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This is to certify that the

dissertation entitled

Measuring the Transverse RMS Emittance and RMS Pulse
Length of a Short Pulse, Photoinjcctor Produced Electron
Beam with the Second Moment of its Image Charge

presented by

Steven J Russell

has been accepted towards fulfillment
of the requirements for

Doctorate . Physics
degree in

 

 

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Major professor

Date 4//"/ a7, ’777

MS U i: an Affirmatiw Action/Equal Opportunity Institution 0— 12771

 

 

LIBRARY

Michigan State
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.57

MEASURING THE TRANSVERSE RMS EMITTANCE AND RMS PULSE LENGTH
OF A SHORT PULSE, PHOTOINJECTOR PRODUCED ELECTRON BEAM WITH
THE SECOND MOMENT OF ITS IMAGE CHARGE
By

Steven J. Russell

A DISSERTATION
- Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPY

Department of Physics and Astronomy

1998

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ABSTRACT

MEASURING THE TRANSVERSE RMS EMITTANCE AND RMS PULSE LENGTH
OF A SHORT PULSE, PHOTOINJECTOR PRODUCED ELECTRON BEAM WITH
THE SECOND MOMENT OF ITS IMAGE CHARGE
By

Steven J. Russell

Radio frequency, photo-cathode injectors are a recent development in the electron
accelerator community. They work by placing a small, photo-emissive surface inside a
radio frequency accelerating cavity. Electrons are stripped from the photo-cathode with a
pulsed laser and immediately accelerated by the cavity fields. These photoinjectors
enable the creation of high charge, short pulse length beams. However, they also create
problems for the electron beam diagnostics. A photoinjector accelerates the electrons to
relativistic velocities very quickly. As a result, the beam does not have time to come to
equilibrium. Its spatial distribution will be unknown and cannot be well approximated by
a Gaussian. Therefore, diagnostic techniques can make no assumptions about the beam’s
spatial distribution. A class of diagnostics that fulfills this requirement look at the image
charge “wake” generated in the metal walls of the beam pipe as the electron beam passes.
These devices are generically known as beam position monitors and are normally used to
measure the first moment of the image charge signal, thereby determining the position of
the beam’s center. However, coupled with a good knowledge of the beam line, they are
also capable of determining the rms emittance of the beam by measuring the second

moment of the image charge signal. In addition, when used in tandem with a deflecting

 

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cavity, beam position monitors can also be used to perform a measurement of the beam
pulse length. Both of these measurements are independent of the beam’s spatial
distribution, making them ideal for photoinjectors. Described here is their theory and

implementation.

To my parents

Jack and Pat

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ACKNOWLEDGMENTS

I extend my thanks to the Michigan State University physics department and the
members of LAN SCE-9 at Los Alamos National Laboratory. To Dr. Bruce Carlsten, for
his guidance, encouragement and supervision during my thesis work, I owe a great debt
of gratitude. I doubt he knew what he was getting into. I owe special thanks to Dr. Brad
Sherrill for being my liaison to Michigan State and for remembering who I am. I thank
Dr. Rich Shefiield for bringing me into Los Alamos in the first place. I thank Steve
Gierman, Dr. Doug Gilpatrick and Dr. Eric Nelson for very helpful discussions. I thank
Mike Webber, John Plato, Scott Volz, Boyd Sherwood, Marty Milder, Ron Sturgis, Dr.
Ron Cooper and Dr. Doug Fulton for putting the accelerator together and keeping it
running. I thank Dr. Dinh Nguyen, John Power, Brad Shurter and Dr. David Oro for
providing valuable assistance and equipment. I thank Drs. Martin Berz, Jack Bass, Ray
Brock and Edwin Kashy for their service on my Ph. D. guidance committee.

I thank my family for their support. I especially thank my parents, Jack and Pat, for
everything they have given me. My uncle Gary and aunt Elaine I thank for support and
friendship during my stay in Los Alamos. I thank my sister and brother, Ann and Tim,
for making my life better.

I also thank Molly, for her love and inspiration.

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I am also grateful to Michigan State University for the financial support they have
given me during my graduate study and Los Alamos National Laboratory for the

opportunity to perform my research.

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TABLE OF CONTENTS

LIST OF TABLES ............................................................................................................. xi

LIST OF FIGURES .......................................................................................................... xii

INTRODUCTION ............................................................................................................... 1

1.1 Background ............................................................................................................... 1

1.2 Measuring emittance ................................................................................................. 4

1.3 Measuring pulse length ............................................................................................. 8

1.4 Beam position monitors ............................................................................................ 9

1.5 Implementation of measurements ........................................................................... 1 1
CHAPTER 1

BEAM PARAMETERS MEASURED BY BEAM POSITION MONITORS ................. 16

1.1 Introduction ............................................................................................................ 16

1.2 Notation .................................................................................................................. 16

1.3 Physical description of BPMS ................................................................................ 17

1.4 Image charge .......................................................................................................... 19

1.5 Model of BPM coupling to image charge .............................................................. 20

1.6 Beam information provided by BPM ..................................................................... 23
CHAPTER 2

MEASURING EMITTANCE WITH BEAM POSITION MONITORS ........................... 26

2.1 Introduction ............................................................................................................ 26

2.2 Emittance ............................................................................................................... 26

2.3 Measuring the emittance ........................................................................................ 30

2.4 Stability of emittance measurement ....................................................................... 32

2.5 Figure of merit for matrix equation stability .......................................................... 33

2.6 Finding a stable implementation of Miller’s measurement ................................... 37

2.6.1 Poor implementations of Miller’s emittance measurement .......................... 37

2.6.2 Singular equations and strong coupling ........................................................ 41

2.6.3 Stable implementation of Miller’s measurement .......................................... 43

2.6.4 Numerical examples ...................................................................................... 45

vii

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CHAPTER 3
MEASURING THE RMS PULSE LENGTH WITH FAST DEF LECTING

CAVITY AND BEAM POSITION MONITOR ............................................................... 51

3.1 Introduction ............................................................................................................ 51

3.2 Trajectory change of a single electron at fast deflector exit .................................. 53

3.2.1 Field components .......................................................................................... 53

3.2.2 Conversion of field components to Cartesian coordinates ............................ 55

3.2.3 Equations of motion ...................................................................................... 56

3.3 Measuring the rrns length squared of the electron beam bunches ......................... 59

3.3.1 F astdeflector off ........................................................................................... 60

3.3.2 Fast deflector on ............................................................................................ 61

3.3.3 Measuring bunch length ................................................................................ 65

3.4 Calibration of measurement ................................................................................... 66

3.5 Estimate of measurement resolution ...................................................................... 66

3.6 Effect of BPM rotated with respect to fast deflector cavity ................................... 69

3.6.1 Pulse length measurement ............................................................................. 69

3.6.2 Calibration ..................................................................................................... 72
CHAPTER 4

CALIBRATING BEAM A POSITION MONITOR ......................................................... 74

4.1 Introduction ............................................................................................................ 74

4.2 Simulating a relativistic beam ................................................................................ 74

4.3 Mapping the BPM .................................................................................................. 76

4.4 Fitting the map data ............................................................................................... 76

4.4.1 Centroid calibration ...................................................................................... 79

4.4.2 Quadrupole calibration .................................................................................. 84

4.5 Simulating a diffuse beam ..................................................................................... 88

4.5.1 Interpolation versus measurement ................................................................ 91

4.5.2 Check of BPM accuracy for measuring (x2 > — <y2> .................................... 93

4.6 Concerns with calibration method ......................................................................... 95
CHAPTER 5

EXPERIMENTAL RESULTS ........................................................................................... 98

5.1 Introduction ............................................................................................................ 98

5.2 Experimental apparatus and data acquisition ......................................................... 98

5.2.1 Experimental apparatus ................................................................................. 98

5.2.2 Capturing BPM signals ............................................................................... 101

5.2.3 Analyzing digitized BPM signals ............................................................... 101

5.3 Stability experiment ............................................................................................. 104

5.3.1 Shot-to-shot stability ................................................................................... 105

5.3.2 Average stability ......................................................................................... 110

5.4 Experiments to check BPM calibration ............................................................... 111

5.5 Emittance measurements ..................................................................................... 117

5.5.] Single emittance measurement ................................................................... 118

viii

5.5.2 On the question of beam scraping ............................................................... 127

5.5.3 Emittance versus bunch charge ................................................................... 133
5.5.4 Emittance versus magnetic field ................................................................. 134
CONCLUSION ................................................................................................................ 141
APPENDIX A
CHARGE DISTRIBUTION INDUCED ON INNER SURFACE OF BEAM
PIPE DUE TO RELATIVIS'IIC ELECTRON BEAM .................................................. 144
A.l Introduction ......................................................................................................... 144
A2 Solution in beam frame ....................................................................................... 144
A3 Surface charge distribution in lab frame ............................................................. 154
A.4 Relativistic approximation .................................................................................. 157
APPENDD( B

SURFACE CHARGE DISTRIBUTION ON INNER SURFACE OF
CYLINDRICAL METAL PIPE DUE TO AN INFINITE LINE CHARGE

INSIDE THE PIPE AND PARALLEL TO ITS AXIS .................................................... 160
B.1 Introduction ......................................................................................................... 160
B2 Constant potential surfaces due to two infinite line charges in free space .......... 160
B3 Electric potential from infinite line charge inside an infinite metal pipe ............ 163
8.4 Surface charge distribution from infinite line charge inside infinite metal pipe.166

B.4.1 Image charge distribution ........................................................................... 166
B.4.2 Expansion of image charge distribution in Fourier series .......................... 167
APPENDIX C

VOLTAGE SIGNAL GENERATED ON OSCILLOSCOPE FROM
BEAM POSITION MONITOR SIGNAL TRANSMITTED THROUGH

TRANSMISSION LINE .................................................................................. 169
C.1 Introduction ......................................................................................................... 169
C2 General Solution .................................................................................................. 169

C21 Solution to circuit model ............................................................................ 169
C22 Expression for beam current ....................................................................... 172
C23 Solution ...................................................................................................... 175
C3 Approximate solution for short beam pulse ........................................................ I75
C.4 Gaussian beam ..................................................................................................... 176

APPENDIX D

EXPANSION OF BPM SIGNAL AMPLITUDE IN CARTESIAN COORDINATES..179
D.1 Introduction ......................................................................................................... 179
D2 Expansion of amplitude in Cartesian coordinates ............................................... 179

D2] Expansion of summation term in Cartesian coordinates ............................ 180
D.2.2 Expressing amplitude as a sum of moments .............................................. 181

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APPENDIX E
FIRST ORDER CALCULATION OF THE TRAJECTORY OF AN

ELECTRON DUE TO IDEAL FAST DEFLECTOR FIELDS ....................................... 185
El Introduction ......................................................................................................... 185
E2 Electric and magnetic fields of fast deflector ...................................................... 185
E3 Transformation of fields to Cartesian coordinates .............................................. 187
E4 Demonstration of dominant field ......................................................................... 193
E5 Solution to the first order equations of motion .................................................... 196
E6 Electron position and divergence at exit of fast deflector cavity ......................... 198

APPENDIX F

COVARIANCE MATRIX AND THE STANDARD ERROR IN THE

CALCULATION OF THE EMITTANCE ...................................................................... 201
El Introduction .......................................................................................................... 201
F2 Estimating measurement errors ........................................................................... 202
F3 Covariance matrix ................................................................................................ 207
F .4 Estimated error in calculation of the emittance ................................................... 209

REFERENCES ................................................................................................................ 215

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LIST OF TABLES

Table 3-1: Value of a? ((1):) vs. drift length, d, and the FWHM
pulse length of the beam .................................................................................. 70

Table 4-1: Table comparing the values of i, y and (x2 ) — (y’) as
measured by a BPM to their actual values for several .
simulated, diffuse beams .................................................................................. 94

Table 5-1: Average measured value of (x2 ) - (y2 > , predicted value

of (x2 > — (yz) (from fit) and the difference between them

for each quadrupole setting in emittance measurement
for charge equal to approximately 0.2 nC/bunch ........................................... 123

