE+

y2

2lH

!!(F.1)

where x and y denotes the coordinates of the aperture dimension and lE and lH the slantheights of the horn. Eects at the outer rim of the horn antenna are not considered. Theeld distribution of Eq. (F.1) is propagated to the detector horn antenna by superimpos-ing spherical wavelets according to the Huygens-Fresnel principle which was outlined inchapter 7. A numerical simulation is used to compute the eld intensity at the opening ofthe detector horn antenna. The detector signal is assumed to be proportional to the inte-gral of the intensity distribution. The theoretical prediction is adapted to the measureddata by varying the eective diameter of the detector horn antenna. The right graph ofFig. F.2 shows the sum of the quadratic dierences

Pi(I

imeas I itheory)

2 of the intensitymeasurement and the intensity simulation at distances di between the source and thedetector. The optimum horn antenna diameter is 32:1mm. The left graph of Fig. F.2shows the comparison of the measurement and the computation for the optimum hornantenna diameter.

The eective diameter of the detector horn antenna is shown in Fig. F.3 in the frequencyrange of 78GHz to 128:5GHz. An eective diameter of 3035mm in the W-band rangeand of about 40mm in the D-band range has been evaluated. Measurements in the G-band were not possible because of a change of the experimental setup. The solid lineindicates the true diameter of the horn antenna (55:9mm).

The G-band measurements have been performed using an optical setup which is depicted

0Distance [mm]

Horn &Lens

Lenses

horn antenna

Source

horn antenna

Detector

Chopperwheel

Pout

200 520 680

Figure F.4: Quasi-optical setup to determine the detector acceptance in the G-bandrange. Frequency-doubled W-band sources are used. Radiation is emitted through a hornantenna, collected and transfered by an assembly of lenses and focused onto the detectorhorn antenna.

in Fig. F.4. The Gigahertz radiation is generated by frequency-doubled W-band sourcesand is emitted through an isolator and a horn antenna. The radiation is collected, trans-fered and focused on the detector horn antenna by an assembly of te on lenses. The beamtransfer can be computed using the transformation laws applied for the propagation ofGaussian beams outlined in chapter 7. The transverse dimension of the GHz radiationpulse behind the third te on lens is computed by assuming a symmetric radiation envelopein between lens #2 and #3, which was the design specication of the optical beam line.The detector is positioned at the waist of the radiation beam which is the longitudinalposition where the largest signal is detected. The damping of the optical setup, which isa function of radiation frequency, has been measured and the radiation power behind thethird te on lens is listed in Tab. F.3.

f [GHz] 156.0 164.8 176.0 203.2

Prf [dBm] 15.7 14.7 12.1 10.0

Table F.3: Expected radiation power behind the third lens of the quasi-optical setup.

The maximum detector signal is recorded and compared to the radiation power reachingthe horn antenna. The transverse power distribution is given by the properties of a Gaus-sian beam.

The detector acceptance is the ratio of the recorded detector signals and the radiationintensity reaching the detector horn antenna. The result is shown in Fig. F.5. Thedetector acceptance is increasing rapidly between 100 and 130GHz, reaches a maximumat f = 156GHz and decreases towards larger frequencies. The error bars originate from

80 120 160 2000

0.2

0.4

0.6

0.8

Frequency [GHz]

Spe

ctra

l Acc

epta

nce

[a.u

.]

Figure F.5: Calibration measurement of the pyroelectric detectors. The acceptance isincreasing rapidly at frequencies larger than f = 130GHz. A maximum is at f = 156GHzand the acceptance is decreasing towards higher frequencies. The decrease is explainableby destructive interference of the radiation between the front and the back side of the100m thick pyroelectric crystal.

read-out errors of a digital oscilloscope, uncertainties of the computed attenuation of thehorn antenna and uncertainties of the radiation transfer in the optical setup.

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Acknowledgments

Without the great support and assistance I was given, both during the experimental phaseof my work and during the evaluation of my data, this thesis would not have been com-pleted.

I want to thank Prof. Dr. P. Schmuser for his eort and enthusiasm in advising my PhDthesis. He supported the formation of this thesis by fruitful discussions and constructiveideas.

I show my gratitude to Priv. Doz. Dr. M. Tonutti, who excited my curiosity about theTESLA project. I thank Dr. Tonutti, Prof. Dr. Ch. Berger and R. Siedling for the con-struction of the beautiful Martin-Puplett interferometer and for making many measure-ments possible.

Special thanks to Dr. H. Weise who introduced me into the technology of the TTF linacand made it possible for me to learn how to operate the accelerator. I enjoyed workingwith him and learned a lot from his experience with superconducting linacs.

I want to thank Priv. Doz. Dr. M. Leenen for the possibility to present my work at inter-national workshops and conferences and for getting the chance to meet and to interchangeideas with experienced scientists.

I thank Dr. Y. Divin and Dr. U. Poppe for spending several weeks at DESY and forthe introduction into Hilbert-transform spectroscopy. I want to thank Dr. I. Wilke andM. Khazan to measure the spectral transmission of the wire grids and the vacuum win-dow. I furthermore acknowledge the help of Dr. J. Freyer and his collaborators to makeacceptance measurements of the pyroelectric detectors possible.

I am much obliged to express my thanks to all the members of the TESLA collaborationfor the opportunity to take part in their discussions and to learn from their activities. Ihad helpful discussions with Priv. Doz. J. Rossbach, Dr. G. Schmidt, Dr. S. Schreiber,Dr. S. Roth, H. Schlarb and F. Ludwig. I thank the diploma students B. Leissner andJ. Menzel for supporting my scientic investigations. Soft- and hardware problems I dis-cussed with and obtained a lot of help from K. Rehlich, O. Hensler, Dr. K. Mess andP. Albrecht. I want to thank Dr. M. Castellano and L. Catani to prot from the opticaldiagnostics system, Dr. S. Simrock, M. Huning and G. v. Walter for supporting the radiofrequency system.

For technical support I want to thank B. Sparr, O. Peters, J. Holz, T. Plawski and B. Sa-rau and the workshops of the RWTH Aachen, the University of Hamburg and DESY.

I nally thank my girlfriend Dr. E. M. Jaeckel for her moral support and understandingfor being up working many whole nights and weekends to collect the data presented inthis thesis.