Table 5-2: Same as Table 5-1, but measurements 2, 3, 5 and 19 are
all discarded because of beam scraping ......................................................... 126

Table 5-3: Value of (x2) — (f) from simulation, predicted value of

(x2 > - <y2> (from fit) and the difference between them for

each quadrupole setting in simulated emittance
measurement with beam pipe ......................................................................... 131

Table 5—4: Values of the RMS beam parameters for different fits in
simulated emittance measurements ................................................................ 132

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LIST OF FIGURES

Schematic of SPA photoinjector ..................................................................... 13
Schematic of SPA beam line .......................................................................... 14
a) Schematic of magnetic chicane. b) Energy versus

beam bunch length from simulation. .............................................................. 15
Schematic of beam position monitor cross section. The

beam will travel along the z axis, out of the page .......................................... 18

Circuit model of BPM electrode coupling to electron beam.
CD is the capacitance of the BPM electrode. Zc is the
characteristic impedance of the transmission line. ib
represents the image induced image charge from the passing

electron beam .................................................................................................. 21
The x phase space of the SPA beam from simulation .................................... 28
Schematic of beam line for implementation of emittance

measurement using only drifts ....................................................................... 38
Schematic of beam line consisting of quadrupole magnet,

drift and BPM ................................................................................................. 40
Schematic of beam line consisting of a triplet followed by

a beam position monitor ................................................................................. 44
Illustration of fast deflector cavity streaking a beam bunch .......................... 52
Schematic of fast deflector cavity .................................................................. 54
BPM rotated with respect to the x and y axes defined by

the fast deflector fields ................................................................................... 71

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Figure 4-1: Schematic of pulsed wire apparatus for calibrating BPM .............................. 75

Figure 4-2: Maps ofthe a) right (0 = 0°) and b) left (0 = 180°)
electrodes of a BPM ....................................................................................... 77

Figure 4-3: Maps of the a) top (0 = 90°) and b) bottom ( 0 = 270°)
electrodes of a BPM ....................................................................................... 78

Figure 4-4: a) R, versus actual x position of wire, b) R" versus x
position of wire as calculated by fitted equation ............................................ 81

Figure 4-5: a) Ry versus actual y position of wire, b) Ry versus y
position of wire as calculated by fitted equation ............................................ 82

Figure 4-6: a) Difference between actual x position of wire and that
calculated from the fitted equation versus radial position
of the wire. b) Difference between actual y position of
wire and that calculated from the fitted equation versus
the radial position of the wire ......................................................................... 83

Figure 4-7: a) Plot of the difference between actual value of x2 — y2

and its calculated value using the fit of Equation (4-6) ,
and b) using Equation (4-7) ............................................................................ 87

Figure 4-8: Schematic showing grid points for simulation of diffuse
beam with a single pulsed wire ...................................................................... 90

Figure 4-9: a) Plot of (x2 > — <y2> versus Q for a number of simulated
beams (32 = y = 0) using the apparatus in Figure 4-1.
b) Identical to a) except that the responses of the BPM
electrodes for the individual grid points are interpolated
from BPM maps ............................................................................................. 92

Figure 4-10: Plot of the difference between the actual value of
(x2 > — <y2> and that calculated from the BPM signals in

mm2 for a single simulated beam as it is move radial

outward from the BPM center ........................................................................ 97
Figure 5-1: Schematic of end section of SPA beam line .................................................. 99
Figure 5-2: Schematic of data acquisition system for capturing BPM signals ............... 102

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Voltage versus time signal from BPM electrode for a
typical beam shot. There are two full beam bunches and
part of a third displayed ................................................................................ 103

a) Intensity (sum of four BPM electrodes) versus

successive beam shots. b) (x2) — <y2> versus successive
beam shots .................................................................................................... 107

a) x position of beam center versus successive beam shots.
b) y position of beam center versus successive beam shots ......................... 108

a) Average intensity (sum of four BPM electrodes) of

beam bunch versus measurement number. b) Average

value of (x2 > — <y2> versus measurement number. Each

measurement is the average over 99 consecutive beam shots ...................... 112

a) Average value of x position of beam bunch center

versus measurement number. b) Average value of y

position of beam bunch center versus measurement

number. Each measurement is the average over 99

consecutive beam shots ................................................................................ 113

a) Another example of the average intensity
(sum of four BPM electrodes) of a beam bunch versus
measurement number. b) Another example of the

average value of (x2 > - <y2> versus measurement number ......................... 1 14

a) Quadrupole moment versus 322 — yz. The slope of

the fit line is 0.95 i 0.021. b) (x2) — <y2> versus

measurement number for the same data points shown
in a). Each point is the average of 99 beam shots ........................................ 116

Figure 5-10: A matrix associated with settings for quadrupoles in

Figure 5-1 used to measure the electron beam emittance ........................... 120

Figure 5-11: Average beam intensity (sum of four BPM electrodes)

at BPM versus measurement number for emittance
measurement a charge equal to approximately 0.2 nC/bunch .................... 122

Figure 5-12: Intensity vs. measurement number for simulated

emittance measurement ............................................................................... 130

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Figure 5-13: A plot of normalized emittance versus bunch charge ................................ 135
Figure 5-14: Emittance versus magnetic field on photo-cathode ................................... 140

Figure A-l: Relativistic electron beam pulse in metal pipe: a) cross
sectional view, b) longitudinal view ........................................................... 145

Figure A-2: Lab and beam frame for relativistic electron beam bunch

traveling down beam pipe ........................................................................... 155
Figure B-l: Two infinite line charges parallel to the z axis with

charge per unit length A and —A, respectively ............................................. 161
Figure 8-2: Infinite line charge inside an infinite metal pipe ......................................... 164

Figure C-l: Circuit model of BPM electrode coupling to electron beam.
Cp is the capacitance of the BPM electrode. Zc is the

characteristic impedance of the transmission line. ib
represents the image induced image charge from the passing
electron beam ................................................................................................ 170

l a) 2
Figure C-2: The fimction e 2L“ J vs. frequency for a 6 mm, or
20 ps, FWHM long electron pulse ............................................................... 178

Figure E-l: Schematic of fast deflector cavity ................................................................ 186

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Figure E-2: a) J l(knr) (blue) and #r- (red) versus radius,

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b) 12(kur) (blue) and §(k”r)2 (red) versus radius .................................... 191
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Figure E-3: 1— 157x vs. x (cm) ............................................................................... 192
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Figure E-4: I? vs. x (cm) for an 8 MeV electron ......................................................... 195

 

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INTRODUCTION

1.1 Background

In 1984 the Strategic Defense Initiative (SDI) was officially launched. Its stated goal
was to develop weapons capable of destroying nuclear missiles in flight, creating an
impenetrable shield around United States interests. Since the technology to develop such
a system was not in existence, substantial funding was provided to the scientific and
engineering community towards its development.

As SDI got under way, it was not clear what final form the weaponry of SDI would
take and many potential solutions were pursued. Of particular interest were the high
powered laser programs. Fast and precise, lasers were a perfect fit to SDI if laser systems
of sufficient power could be developed. The free electron laser (FEL) was one candidate
that held great potential to satisfy SDI needs[1].

Although a rigorous description is quite complicated[2], F ELs are conceptually
simple devices. A high energy beam of electrons is directed down the axis of an
alternating magnetic field. In this field, the electrons move back and forth across the path
of their initial trajectory, generating light that can be used to amplify a conventional laser
beam or to create a coherent light beam inside a resonator cavity.

At the time that SDI came into being, FELs had been in existence for a little over a

decade[3]. However, this was the first time that significant resources were applied

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toward their development. Because their physics can be described classically, powerful
simulations were developed that indicated fantastic performance was possible using
F ELS. In fact, two machines boasting an average laser power to be measured in mega-
watts and operating over a broad range of wavelengths were proposed[1]. However, in
the end, neither of these devices were constructed. SDI came to a halt and FEL funding
was sharply curtailed.

The SDI program advanced the knowledge of F ELs very rapidly in a short time.
F ELs still hold great promise as high powered, tunable light sources and the knowledge
gained during the frenetic SDI years has proved very valuable. Although in hindsight it
is apparent that the attempts to build the huge, mega-watt machines was premature, many
other technical advancements that came out of the programs of the past are still very
much alive. Of particular importance to the electron accelerator community was the
invention of the radio-frequency (rf) photoinjector[4], [5].

Throughout the history of FELs, it has been apparent that the success of an F EL
experiment depends strongly upon the quality of the electron beam used to drive the
device. This fact was fitrther underscored for those who worked in the SDI FEL
programs. Therefore, there was a significant effort to advance electron accelerator
technology to achieve better beams.

The electron source, as one might imagine, is a very important part of any electron
accelerator. For many years, the only choice was to use a thermionic cathode[6]. This is
a simple device where the cathode material is heated to the point that electrons on its
surface obtain enough energy to overcome the material work function. Immersed in a

static electric field, the electrons are accelerated as they boil ofl‘ the cathode surface to

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form a dc beam. Since high energy accelerators are pulsed devices, this do beam of
electrons is not appropriate for immediate injection into the accelerator. First, a pulsed
structure that matches that of the accelerator is imposed. This is done by bunching the
beam with time varying electric fields. Only then are the electrons accelerated to high
energy.

Although thermionic cathodes are very reliable, they severely limit the command one
has over the shape of the electron beam that is injected into the accelerator. While being
bunched, the electrons are at low energy and the repulsive force between them works
against the bunching process. The end result is that an electron beam from a thermionic
injector will approach an equilibrium shape[6]. Although not obvious, experimentally
this means that the spatial distribution of the beam will be very nearly Gaussian. The
combination of low energy and repulsive space charge force will wash out any other
structure that might be dictated.

The rf photoinjector was invented by Richard Sheffield and John Fraser[4], [5] as an
alternative to the thermionic injector for rf accelerators. The idea was based on the
lasertron concept[7]. Instead of a heated filament, the photoinjector uses a photo-cathode
as its electron source. The photo-cathode is a photo-emissive surface located inside the
accelerator. A short pulse, high energy laser impinges upon the photo-cathode, stripping
electrons from the cathode surface. Once free, the electrons are quickly accelerated to
relativistic velocities.

The inventors of the photoinj ector had two powerful insights. First, they realized that
our ability to manipulate pulsed laser light is much greater than our ability to manipulate

pulsed electron beams. By using a laser to make the electron beam, much of this greater

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capability is transferred, improving our capacity to impose a desired initial shape for the
electron beam. Second, since the electron beam is already bunched, the photo-cathode
can be placed inside the first cell of the accelerator. Here, the very high electric fields
accelerate the electrons to relativistic velocities over a distance an order of magnitude
shorter than that required by a thermionic injector. Therefore, because of relativistic
effects, much of the original structure of each electron beam is preserved. This allows a
great deal of control over the spatial distribution of the final electron beam. No longer is
a Gaussian-like shape inevitable.

The ability to control the final beam distribution, at least partially, is the most
significant advancement that the photoinjector brings to the electron accelerator
community. Employing this control wisely can improve the beam quality and enable
longitudinal compression of the electron beam that is much more effective than what can
be achieved with a thermionic beam[8]. However, this also creates new challenges for
the beam diagnostics. Because the beam is no longer GauSsian-like and because our
control over its shape is not complete, we can no longer make accurate assumptions about
the spatial distribution of the beam[9], [10]. Therefore, it is important that our

diagnostics make no assumptions about the beam distribution.

[.2 Measuring emittance

Each electron in a beam bunch is described by six coordinates: the three spatial
coordinates and their associated momentum. Taken as a whole, the bunch occupies a six
dimensional volume. Ideally, we could know the beam distribution in this six

dimensional phase space at any given time. However, the ability to make such a

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5

measurement has not yet been realized. Instead, we look at projections.

If a screen is inserted into the beam path we see the projection of the six dimensional
phase space onto the x, y plane. This is a very useful diagnostic that tells us the
transverse shape of the beam at a given point in its transport. However, the projections
onto the x and y phase spaces tell even more. These are the planes with one axis defined
by the position coordinate and the other by the respective momentum. In principle,
knowledge of these projections at any given point in the beam transport allows us to
determine the shape of the beam at any other point in the beam-line if the focusing forces
are linear[6].

There are schemes for mapping out the complete transverse phase space of an electron
beam using slit and collector type schemes[9]. However, for high charge beams with
energies of more than a few MeV, it is questionable how efficacious these methods are.
The slit, which is used to select slices of the beam while blocking the rest, becomes less
effective as the beam energy increases. The electrons start to punch through the material
that is meant to stop them and at the same time produce copious x-rays. This results in a
large amount of noise in the collector necessitating some scheme for subtracting the
background out of the desired signal. At high charge and high energy this becomes
difficult.

For higher energy electron beams, complete maps of the transverse phase spaces are
usually abandoned in favor of an envelope description. This simplifies the measurement
greatly and still provides useful information. In this scheme, the x and y phase spaces are

characterized by the x and y root-mean-square (rms) emittances defined by

m
m
J

mi

11: 3.15ch t
fiergcncies.
beans. ln If.
ellipse. Tried
0525: }' emit.

From the l

and

 

a. J<x2><x'2>-<xx'>2

and

 

8, \/<Y’><y">-<yy'>2 .

The angled brackets indicate an average over the beam distribution. The x and y

divergencies, x' and y' , are proportional to the x and y momenta for relativistic electron

beams. In the x phase space, the three numbers, (x2), (x'z) and (xx'), describe an
ellipse. The emittance is the area of this ellipse divided by the number 1:. The meaning
of the y emittance is similar.
From the rms emittance, we can define the normalized rrns emittances as

8,... E 1578,.
and

syn 5 Bye,
where [3 and y are the usual relativistic parameters associated with the average energy of
the beam. The normalized emittance is a useful metric because it has the property that it
is a conserved quantity in a linear focusing channel with acceleration[6]. Because of this
property, it is an excellent indicator of unwanted nonlinear processes. Also, the emittance
is to charged particle optics what wavelength is to light optics: it provides a fundamental
limit on how tightly a beam can be focused and indicates how fast it will diverge from
that focus. Since the main goal is often to pack as many electrons as possible into a given
area, normalized emittance is widely used as a metric for beam quality. Accelerator

facilities will always quote normalized emittance as a standard for machine performance.

 

an optical In“;

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hem SN”

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and j)". I‘"
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7

A typical procedure for determining the rrns emittance is to measure (x2) and (y’)

at several points in a linear focusing channel. If, for example, we wanted to know the rms

x emittance, (x2) would be measured in at least three different beam line locations. This
results in a set of three, or more, linear equations that can be solved for (x2 ) , <x'2) and
(xx'). Similarly, we could measure <y2> and obtain <y2>, <y’2> and (yy').

Measuring (x2) and <y2> can be a challenging task. A common method would be to

insert a screen in the beam path. This screen can have a phosphor coating that generates
an optical image of the beam spot. A series of mirrors and lenses is then used to direct

the beam spot image to a camera where it can be captured.

If the beam spot image were perfect, a direct numerical integration would give (x2)

and (f). For a real image, however, integration is a poor option. There are two main

reasons that it fails: image noise and the limited dynamic range of the camera. Noise
spikes in the image, whatever their origin, are always positive. That is the nature of the
measurement, there is never a negative intensity. Because integration sums the
intensities, the noise will add together, causing substantial error. The limited dynamic
range, on the other hand, will cause some of the beam image to be lost. Away from the
bright central core of the beam, the light generated by the electrons is too faint for the
camera to see. This might seem unimportant since a faint image indicates a low electron
density. However, because the area is large, the total number of electrons in these faint
regions can be a large fraction of the total number in the beam. Leaving them out of an

integration would also result in a large error.

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8
Instead of integrating the image, the problem can be greatly simplified by assuming

that the beam has a particular type of distribution. Then, the obtained beam image can be

fit to this distribution and (x2) and <y2> fall out naturally. Noise spikes essentially have

no affect on the fit and the central region of the beam image is enough information for
this type of processing if the distribution assumption is a good one. Thennionic beams,

with their Gaussian nature, are ideal candidates for this technique and accurate values for

(x2) and <y2> result. For photoinjector beams, however, the distribution is not known.

Any distribution assumption, Gaussian or otherwise, will generally be a poor one.

Inaccurate values for (x2) and <y2) are the consequence. In turn, because it depends

upon these values, poor estimates of the rrns emittance result.

1.3 Measuring pulse length

Measuring the length of the beam pulse is another pressing diagnostic problem.
Recently, photoinjector beams of substantial charge have been compressed to lengths of p
less than 0.3 mm[8]. Traveling at nearly the speed of light, this is less than 1 picosecond
in time. Diagnostics with such time resolution are hard to come by and often expensive.
An economical solution to this problem utilizes a radio frequency (rf) cavity and a simple
screen[8].

A cylindrical cavity operating in its TM”0 mode will produce a time varying,
magnetic dipole field. If a beam pulse is directed down its axis and its arrival and the
phase of the cavity fields are synchronized correctly, the head and tail of the pulse will be

kicked in opposite directions while the trajectory of the beam center remains unchanged.

9

This will spread the beam in one dimension. By placing a screen downstream from the
cavity (called a fast deflector), the increase in width that results can be measured. In turn,
this will give the length of the beam pulse.

This technique is quite good for low charge beams. As the charge per pulse is
increased, however, its effectiveness breaks down. After passing through the fast
deflector cavity, the beam continues to expands as it drifts. Therefore, the increase in
beam width will be greater the longer the beam is allowed to drift. When the beam pulses
are very short, it is important that this drift be of sufficient length to obtain the necessary
resolution. At the same time, how tightly the beam can be focused at the position of the
screen is lessened the longer it is allowed to drift, especially for high charge beams
because of the repulsive space charge force. From a practical point of view this is
important because a large beam spot at the screen results in a low signal to noise ratio,
making it difficult to measure any width change. At low charge, this is not an issue.
However, as the charge per bunch is increased, the goal of adequate resolution and a

tightly focused spot become mutually exclusive.

1.4 Beam position monitors

The term beam position monitor (BPM) refers to a generic class of devices that
determine the position of a charged particle beam’s center. They work by coupling to the
electromagnetic fields that accompany the beam. One particular class of these devices
sees the image charge that these fields induce in the metal of the beam pipe walls.

A dual axis BPM of this type is a BPM that measures both the x and y positions of

the beam center. Without going into too much detail now, it will consist of four

10
electrodes placed on the inner beam pipe wall at 90 degree intervals. What will be shown

in Chapter 1 is that, when combined properly, the four electrode signals measure four
beam properties: the total charge in the beam, the x and y center positions of the beam
and the value of (x2 > — (yz). When the beam pulses are very short, as is the case for our
photoinjector beam, the determination of these quantities is independent of the beam’s
spatial distribution.

BPMs have been used for quite some time for measuring the beam position[11].
They have not been used extensively to determine (x2 > — <y2>. However, knowledge of

this quantity can be exploited to measure both the rms emittance of the beam and the rms

pulse length.

To measure the emittance, we perform essentially the same measurement as was

described earlier. It was first proposed by Miller et. al.[12] to measure (x2>—<y2>,

instead of (x2) or <y2> , at various points in a well characterized, linear focusing channel.
If this is done at least six times, a set of linear equations results and we can solve for
(x2), <x'2>, (xx') , (yz) , <y'2> and <yy') , determining both the x and y rms emittances.

The measurement of the pulse length is also the same as described before, but with
the screen replaced by a BPM. By measuring the value of (x2 > — (yz) before and after

the cavity is turned on, it can be shown that the difference between the two measurements
will produce the rrns length of the beam pulse. The advantage that the BPM brings to the
method is that, unlike the screen, a good focus is not necessary at the BPM location. This

allows us to increase the drift, and therefore the resolution, between the BPM and the

ll

cavity indefinitely as long as the beam does not intercept the beam pipe walls.

1.5 Implementation of measurements

The Sub-picosecond Accelerator (SPA) facility at Los Alamos National Laboratory is
an 8 MeV, rf photoinjector operating at a frequency of 1300 MHz[13]. Its primary
mission is to explore the uses and dynamics of compressed electron beams. State of the
art in this field, SPA is capable of compressing electron pulses containing more than 1 nC
of charge to sub-picosecond lengths with a magnetic chicane[8]. In its previous
incarnation, the SPA photoinjector was the electron source for the High Brightness
Accelerator FEL (HIBAF) facility[14]. Schematics of the important components of the
SPA facility are shown in Figure LI, [-2 and I-3.

The chicane, shown in Figure I-3a, is a series of four dipole magnets whose field
orientations are such that an electron will travel a path through them like that shown in
the figure. Utilizing the unique capabilities of the photoinjector, electron beam pulses of
an appropriate length are injected into the accelerating structure at a phase in the rf cycle
so that an energy versus phase correlation is introduced. This energy slope, shown in
Figure I-3b, is largely linear and is such that the electrons in the front of the beam bunch
have lower energy than those at the back. Because of their lower energy, the electrons at
the front of the pulse are directed along a bigger are through the chicane than those at the
back. Therefore, they travel a greater distance. If the energy slope and initial bunch
length are chosen correctly, the particles at the back of the bunch will catch those at the

front as the beam exits the chicane, compressing the beam.

12

Compression technology is an important issue in the electron accelerator community.
The delivery of large amounts of charge in small volumes is a critical ability for both
applications and fundamental research. However, recent work has predicted an
unfortunate degradation of rrns emittance as a bunched beam passes through a bend[15].
Since a chicane is a series of bends, it is very likely that, at the same time it compresses
the beam, it causes serious harm to its quality. What is needed are the tools to perform
the relevant measurements of the SPA beam properties.

The two diagnostic techniques described here will fill important diagnostic needs for
SPA, if they prove practical. What is presented here is the theory and implementation of
the nns emittance measurement using BPMs and the theory behind the rms pulse length

measurement using a cylindrical cavity and a BPM.

Solenoid 1 Solenoid 2

 

Laser

   

 

 

   

Il|>r

Figure I-l: Schematic of SPA photoinjector.

14

        
     

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Electron Energy (Mel)

 

 

Electron Position Relative to Bunch Center (cm)

b)

Figure I-3: a) Schematic of magnetic chicane. b) Energy versus beam bunch length from

simulation.

Chapter 1

BEAM PARAMETERS MEASURED BY BEAM POSITION MONITORS

1.1 Introduction
Before beginning any other discussion, it is important to demonstrate what it is that
BPMs measure. A complete understanding of the signals they produce and how they can

be exploited to obtain important beam parameters is presented in this chapter.

1.2 Notation

Before proceeding, a brief statement conceming notation is in order. Throughout this
document, quantities will be averaged over the electron beam distribution. This will be
done in two ways. The first is called an ensemble average. This is the average of a
quantity over the beam distribution in the coordinate system whose origin corresponds to

the center of the beam. An ensemble average is denoted by angled brackets:

jjjquanfity x p(sz)dV

Origin conesponds to beam center

mas-w ’

Origin corresponds to beam center

 

(quantitY) 5 (14)

where p(x) describes the electron beam’s spatial distribution. By definition

(X)=(y>=(2>=0.

16

17
The second way in which the average of the beam distribution is taken defines the

transverse, or xy, origin to correspond with the center of the beam pipe. The
longitudinal, or z, origin corresponds to the longitudinal center of the beam bunch. This
average is important when describing the signals produced by the BPM. This average

will be referred to as a BPM average and is denoted in the following way:

”Iquantity x p(x)dV

xy origin corresponds to beam pipe center, 2 origin to beam bunch center

WW

xy origin corresponds to beam pipe center, 2 origin to beam bunch ccnte

 

(quantity)BPM a (1-2)

The BPM averages of x and y yield the x and y position of the center of the beam with

respect to the beam pipe and

<Z>BPM = 0-

1.3 Physical description of BPMs

The BPMs employed on SPA are dual axis, capacitive type probes that differentiate
the beam image charge[16]. Physically, they are short sections of beam pipe in which 3.2
mm long slots that subtend an angle of 45° have been cut out of the beam pipe walls.
Metal electrode are inserted into the gaps created by the cutouts. Somewhat smaller than
the slots and electrically isolated from the rest of the BPM, the electrodes are slightly
inset from the pipe wall. They are placed at 90° intervals around the pipe circumference:

two on the x axis and two on the y axis. Figure 1-1 is a schematic of the BPM geometry.

Top Lobe

 

Left Lobe Right Lobe

20. = 45° x

F

 

  

Bottom Lobe

 

Figure 1-1: Schematic of BPM cross section. The beam will travel along the z axis, out

of the page.

19
1.4 Image charge

As an electron beam bunch travels down the z axis (out of the page in Figure 1-1) it is
accompanied by a “wake” of image charge in the beam pipe walls. The distribution of
the image charge depends upon the shape of the beam bunch.

Assuming that the BPM electrodes introduce a negligible perturbation, the image
charge distribution that they see can be well represented by the analytical result obtained
for a smooth metal wall. For a bunch of charge moving at a relativistic velocity along the
z axis of the pipe, this is

r!
n m a —flm|z-2'l
X

c(0,z,t) = $177 ij(r',0’,z',t)Z;an cos[n(0 —0')]ZlT-Te “ d3x'
V n= m= n nm

where

1
a : E’nzo. (1'3)

1, n :t 0
The radius of the beam pipe is a, the xnm’s are the Bessel function zeros and y is the usual

relativistic parameter. The beam bunch has a density distribution function pL(r,9,z, t) in
the lab frame. (See Appendix A)
When the beam is highly relativistic a useful approximation can be made. One

definition of the Dirac delta function is

_lx-X'l
e 8
28

 

6(x — x') a ling

Making the definition

20

 

 

 

 

 

a
8:
YXnm
gives
_|z-2'|
_a__ _Iz—z'l
“Y‘X—mllz-z'l a 6 Wm" a e 8
e a = =2
YX... a m... 28

Yxnm

When 7 is large one can make the approximation

X
-r :"lz-Z'I a

:2

 

 

8(z — 2'). (1-4)

nm

Substituting (1-4) into (1-3) gives

O'(9,Z,t) E 5;; I

area of
Pipe

pL(r',9’,z,t){l+2i(-;-') cos[n(6—9')]}r'dr'd0'. (1-5)

The geometrical term in parentheses is the distribution function for an infinite line charge
inside a cylindrical pipe. (See Appendix B) This is a result of the well known “pan-
caking” effect. When the beam hunch is relativistic, the electric field lines are almost

perpendicular to the direction of motion.

1.5 Model of BPM coupling to image charge

The coupling of the BPM electrodes to the beam image charge can be modeled by the
circuit in Figure 1-2[17]. The image charge is represented by the current, ib. Z6 is the
characteristic impedance of the transmission line and CD is the capacitance of the BPM

electrodes. In the frequency domain, this model is accurate up to approximately 2

GHz[16].

21

Transmission Line

 

 

 

/\

U

K)
O

 

 

 

 

 

Figure 1-2: Circuit model of BPM electrode coupling to electron beam. Cp is the

capacitance of the BPM electrode. Zc is the characteristic impedance of the transmission

line. ib represents the image induced image charge from the passing electron beam.

11): signal

an OSClilOSCI‘P‘

 

short a large
inaccurate rho
ensure that in
modified sight
functions are

Ate) for the c

22
The signal generated by a BPM electrode travels down a transmission line to where

an oscilloscope displays the resulting voltage signal. Because our beam bunches are very
short, a large part of the signal content is at very high frequencies. Since the model is
inaccurate above 2 GHz, low-pass filters are placed on the oscilloscope inputs in order to
ensure that we are far from that regime. The end result is that the original signal is

modified significantly by the time it is displayed on the oscilloscope. Therefore, transfer
functions are assigned in the frequency domain: 0(0)) for the transmission line and

A((o) for the combination low pass filter and oscilloscope.

As already mentioned, the electron beam bunches on SPA are quite short. The
maximum expected full width at half maximum length (FWHM) is 6 mm, or 20 ps.
Under these conditions the beam bunch is essentially a delta function longitudinally for
all frequencies of interest. Taking this into account and using the circuit model in Figure
1-2 and the expression for the image charge given in equation (1-5), the voltage signal

seen by the oscilloscope is

 

area of -ao n=l
Pipe

1 +ao co , n .
vosc(t) = 4? I da' jdz'pL(r',0',z’){2a + 42 (21-) srnnnct cos[n(0 — 90]}

 

 

m afizo
® 3‘6 2n

w

[1 _ 6-1;?) Lam/1(a))"

, ujmtdm , (1‘6)
1+ JCOCPZC

where L is the length of the electrode and 20 its longitudinal position. (See Appendix C)
Without knowledge of the form of the transfer ftmctions 0(0)) and A(co) or the
values for the various constants, the shape of the time signal cannot be predicted.

However this does not present a problem. As will be shown, it is the amplitude of the

 

'1,
nmesumtm-

Mmemh

firmmk

Mumnm;

Al‘xndtx D}

23

time signal that provides the useful information about the electron beam.

1.6 Beam information provided by BPM

The amplitude of the time function in (1-6) is given by

 

1 M °° r' " sinnct
A = Fatima ldz pL(r ,9 ,2 ){2a +421?) n cos[n(0—9 H}. (1-7)
pipe

The term in parentheses can be expanded and converted to Cartesian coordinates. (See

Appendix D) Then, the integral can be taken over each term in (1 -7) yielding

4 .
A = 4:, {2a +—S;flx-(Ycos0 +YSin0)

 

2 sin 2a

2 [((x’ >391“ _ (yz >BPM)coSZ¢ + 2<xy>BPM sin2<|>]

 

a

+ g 8:13“ “()8)an — 3(xy2 >BPM)COS3¢ + (3<x2y>BPM — <y3>BPM)Sin 3¢]

 

+ Okla-j} . (See Appendix D) (1-8)

The x and y coordinates of the beam center with respect to the beam pipe are given by x
and y respectively.
The BPM electrodes are placed every 90 degrees around the circumference of the

beam pipe at 0, 90, 180 and 270 degrees. Substituting these values for 0 into (1-8) gives

 

 

 

4 ' 2 ' 2
A(e.=0)EAR =42, {2a+ 81:“;th 8;”, 010(2),,le —<y’>B,M)

 

estate--3<xy2>..>+ot.al

Altl :

 

/
A 9:
at
A' 9 =
“16 SUbSCnva\

24

4sinoc 2 ' 2
A(9=“)EAL=fiz—{2a‘ a i+ 8:12 “(<XZ>BPM—<y2>BPM)

 

 

 

‘éSiZEa (<x3>BPM —3<Xy2>BPM)+O(;:T)}’ (1-10)

 

 

1r q 4sina_ 23in2a
A(0 :2) EAT 23:7{20H a y- a2 (<X2>BPM -<y2>BPM)

 

4 ' 3 1
“381:3a (3<x2y>BPM -<y3>BPM)+O(_a_4-)} ’ (1-11)

and

 

 

 

31; q 4sina_ 23in20t
A(G=—2_)EAB=41:2 {2a— a y— a2 (<x2>BPM_<y2>BPM)

 

a4

+§Si:3“(s<x2y>...-<y3>...)+0(-‘-l}- W

The subscripts R, L, T, and B identify the right, left, top and bottom lobes respectively

(Figure 1-1). It is easily shown that the charge in the bunch is given by

1:2 (beam size)4
(1:2;(143 +AL+AT+AB) to order a4 , (1'13)

 

and that the centroid locations are given by

 

 

 

 

an AR —-AL (beam size)3
-= t d , 1-14
x a28inct AR+AL oor er a3 ( )
_ at AT —AB to order (beam size)3 (1 15)
= a . . "
y Zsmct AT +AB a3

The last bit of information that can be extracted is the quadrupole moment, given by

2 . 4
a a AR +AL —AT —AB (beam srze)
t d . 1-16
sin2ct AR +AL +AT +AB o or er a4 ( )

 

 

(x2 >BPM — (yz >BPM =

In Appendix I

measure of r

25
In Appendix D it is shown that

(X’lm —(y’),,,, = (fl-(WNW -Y’-
Since i and y are determined by (1-14) and (l-15), equation (1-16) is essentially a

measure of (x2 > — <y2>.

ME

2.1 lntrodu

This err
deems “h?
BPM emitt'd
numerical ch

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22 Emittance

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Chapter 2

MEASURING EMITTANCE WITH BEAM POSITION MONITORS

2.1 Introduction

This chapter will first give a definition of the x and y rms emittances and briefly
discuss why they are important. It will then move on to describe the theory behind the
BPM emittance measurement. Ending the chapter is a valuable discussion of the
numerical characteristics of this diagnostic.

There is a strong tendency for this emittance measurement to be numerically unstable.
If the technique is implemented improperly, the measurement errors in the BPM data are
amplified greatly in the fitting process. The last part of this chapter addresses this

problem and describes how it is avoided.

2.2 Emittance

Each electron in a beam bunch is described by six coordinates: the three transverse
coordinates and their associated momentum. Taken as a whole, the bunch describes a six
dimensional volume. The x and y nns emittances are derived from the projections of this
six dimensional volume onto the x and y phase spaces.

The x phase space is two dimensional with one axis defined by the x position and the

other by the x momentum. The definition of the y phase space is similar. In charged

26

{1331le 11:12

djtergenee. \\

 

plane. It is rei.

where (i and '5
electron mass.
By will essest.
The same It’hr
formulation of

A numeric
accelerator is 5

al'fmges (X:

(3.x2 ~
Thefim three.
4',"
B. s \
E
(1 =3
, a
t
and
/

27
particle beam dynamics, the momentum is replaced by the divergence[6]. The x

divergence, written as x' , is the angle that the electron makes with the z axis in the x
plane. It is related to the x momentum according to the equation

dx dzdx v2
= v = —= ——=
Px ym x Ymdt Ymdt dz me e

 

x' = Bymcx'
where B and y are the usual relativistic parameters, c is the speed of light and m is the
electron mass. If the energy spread within each electron bunch is small, then the value of
By will essentially be a constant within the beam bunch and x' will be equivalent to px.
The same relationship exists between y' and py. Using the divergence facilitates the
formulation of the beam dynamics as an optical, or geometrical, system.

A numerical computation of the x phase space of the SPA beam as it exits the
accelerator is shown in Figure 2-1. The superimposed ellipse is defined by the ensemble

averages (x2 > , (xx') and (x’2 ). The equation for this ellipse can be written as

2 t '2—
Bxx -(XXXX +yxx —1re .

X

The first three constants, or Twiss parameters, are given by

 

 

and

\' (”Hill

I

,
. I
l I
it .
'1’!

.
xi-
‘.-'

4.00 ‘r’

3.00 -»

2.00 -.

x' (mrad)

-1.00 ~

-2.00 -

 

   

28

Simulated x Phase Space of SPA

  

‘ ll. J"
Int-
.3, EE'J
r.’l

    

 

I. I.
-3.00 4» I - '
I
4.00 .L + l t l e a H
4.00 -3.00 -2.00 -l.00 0.00 1.00 2.00 3.00 4.00
x (mm)

Figure 2-1: The x phase space of the SPA beam from simulation.

The) half the
l3 '4 -
II I

The this x en:-

E‘ \ \‘
ltisthe area

giten by

m
"I
.a

“£111ch

and

I"

“here B? is t
comm“ qu;
change in the

nonlinear Proc

ml) eminanch

“fees.

1

till he at a foo

29
They have the property that

BxYx —a: =1'

The rrns x emittance, ax , is given by

 

a, EJ<X2><X'2>—(xx'>2 . (2-1)

It is the area of ellipse divided by the number n. Equivalently, the rrns y emittance is

given by

a, ((y2>(Y">-(yy'>2 - (2-2)

It can be shown that the normalized rrns emittances, defined as

 

s .. 2 Bye. - (2-3)
and
am a Byey (2'4)
where By is the relativistic parameter associated with the average beam energy, are
conserved quantities in a linear focusing system with acceleration[6]. Therefore, any
change in the value of the normalized x and y rms emittances are necessarily due to
nonlinear processes in the beam transport. For this reason, knowledge of the normalized
rrns emittances is a valuable diagnostic. Changes in their value, unforeseen or otherwise,
can indicate misaligned focusing elements, nonlinear interactions and other undesirable
effects.
The normalized rrns emittance is also a valuable measure of beam quality. The beam

will be at a focus in x when the value of ctx is zero. When this is true,

 

I
Substituting 1 _

 

in essence. th-

“iil diverge it

23 Sleasurin

Roger M;
prohefll]. L:
tithout refer:
photoinjector

The trans.
Channel can t.

the end of t}
channeh b)" If

M115. “near

30

 

In essence, the emittance indicates how tightly a beam can be focused and how quickly it

will diverge from that focus.

2.3 Measuring the emittance

Roger Miller et. al. first proposed using BPMs in a non-intercepting emittance
probe[l2]. Later, it was demonstrated that this technique measures the rrns emittance
without reference to the spatial distribution of the beam[19]. Because of the difficulties
photoinjectors present for diagnostics[9], [10], this technique is ideal for SPA.

The transverse motion of a single charged particle traveling down a linear focusing
channel can be characterized by a set of linear equations. The final particle parameters, at
the end of the channel, are related to the initial parameters, at the beginning of the

channel, by the matrix equation

(xf) "R“ Rl2 0 07m
0 0

X; __ R21 R22 xi [6]
Yr 0 0 R33 R34 Yr .

0'” 0 0 R43 RMXYU

L.

 

 

 

 

 

The transfer matrix is determined by mapping the electric and magnetic fields of the
focusing elements. A focusing channel that consists of quadrupole magnets and drifts

will be linear. The section of beam line that is used to measure the emittance on SPA

consists 0t

-—_.
r4

:7
a
F,"

x

Where the

than rel em

Jpn
IN

MI
III

and

31

consists of only these two elements.

It is easily shown that

(x2)f —<y2>f = (Rll)2<x2>i +2R11R12(XX'>i +(R12)2 (x'2>i

‘(R33)2<y2>, ‘2R33R34 <33"), ‘(R34 )2 (VIZ), [12],
where the f subscript refers to the BPM location and the i subscript to the focusing

channel entrance. Changing the transfer matrix m times, where m 2 6, and measuring

(x2 > f — (y2 > f for each change, results in the matrix equation

A:

II
a

(2-5)
where

Ml treat]. PM 4““)(5

(R33)2 L "' [2R33R34 L " [(R34 )2

N
I
~

2

>11
III

9(2'6)

 

 

_[<R..°)2],[2R..ie.i.[(R.;)2]m-[(R;.r]m—[2R.;R. ([0502],

(<x2),\

(XX’>
PK”)
0 >
(W).
( >

yt2i)

“I
III

N

 

 

\

and

32

5'1
111

 

 

t[<x2>. -'<y2>.]m.

Solving for i , in the least squares sense, estimates the rrns beam parameters (x2)i ,

(x'z)i , (xx')i , (y’)i , (y'2>i and (yy'>i. In turn, we can estimate the x and y emittances

at the start of the focusing channel using (2-1) and (2-2).

2.4 Stability of emittance measurement
Because 5 in (2-5) is determined by measurement, there will be errors associated

with it. These errors propagate to the estimate of it. How accurate this estimate will be

depends upon the stability of the matrix equation.

Consider the true value of i as existing at the bottom of a potential well. A
determines the shape of that well. When the stability of the matrix equation is good, the
well is deep with sheer walls and it is difficult for the errors in 5 to move the estimated
value of i far from the true value of it. However, if the stability of the matrix equation
is poor, the well will be very shallow and the errors in b will push the estimated value of
it far from the true value of i .

A matrix equation’s stability is inherent and cannot be improved using clever data

processing techniques. However, stability is also relative. The stability demanded of the

matrix equation depends upon how accurately b is measured and how precisely we need

to lino“ i.

the BPM me

we could act

 

enough to toi'

The quest
equations. so:
is often enou
this problem
finding a st

applicable 10

33

to know i . Because the emittance of the SPA beam is small, the accuracy demanded of
the BPM measurements for typical implementations of Miller’s method was more than
we could achieve. A realization of Miller’s technique was required that was stable
enough to tolerate a BPM’s limited precision.

The question of stability can arise in any diagnostic technique that involves linear
equations, sometimes rendering a clever measurement approach useless. However, there
is often enough flexibility in the way a given measurement can be implemented so that
this problem is avoided. Miller’s technique is such a measurement. The approach for
finding a stable implementation of Miller’s method that is outlined here may be

applicable to other diagnostics.

2.5 Figure of merit for matrix equation stability

In this section, a measure for the stability of a general matrix equation is derived.
This metric will be used in the next section to compare different implementations of
Miller’s emittance measurement.

Consider the general matrix equation,

‘
— ‘

Ar'i = b ,
where A has m rows and n columns, it has dimension n and b has dimension m.

Assume that m 2 n and A has rank n. The least squares solution for it is

(

1 z: :: ::
AT is the pseudo-inverse[20] of A and is denoted by A’.

>11
>11
-1
5'1
ll
>11
5:1

- -1
i TA) (2-6)

>11

(

TR)

|
To. met»;

and the resul

 

normal and 1

elements of i
Th6 mat

damp—1,11 ;,

3'11
11

34

To measure the stability of the matrix equation, a relationship between the errors in b

and the resulting errors in it needs to be established. Assuming that the errors in b are
normal and have value i o , an upper bound on the errors for the estimates of the

elements of it is established.
The matrix A can be factored into three matrices called the singular value
decomposition of A [20]:

‘
A

$6. (2-7)

011

O, is an m by m orthogonal matrix. Like A , E is an m by n matrix. Its first It entries

along the main diagonal are A, , where the A, ’s are the eigenvalues of the matrix

—.
1‘

ATA , and all other elements are zero. (:22 is an n by n orthogonal matrix whose

>11

columns are the n orthonormal eigenvectors, vi , of AT

Using (2-7), the pseudo-inverse of A can be written as

,6?

>11
II

011
M11

Because they are orthogonal, the inverses of O, and O, are their transposes. E’ is the

pseudo-inverse of E and is an n by m matrix with the first n entries along its main

diagonal given by

m

and all other elements zero[20]. Writing Ii as

The cut

hare maenit

%

approximate

35

c‘ ‘

b = b0 + e, ,
where 50 is the ideal value of b and e, is an error vector, the error in it is given by
e, = 6,i+6, a, (2-8)
The exact form of 6,, is unknown. However, the elements of 6b are estimated to

have magnitude 0, the error in the measurements that determine 5. Therefore, the

approximate magnitude of e, is given by

|é,| = ,/é, re, 5 x/mo'z = m/m
and the approximate value of 6,, can be written as

was ,

Ill

éb
where ii is some unit vector that is unknown. Substituting this into (2-8) gives

(st/$6,

M11

Ill

6, *Offi.

Because O, is orthogonal, its transpose will only rotate ii into some other unit

vector, 11' . Therefore,

0756,

M11

I it ' . (2-9)

III

as
Like ii, 11' is unknown. However, it is assumed that the measurement errors in b are

random. Therefore, the elements of ii' , on average, have magnitude 1/ 75 . Recalling

the definition of E’ and O, , (2-9) becomes

“here \

multiplies:

“here )
V.1'1'1111 1‘

II Should I
”are that it <

ch00
Sem
”1 an

36

v12 v22 . . . v"2 iAmTz

:9:
III
Q
a

 

 

 

_vIn v2n . . . vnn_ i 1
k m)»,;

where vij is the jth element of the im eigenvector of IVA. Doing the matrix

multiplication and adding in quadrature gives

e . 50' A (2-10)

where ex). is the error in the jth element of i .

Since the eigenvectors V, are orthonormal, the maximum value of any given element
is 1. Therefore, from (2-10), the maximum magnitude for any particular ex]. is

o
A.

 

It is not unreasonable, then, to define a figure of merit (FOM) to measure the stability of
the matrix equation,
1

V Amin .

is the smallest eigenvalue of 3T; .

FOM =

 

(2-11)

where Ami"
It should be pointed out that the FOM as defined here is only a useful tool if one is

aware that it does depend upon the units chosen for the vector i. For instance, if we

choose mm and mrad for our length and angle units in Miller’s method, we are making

>,
oll
--

2.6

ms;
couv
EX};

the]

37

the choice that an error of :1 mm2 (in (x2)i and <y2>i) is as severe as an error of
:t:1mrad2 (in (x’z)i and <y’2>i) and as severe as anerror of i1 mmmrad (in (xx')i and
<yy')i). If we wish to measure <x2)i and <y2>i to i0.1 mm2 and <x'2>i and <y'2>i to

i 0.1 mrad2 , then this choice of units is appropriate. However, if we wish to determine
(x2)i and (yz)i to i0.l um2 and <x'2>i and <y'2>i to i0.1rad2, then choosing mm and

mrad will cause the FOM to provide inaccurate information when comparing different

implementations of Miller’s method.

2.6 Finding a stable implementation of Miller’s measurement

This section will first present naive attempts at implementing Miller’s emittance
measurement that fail. It will then move on to discuss near singular equations and
coupling, the cornerstones of making Miller’s method stable. At the end, three numerical
examples from SPA are presented and are compared using the figure of merit derived in

the last section.

2. 6.1 Poor implementations of Miller ’3 emittance measurement
Figure 2-2 shows a schematic of a very simple attempt to implement Miller’s
measurement: several BPMs separated by drifis. The number of BPMs, m, is at least six.

The linear transfer matrix for a drifi is

38

 

A

 

 

.matu Eco m5? #:0805308 oocwEEo mo couficofioag How on: :83 mo unmaonom ”N-N 8:me

 

 

 

 

 

 

 

lthrva

 

 

 

 

Enema

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

88802 :oEmom Edam

TEE

T.
til

3501

2711

39

wit
6.
II
Ot—‘OO
~O‘oo

 

OOOF‘
COD-I'D-

 

I— _

where d is the length of the drift. From (2-6), then,

r 1 2d, cl? —1 -2d, —df
1 2d, d; —1 —2c12 —d§
X:
_ 1 2dm dfn —1 —2dm —dfnd

 

 

The second three columns are linearly dependent upon the first three columns and it is

readily obvious that this A is singular and the matrix equation has no solution.
Therefore, the emittance cannot be measured by BPMs separated by drifts.

Figure 2-3 shows a schematic of a second configuration that often already exists in an
accelerator beam line: a single quadrupole magnet followed by a drift and a BPM. The
quadrupole magnet acts like a lens that focuses in one direction and defocuses in the
other. Although a thick lens formula is more accurate, it is assumed that the effect of the
quadrupole can be approximated by a thin lens. The transfer matrix for a thin lens that

focuses in one direction and defocuses in the other is

'1 o o o“
: -%1 O 0
R:
t... 0 o 1 0’
o o %1

 

 

where f is the focal length of the quadrupole. The total transfer matrix of the drift/lens

combination is given by

40

.Emm can and .8:me 2833:: .«o 9:63:00 0:: :83 mo oumaonom

 

- .. . AIIII Sacco

 

 

 

 

 

«2&va
.8552
:oEmom 88m 283630

”3 2am;

mu

[sing {2-6

results in

3.”
ll

2.6.2 Singll/a

Ending a
Ln?

410
[ma] proc

El \‘en
Co r
“figUl

41

p—

 

 

1-% d o o
: = _—. —1f 1 o 0
R”‘=R"R'““= 0 0 ”Cyf d'
0 o y, 1

Using (2-6) and varying the focusing strength of the quadrupole m times, where m 2 6,

-(1—%): 2d(1—%) c12 —(1+%.): _2d(1+%1) -d2_
(1%) 2.10%) d2 (1%) 441%) _,2

2

 

 

. . . . 2 . .

4‘) {—dJZ—(dl—(dJ—z
-0 A“ 2d] A“ d ”A“! 2d ”A“ d -
Because column three and column six are linearly dependent, this matrix is also singular.

If the thick lens formula is used to describe the quadrupole magnet, A in the second
example would not be singular. However, it would be close to singular and the matrix
equation would be very unstable. What these two examples illustrate is that stable
configurations of Miller’s measurement do not necessarily occur naturally. Typically,

they must be searched out.

2. 6.2 Singular equations and strong coupling

Finding a stable implementation of Miller’s emittance measurement is a somewhat
informal process. Although the figure of merit, (2-11), will indicate whether or not a
given configuration is good enough, it does not disclose how a stable implementation is

to be found. That task requires some trial and error.

There a
first has 33:
consider \\
Miller's to
3110“ trans
R) R:; an.

R3: 31’? zen

3’”

42

There are two aspects to consider when searching for a stable matrix equation. The

first has already been discussed: avoid nearly singular matrix equations. To do this, first

consider what control is available over the elements of the matrix A. For instance, in
Miller’s technique assume that we can implement the measurement in such a way as to
allow transfer matrices with the following four characteristics: in the first, the elements
Rn, R33 and R3,, are zero, in the second, R”, R33 and R34 are zero, in the third, R“, Rl2 and

R3, are zero and in the fourth, R11, Ru and R33 are zero. Substituting these into (26) gives

q

r[(Rrr)2]l 0 0 0 0 O
o o [(R,,)’]2 o 0 o
3: Z Z O -[(R(:3) L 2 -.(R0 2‘ . (2-12)

[11ml],[2R.,R..1.[(R.)2]5[(R..)]-[2R..R..5a.):

_[(R11)2]6[2R11R12]6[(R12)2] [—(R33) 26] -[2R,,R,, -:(R34)

The first four rows are perpendicular to each other. The settings for the last two rows are

:5

 

 

 

 

6 .6

—I

now easily chosen to avoid a singular matrix.

The second aspect to be considered when searching for a stable matrix equation is the
coupling of the desired parameters. Good coupling is when ASE is much larger than the
error in the measurement of b. For the matrix equation to be sufficiently stable, each of
the initial parameters, (x2), (x'2 > , (xx') , <y2> , <y'2> and <yy') , must couple strongly
to the final value of (x2 > — <y2> in at least one of the measurements.

Consider the motion of an electron in the fields of a quadrupole magnet. As it passes

through the magnetic field of the quadrupole, the strength of the field that the electron

La

IF.
it;

43
sees depends upon the electron’s position in the magnet aperture. The resulting change in

the electron’s trajectory will be strongly dependent upon its initial position in the

quadrupole field and much less dependent upon its initial direction, or divergence.
Therefore, the final value of (x2 ) — <y2> will depend very little upon the initial values of
<x’2> and <y'2> but very much on the initial values of (x2) and <y2>. A quadrupole
magnet will provide strong coupling between the initial values (x2) and <y2> , and the
final value of (x2 > — <y2>.
The transfer matrix for a drift of length d has
Rn=Ru=d.
R12 and R 34 in the total transfer matrix of the beam line determine the coupling of the

initial values (x'z) and (y'2 ). Therefore, long drifts tend to amplify the coupling of the

initial values of <x’2> and <y'2> to the final value of (x2 > — <y2> .

2. 6.3 Stable implementation of Miller ’s measurement

Taking into consideration near singular matrices and coupling, an excellent
implementation of Miller’s emittance measurement is that shown as a schematic in Figure
2-4: a triplet followed by a BPM. The triplet consists of three quadrupoles separated by
roughly equal drifts. Having two quadrupoles ensures enough control over the transfer

matrix elements to avoid a nearly singular matrix equation. The third quadrupole adds

another degree of freedom to guarantee satisfactory coupling of the initial values of (x2)

and <y2> to the final value of (x2)— <y2>. Sufficient coupling of the initial values of

8:88 838m 853 a 3 330:8 833 a .8 wfiuflmaoo on: :83 .8 £228:an ”arm 0.8me

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

All Me Ilv Allan llv ALe IV
4 w J 1 a
no No 8 1| :aoe o
k f k F k
.8882
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examples

They;

and
The quad.

beam em

“'11 UQS of

and

45

(x'z) and <y'2> to the final value of (x2)— <y2> is assured by making the drift lengths

between the quadrupoles long enough.

2. 6.4 Numerical examples
To demonstrate the effectiveness of the triplet configuration, three numerical

examples using the parameters from SPA are presented and compared.

The values for the drifts in Figure 2-4 are

(11 = 83.1 mm,

(12 = 326.75 mm,

(13 = 425.2 mm
and

cl4 = 236.6 mm.

The quadrupole magnets are electromagnetic and have an effective length of 86 mm. The
beam energy is 8 MeV. From simulations of SPA with the code PARMELA[21], the

values of the beam parameters at the end of the accelerator for a 3 nC beam are
(x2) = 7.05 mm2,
(xx') = —3.15 mm mrad ,
(x'z) = 152 mradz,
<y2> = 050 mm2 ,
<yy') = —O59 mm mrad

and

1th

33’;

cl

1'6

46

(y'z) = 229 mradz.

The emittance can be calculated from the parameters using (2-]) and (2-2) giving
ex = 1.25 1: mmmrad ,

and

8), = 0.892 1: mm mrad.

It is expected that the BPM will measure the value of (x2 ) — <y2> to o = i05 mmz.

Six quadrupole settings will be used in each example. According to simulation, each
setting transports the beam past the BPM location. The transfer matrices are calculated
using the linear beam transport code Trace3D[22].

The first example is a redux of Figure 2-2 using only the last quadrupole in Figure
2-4. This time, however, the proper transfer matrix for the quadrupole magnet is used,
not the thin lens approximation. The second example employs all three quadrupoles in
Figure 2-4 with random field strengths. In the third example the matrix is made stable
using the concepts discussed previously.

In example one, three arbitrary values for the current in the third quadrupole are

chosen. The signs of the three currents are then reversed to get to six settings. A typical

result is
_ 3.6529 1.3 725 0.1289 - 0.0260 — 0.0927 - 0.0828‘
2.7403 1.1553 0.1218 — 0.1466 — 0.2275 — 0.0882
= 1.9742 0.9526 0.1 149 — 0.3720 - 0.3739 — 0.0940
A = 0.0260 0.0927 0.0828 — 3.6529 — 1.3725 — 0.1289 '
0.1466 02275 0.0882 — 2.7403 — 1.1553 — 0.1218
0.3720 0.3739 0.0940 — 1.9742 - 0.9526 — 0.1 149_

 

 

I-..
1“
CEJLMJ'.

Tile exp

and

ObViOus
COmPlEit
313m Oxj n

Fort

leagr QHC

T.”

47

Calculating the eigenvectors and eigenvalues of AT}: and using (2-11) results in

 

1 4
- 3.05x10 .

M-

The expected errors in the estimated values for the beam parameters are, from (2-10),

FOM =

err(x,) = $190 mm2 ,
errm.) = :20 x 103 mm mrad ,

err(x,,) = :11 x 10" mrad2 ,
err , = $190 mm2 ,
(Y)

err , =i2.0x103 mmmrad
(yr)
and

err(y,,> = $1.1 x 104 mrad2 .

Obviously, when compared to the actual values of the beam parameters, these errors are
completely unacceptable. Also note that the magnitudes of the two largest errors are
approximately 6 times the FOM.

For the second example, six settings were chosen at random, using all three quads at

least once. A typical result is

P 0.0104 — 0.1505 0.5419 — 4.6088 — 8.3564 — 3.78791
0.0191 0.3555 1.6529 - 0.6651 — 1.3695 - 0.7049
7.5388 8.5015 2.3968 — 0.4586 15146 — 1.2506
05010 — 0.3159 0.4098 — 0.1028 — 1.4157 - 4.8723 '
0.0550 0.2471 0.2776 — 3.3319 — 8.0093 — 4.8132
3.6529 8.7298 5.2157 — 0.0260 — 0.1450 - 0.2025

>11
II

 

 

 

 

“f1

3:

ll:

48

Again calculating the eigenvectors and eigenvalues of AT; , (2-11) yields
FOM = 3.0.

From (2-10), the expected errors in the parameter estimates are

err<x,> = i071 mm2 ,
errm.> = 3:053 mm mrad ,
err<x,,) = :04] mrad2 ,
err , = i105 m2,
0 )
err , = $0.66 mm mrad
(yr >

and

err ,, =:l:0.33 mradz.
(y )

This is much better than the first example. However, the errors are still bigger than some

of the beam parameters. Again, note that the largest expected error is approximately 0
times the FOM.

In the third example, the six settings were carefully chosen to make the matrix stable.
In this case, the matrix A is

P 0.0000 0.0000 4.7779 — 0.0454 0.0000 0.0000 ,
0.0454 0.0000 0.0000 0.0000 0.0000 — 4.7779
0.0178 0.0000 0.0000 — 8.9987 0.0000 0.0000
8.9987 0.0000 0.0000 — 0.0178 0.0000 0.0000
0.0003 0.0096 0.0887 — 4.3470 - 102809 — 6.0787
4.3470 10.2809 6.0787 0.0003 0.0096 0.0887 _

>11
ll

 

 

b

Calculating the eigenvectors and eigenvalues of AT; , (2-11) yields

From 12-
e
e
e
8:
er

and
er

Thfse CIT

again. the

In 111
meaSUIErr
the ratio (

FOM for l

49
F OM = 025.

From (2-10), the expected errors in the estimated parameters are

x2

err< ) = $0.056 mm2 ,
cum.) = i0.070 mm mrad ,

err< > = i0.105 mrad2 ,

x12

err , = 350.056 mm2 ,
(y )

err , = £0070 mm mrad
(yr )

and

err ,, =i0.105 mradz.
(y >

These errors are very reasonable and much better than the previous two examples. Once
again, the magnitudes of the two largest errors are approximately 0' times the FOM.

In the examples, the FOM, together with the expected error in the BPM
measurements, provides an accurate indicator of the maximum expected error. Therefore,
the ratio of the FOMs is a good comparison of each implementation. Taking the ratio of
FOM for examples one and three gives

FOMr-zx.r s
FOMEH — 1.22 x10 .

The ratio for examples two and three gives

FOMEXJ =120
FOMEL3 ° °

So, although the BPM measurements are equally accurate, the estimates of (x2 > , (x'2 > ,

50

(xx') , <y2> , (y'z) and <yy') from the third example are an order of magnitude better

than the estimates from the second example and five orders of magnitude better than the
estimates fiom the first. This is a significant improvement.

It should be noted that the nature of the matrix equation is not the only factor that
limits the accuracy of Miller’s emittance measurement. 1 do not want to imply that its
accuracy can be improved without bound by rearranging the quadrupoles or increasing
their strength. Quadrupole settings that cause the beam to intercept the beam pipe walls,
chromatic aberrations due to energy spread in the beam, imperfections in the quadrupole
fields, space charge forces and other factors also limit the performance of Miller’s
measurement. The matrix stability is only one facet of the problem.

There are many measurements that require that a set of linear equations be solved. If
this question of stability is an issue, considerable gains can often be realized if the
problem has sufficient flexibility. Techniques that might seem hopeless can be salvaged

and the accuracy of proven measurements can be made better.

.‘lEA

 

3.1 lntrodl
In this
electron hp,
Principle be
IMHO mod;
Varying 01.1
and the Phi:
deilccred ll‘.
unchanged.
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lenglh 01th

0cm; in ma},
quadmpgle n

 

Chapter 3
MEASURING THE RMS PULSE LENGTH WITH FAST DEF LECTING

CAVITY AND BEAM POSITION MONITOR

3.1 Introduction

In this chapter a technique is describe for measuring the rms length of the SPA
electron beam bunches using an rf cavity and a BPM. Figure 3-1 demonstrates the
principle behind the measurement. The fast deflecting cavity is designed to operate in a
TM”o mode. In this mode the electric and magnetic fields are dominated by a time
varying, magnetic dipole field. When an electron beam bunch passes through the cavity
and the phase of the cavity fields is set properly, the front and tail of the bunch are
deflected in opposite directions while the trajectory of the bunch center remains
unchanged. This “streaking” effect causes the beam width to increase as the bunch drifts
away from the cavity. The rate of the width increase is dependent upon the longitudinal
length of the beam pulse. If the cavity fields are orientated properly, this increase will

occur in mainly one plane, x or y. Measuring the position of the beam center and its
quadrupole moment, (x2 >an — (y2 >BPM , with the BPM when the deflecting cavity is on
and when it is off determines the rrns length of the beam bunch.

The rrns length of the beam bunch is defined as the ensemble average (22). It will

prove convenient to replace the longitudinal coordinate, z, with a phase angle, 41,. This
51

‘

>223 matrix: : we :25: 53.5.3
~35: A: :99: 232.495.? 3:: 53.9.3.
874: 0.2.42.4.22LC 28:71.2; 02:5

52

.855 Enos m map—«ohm .038 882.88 “mam 8 88.33: ”Tm onE

$.60 882.80 5mm

1 3

 

 

 

 
 

 

 

 

 

 

 

88m 88m

one: x a

a 888m 58m 88m f / \ k
<

538 88253 x mm 855 Enos .8
888 3 zoom acreage Ea fimaobm
Bow 38:me .8 8an8% 08C.

 

 

phase an;

c3111}: T

The freq;

3.2 Trait

To 5:.
relativism
Calculalio

The 1':
PICSemed
is a lime ,

0f mOIlon
3'2" F18].

The fa

is ShO“'11 i

and

53
phase angle is referenced to the wavelength of the rf frequency of the fast deflecting

cavity. The rms length can then be written as

<¢:>=[-§§)2<zz>=[31‘5)2<22>. (3-1)

C

The frequency of the cavity is f and the speed of light is c.

3.2 Trajectory change of a single electron at fast deflector exit

To start off the discussion, this section presents the change in trajectory of a single,
relativistic electron as it travels through the fields of the fast deflector cavity. This
calculation is to first order.

The results of this section are essentially a summary of a more complete derivation
presented in Appendix E. It begins by demonstrating that the dominant field in the cavity
is a time varying magnetic dipole field. Then, using only this field, the simple equation

of motion is solved in the appropriate coordinates.

3.2.1 Field components
The fast deflector is a cylindrical cavity that operates in a TM“o mode. A schematic

is shown in Figure 3-2. The ideal electric and magnetic fields are

132 = E0J1(k”r)cosecos(cot +¢) 9
2
B =wfiEoJKknrfiinOsinbt‘tl’)

1'
11

and

 

 

 

//\

54

.538 882.88 “mm.“ 8 otmEonom ”mrm PEEL

 

 

A

 

 

CD

 

 

 

 

 

All other

there x
of the cat

(3.)

Where the

Thfift‘forc

“16 160g:
The mini:
kinellc Cnr
apeIIUre O

or 1.27 CIT

3'2'2 (Om

Using 1

-n
l/

55

B6 = a) fiEOJKknrfiosO sin(o)t + 11)).

11

All other field components are zero. The constant kll is defined as

1(11

x11
3

where xu , equal to 3.8317, is the first zero of the first Bessel function and a is the radius
of the cavity. The angular frequency, (1), is given by

or = 27tf
where the frequency, f, is 1300 MHz for our cavity. The relationship between the angular
frequency and the geometrical properties of the cavity is

0) = knc.
Therefore, the cavity radius is

a—i‘iE—flg-OMrn
_ co -21tf-' °

The length of the cavity, L, is independent of frequency and on SPA measures 14.48 cm.
The maximum amplitude of the electric field, E0, is 24 MV/m, from measurement. The
kinetic energy of the electrons as they enter the cavity will be approximately 8 MeV. The
aperture of the cavity, the opening that the beam travels through, is one inch in diameter,

or 1.27 cm in radius.

3.2.2 Conversion of field componena' to Cartesian coordinates
Using the relationships

f: xcose + ysin0

and

13-
the clectr:
E.
B
and
B I
SlflCC [he
dfflct‘tor t“.
El
Bl
and
Bi. .
3'2'3 Equal

ofthe r3110,

 

56

and
0: -xsin0 + ycost) ,
the electric and magnetic fields can be converted to Cartesian coordinates, giving

E = EOJl(k“r)cosecos(wt + (1)) ,

Z

Bx = 3(02 J2(k“r)E0 sin208in(o)t+¢)
xnc

 

and

By = $E0[fi;ll(knr)— J;,_(k”r)cos2 0] sin(ort + (it).

Since the beam is limited to the transverse region defined by the aperture of the fast

deflector cavity, these expression are well approximated by

Er;

EEEO2

xcos(cot + (h) ,

(0x11
Bx E 2
4ac

 

ony sin((ot + 111)
and

2 5:281:73, sin(cot + (1)). (See Appendix E) (3'2)
11

3. 2.3 Equations of motion

Of the three field components, By is dominant. In the aperture region, the magnitude

ofthe ratio of B, to By is

11

W

 

 

showing t
show to]
8 MeV b
alloxxingl

10 fir

and

“here 2

This eq

% 11 mg

and

57

(0X11

 

 

 

 

 

 

B, 3,02 onmaxymax xfirjm (3.8317)2(1.27cm)2
s = , = , =0.03,
lByl ma E 2a 4(14.0cm)
2xnc2 0

showing that Bx can be ignored to first order. The electric field along the axis can be
shown to produce a relative change in an electron’s energy of less than four percent for an
8 MeV beam. (See Appendix E) In most cases it will be much smaller than this,
allowing the electric field to be ignored as well.

To first order, then, the only equation of motion that is of consequence is

" - eZ—(BE-E sin((ot + 111)
m ZXHCZ 0 °

Using the relationships

and

t = 133’ (33)

where 2 equal to zero is defined as the entrance to the fast deflector cavity, gives

x"- e coa E sin[9-z-+¢)
—BYmc 2x11C2 0 BC .

 

This equation can easily be integrated to give the divergence and position of the electron

as it moves through the fast deflector cavity:

6 3 (DZ
X'(Z) = —-'Y;EX172_EO COS[B: + 4)) + Cl (3'4)

and

 

“here

and

 

 

 

The taluc:
The chap.é

can be for

and

It Ml

Quit}. fig
10 be mm

the bunch

AS m:

 

e a [3c , (oz

X(Z) = —RZXITEo—QTSIH[E+¢) +CIZ+CZ, (3-5)

where
' e a

C, = Xi +7m—m2—EO COS¢ (3-6)

and
+ e a E 5° ' 3 7)
=x. — -—— , -
C2 I szxllcz 0 (0 81nd) (

The values of x and x' at the entrance to the fast deflector are xi and x; , respectively.

The change in the electron’s trajectory caused by the action of the fast deflector cavity

can be found by setting 2 equal to L, the length of the cavity, in (3-4) and (3-5), resulting

in
e a Bc , (9L
XL =X(L)=——YI;WEo-Eo_sm(fig+¢)+C1L+c2 (3-8)
and
e a coL
XL = X'(L) = —-y;n—'2—x;:2—Eo COS(BC_+¢] + Cl . (3-9)

It will prove useful to rewrite the phase angle, (11, as
41 = «l. + Al + «1..
The angle (110 + A111 is defined as the phase of the beam bunch center with respect to the
cavit)l fields. The angle Ad) is included for calibration purposes. Its magnitude is defined
to be much less than one. The angle (11, is defined as the phase of a particular electron in
the bunch with respect to 410 + All) .

As mentioned in Chapter 1, the beam bunches on SPA are 6 mm FWHM in length or

 

less. 11.:

1 will ht

“18801;

[I

Suhsrltulir

aPPIOXims

and

 

 

 

 

3'3 ”tag",
After fix
plpe 10 [he ]

Chime} 50 U

59
less. This is a time duration of 20 ps. At the frequency 1300MHz, then, the magnitude of

02 will be
l¢.| S (10 ps)(2rrf) = 0.082 .
Therefore,
IA¢ + 02' << 1.
Substituting (3'6) and (3'7) into (3'3) and (3-9), this condition makes the following

approximation possible:

xL = x +fo +-Ye;XEé-;Eo%g{sin¢o +%cos¢o —sir{%:1+¢0)
+(A¢+¢z)|:cos¢o -%::sin¢0 —cos(%+¢0]]}, (3-10)
and
x' 2x: +-— e a 213,,{c05111 —cos(——+¢0)
L — szx xcll 0
, (0L , .
+ (A4) + ¢Z)[srn(B(-:- + 410] — 81ml)0 J}. (See Appendix B) (3-1 1)

3.3 Measuring the rms length of the electron beam bunches
After exiting the fast deflector, the electron beam will travel down a section of beam
pipe to the location of the BPM. It will be assumed that this region is a linear focusing

channel so that a linear transfer matrix can be assigned,

[sing R

 

eiectronh
1011-141;

in Figure

3.3.1 Fas.
“hen
Therefore

the BPM

xi!

”11
ll

Using It , and equations (3-10) and (3-11), it can be shown that the rms length of the

electron beam bunches can be determined by measuring the values of i and y , according
to (1-14) and (1-15), and (x2>BPM — <y2>BPM , according to equation (1-16), with the BPM

in Figure 3-1 when the fast deflector is on and when it is off.

 

PR1] R12 0
R21 R22 0

0 R33
_ 0 0 R43

3. 3.1 Fast deflector off

When the fast deflector cavity is turned ofi‘, it is nothing more than a section of drift.

Therefore, the total transfer matrix between the entrance to the cavity and the location of

the BPM is

”11
-1

”It
-1

Therefore, the final parameters of a single electron at the location of the BPM will be

 

FRI] R12
= R21 R22
0 0 11,,
_ 0 0 R43
R11 R12 + L1(11
_ R21 R22 +1"‘1121
' 0 0
L 0 0

 

60

 

 

0T1L0
0010
R3400]
R,,__000
0 0
0 0

R33 R34 +LR33 .

 

R43 R44 + LR34,

--t-'oo

 

 

and

The BM

locailglx

and

61

 

 

xf pR” Rl2 -l-LRll 0 0 7x,
x; R2, R22 + LR2| 0 0 x;
Yr _ O 0 R33 R34 + LR33 Yr .
Yi case, _ 0 0 R43 R44 + LR34AY1'

Doing the matrix multiplication yields

(x,)mfi = Rnxi +(R12 + LR,,)x; (3-12)
and

(y,)mflr = R3,)», +(R,3 + LR,,)y;. (3-13)

The BPM in Figure 3-1 will be used to measure x, 7 and (x2 >am — (y2 >BPM at the BPM

location. That is,

(Etienne = «sheer-hm, (3-14)
(Valera.r = (00.0%),”, (3-15)
and ,
(txiiaai... -<(y%)....>.... = (tuna-(tyne...)
+002)... —(rr)i..er- (346)
3. 3.2 Fast deflector on

When the fast deflector cavity is on, the values of x, y and (x2 )3,“ " (y2>3p~1 will

change. To calculate that change, start by noticing that, according to the approximate
equations of motion, the cavity fields will have no effect in the y plane of motion. The

fast deflector is still a drift of length L. Therefore, the results for the y center of the beam

 

GI
13

Therefore

in the

the 10*:th

Making 11

62

are identical to those for the first case,

(3%)“)0“ = R33Yi +(R33 + LR34 )Yi

Ol‘

(Yr)FD0n =(Yr)FDOff- (3-17)

Therefore, the BPM will measure

(mm, = (7.)“,0... (3-18)

In the x plane, we must use equations (3-10) and (3-11). The values of x and x' at

the location of the BPM are given by

(xi) _ [R11 R12 ](XL]
Xi R21 R22 xi .
Doing the matrix multiplication yields

' e a BC , c0L
(xf)FDOn = Rllxi +(R12 +LRll)xi +fo—“EE'E0{R11 '0:[Sm¢o +Ecos¢o

_ sin(%:; + 410)] + R,2[cos¢0 - cos(-(E%- + ¢0)] +(A¢1 + ¢z){R11 %[cos<j>0
_—(;%sin¢o —cos(%+¢o]]+ R,2[sin(%+¢o) —sin¢0]}}. (3—19)

Making the definitions

ad

 

nmphhesl

(x,

[Sing first;

(x,

 

I
l

0

meleftilrel

Tofinr

13.17) and

 

R [% j ' 320
+ 12 srn Bc+¢10 —sm(j)0 (- )

and

__e_ a E RB: -¢+% ¢ (93.4,)
32—m2Xllc2 0 ”(0 srn 0 BCcos 0 srn BC 0

+R,2[cos¢10 —cos(%cli+¢0)]}, (3-21)

simplifies (3-19) to
(xf)FDOn = Rnxi +(R12 + LR”)xi' +a,(A¢ +¢z)+a2.
Using equation (3-12) this becomes

x = xf +al A111+1hz +a . (3-22)
( f)n)on ( )FDOff ( ) 2

Therefore, the BPM will measure

(XIX-‘00“ = <(Xf)mon>BPM = <(xf)n)off>8pM +<31A¢>Bm +<al¢Z>BPM +<az>3pM-

Since 412 is proportional to z,

<¢Z>BPM = 0'
Therefore,
(if)FDOn = (2,)mofi + a,A¢ + a,. (3-23)

To find (x2 >am — (y2 )3?” at the BPM location with the fast deflector on, first square

(3-1 7) and (3-22) then subtract the results to obtain

 

 

 

{Sing (3.]

N0“; beCar

64

(x2)... —(yi)...,. =02)... ”termite+tl+2[(x.)....]a.

+a,2(riq>2 +2A¢¢z +¢§)+2a,(A¢ +¢,)a, +a§ —(y§)mm.

Taking the BPM average of this expression yields

<(xiimonirm ‘<(Yi)mo.>,m = <(xiirnoe>,,m ‘<(Yi)rnoe>anr

+ 2a,A¢<(xf)FD0fi>BPM + 2al <(xf)mofi¢z>8m + 2a2 <(Xr)mofi>3m

+312A¢2 +2afA¢<¢z>BPM +af<¢i>8m +23132A¢+23132<¢Z>BPM +ag.

Using (3-14), (3-15) and (3-16), and realizing that

<(xf)morr¢z> = 0

gives

(00%).... -<(Y?)m.>,m = (Wiener-ti -<(y%)reer>+(ir):pea -('y‘r):ear

+2a,A¢(r,)mm +2a,(r,)mfi +afA¢2 +af<¢§)m +2ala2A¢ +a§,
<(x§)FDOn>BpM _ <(yf)m0n>BpM = <(xf)roorr> _<(y§)rporr>

+[(if)m0ff +alA¢+32r "(yfx'non +af<¢i>BPM°

Using (3-18) and (3-23), this becomes

<(xi)FDOn>BpM _ <(yi)FDOn>BPM = <(xf. )roorr> - <(yi)morr>

_ 2 _ 2
+(xf)FDOn _(Yf)1=n0n + a? <¢i>BPM '
NOW, because 11), is a longitudinal coordinate and because the longitudinal origin defined

in the ensemble average, equation (1-1), is the same as that defined in the BPM average,

equation

Therefore-

3-3.3 Men
[Sing

13°34), 11-;-

 

 

 

.\l
and

M ,
Sui‘tractinrt

M ,
harm,

"sf.

Rita” that

bUHChes.

 

65

equation (1-2) , we have

0:)... = <¢i> -

Therefore

<(xi)m0n>3m - <(yi)r00n >BPM = <(xi )roorr> — <(yi )morr>

3.3.3 Measuring bunch length
Using the six measured quantities given in (3-14), (3-15), (3-16), (3-18), (3-23) and
(3-24), we can define the following quantities:
_. 2 2 _ 2 _ 2
MFDO“ = «x1 )roorr >BpM _ <(yf )rnorr >BPM - (Xf)roorr + (”Loon (3'25)

and

Mae. 2 <(xi).,,.,,,>m —<(y%),.,o,,>m —(ir):,,o,, +(rr):,,o,,. (3-26)
Subtracting these two expressions yields

MFDOn _MFDOff = ai<¢i>

Therefore

®9=Mmm;Mmm- can
1

 

Recall that, from (3-1), (412) is equivalent to the rrns length of the electron beam

2

bunches.

 

3.401111
10c.
This is .j
shitting '.
“he:
center 01 _

mum}

B." ”13111::

HT

and

1ilefeltlre.

3'5 [still];

The re.
can be 13111
“Mmm
BPM H0

 

66

3.4 Calibration of measurement

To calibrate this measurement, the value of al , defined in (3-20), must be determined.
This is done easily by measuring the position of the beam in the BPM aperture while
shifting the phase of the fast deflector cavity.

When measuring the bunch length, the phase of the cavity fields is set so that the
center of the beam is very near the center of the BPM. By shifting this phase slightly, the

value of Ad) is changed, steering the x coordinate of the beam according to (3 -23):

(alum!) = (EJFDOE +aIA¢ +a2.

By making measurements of x for two values of A11) we get

a, = (2,)mofi + a,Atll + a, (3-28)
and
x, = (20mm +a,A¢, +a,. (3-29)
Therefore, by measuring x, , i2 and A11)l -— A02 , we have

x1"‘xz

a =———.
l A¢l—A¢2

(3-30)

3.5 Estimate of measurement resolution

The resolution of this measurement depends upon several factors: how much power
can be put into the fast deflector cavity before it breaks down, the accuracy of the BPM
and the nature of the transfer matrix between the fast deflector and the location of the
BPM. However, by making some reasonable assumptions, it is possible to obtain an

estimate of the technique’s resolution.

67
Assume that the region between the fast deflector cavity and the BPM is nothing

more than a drift. Therefore,

and

where d is the length of the drift. From (3-20), then,
e a BC 00L , S(00L ]
a1 - 7m 2xnc2 E°{m [cosdl0 — BC smd)0 —co BC +d)0 :I

. wL .
+ d[sm(Ec- + 00) — 81nd,,” (3-31)

When measuring the beam bunch length, the phase of the fast deflector cavity fields is
set so that the beam is nearly centered in the BPM aperture. This condition is satisfied

when the y magnetic field given by (3-2) is zero when the center of the beam bunch

arrives at the center of the cavity. Using (3-3), (3-2) can be rewritten as

(03 , (DZ
By = 38702-130 Sln(T3; + (1)0)

where Ad) and d)z have been set to zero. At the center of the cavity,

N
II
NH"

Therefore, By is zero when

_ 211:.
(ho—-208

Substituting this into (3-31) gives

e a
a =—
' ym2xuc2 0
e a
e a
a =——
' ym2xuc2°

y =16.63.

kinetic energy of 8 MeV yields

ti
m _

68

“S"‘izli—c XE: 2“;

As mentioned at the start of this chapter, the maximum value of E0 is 24 MV/m. A

a1 = 0.051 x (0.037 meters+ 2.0 x d) .

FWHM = 2.35% .

ED-L- sin(—
[3c

in time yields a compression from 20 ps to less than 1 ps.

31:)
20c _

00L coL

T3?“ 20c

Looking up the other constants and substituting them into (3-32) gives

relationship between the FWHM and the rrns length of the beam bunch is

——li

(3-32)

(3-33)

SPA is capable of compressing the electron beam bunches from 6 mm in length to

less than 0.3 mm in length. These are FWHM measurements. Expressing these distances

The longitudinal distribution of the beam bunches is unknown. However, for the sake

of this estimate, and for simplicity, we will assume that it is Gaussian. Therefore, the

69
Using (3-1) this becomes

((1):) = (L22 X 10-5) X [FWHM (in picoseconds)]2 . (3-34)

Table 3-1 shows the estimated value of af<d)2> using (3-33) and (3-34) for various

2

beam pulse lengths versus the length of the drift after the fast deflector cavity. It is

expected that the BPM will have an accuracy of i 05 m2. Therefore, with a drift length

of two meters, a one picosecond resolution is feasible.

3.6 Efl'ect of BPM rotated with respect to fast deflector cavity

As a final note it should be mentioned that it is not always easy to align the BPM and
fast deflector. Therefore, the BPM will often be rotated slightly with respect to the x and
y axes defined by the cavity fields. As will be shown, as long as the angle of rotation is

small, this is not a significant effect.

3. 6.] Pulse length measurement
If the BPM is rotated as shown in Figure 3-3, then a point in the BPM frame is related
to a point in the fast deflector frame by
xm,M mm = xcosor + y sinor (3-35)
and
yBPM Fm, = —xsinot + ycosor. (3-36)

When the BPM is not rotated with respect to the fast deflector, it was previously

shown in equation (3-27) that the rms pulse length squared is given by

 

70

Table 3-1: Value of af ((1):) vs. drift length, d, and the F WHM pulse length of the beam.

 

FWHM of Beam Pulse (in picoseconds)

 

 

 

 

 

 

d (in meters) 20 10 l 0.1
0.5 13.65 mm2 3.41 mm2 0.0341 mm2 0.000341 mm2
1.0 52.67 mm2 13.17 mm2 0.1317 mm2 0.001317 mm2
1.5 117.1 mm2 29.28 mm? 0.2928 mm2 0.002928 mm2
2.0 206.9 mrnj2 51.73 mm2 0.5173 mm: 0.005173 mmf

 

 

 

 

 

 

71

 

 

 

Figure 3-3: BPM rotated with respect to the x and y axes defined by the fast deflector

fields.

72

 

_ MFDOn "Mr‘norr
<¢:>- a].

where MFDOfr and MFDO" are defined in (3-25) and (3-26) respectively. When the BPM

is rotated, it is easily shown using (3-3 5) and (3-36) that this equation is modified to

 

< 2>_ MFDOn—MFDOff
z af(cosZOL—sin2 or)

The definitions of Mmopf and MFDOn do not change. Therefore, small rotations will

have no significant effect.

3. 6.2 Calibration

The calibration with the rotated BPM proceeds just as before. However, instead of
just measuring the x position of the beam when the phase of the cavity fields is shifted, as
in (3-28) and (3-29), the y position must also be measured. Using (3-35) and (3-36), this

results in the four measurements

it, = (if)FDOflr +(a,Atlrl + a,)cosot, (337)

'y‘, = 6,)an +(a,A¢, +a,)sinot, (3-38)

x, = (if)FDOPf +(a,A¢r2 +a,)cosot (3-39)
and

y, = (mumr +(a,A¢, +a,)eoso. (3-40)

Keeping track of the phase shift Ad)I — Ad)2 , (3-37), (3-38), (3-39) and (3-40) yield

X]—

 

(A11)! -

22
Ad)2 )cosa

 

73

Chapter 4

CALIBRATING A BEAM POSITION MONITOR

4.1 Introduction

In general, a real BPM will have flaws in its construction. The electrodes will not be
identical and the actual image charge distribution will be perturbed from its ideal. For
these reasons, each BPM needs to be calibrated to understand its response and to
demonstrate that we can measure the desired beam quantities with accuracy.

To calibrate a BPM, a thin wire is placed inside its aperture. A current signal on the
wire simulates a relativistic, pencil beam[23]. Since the position of this wire can be
controlled very accurately, the response of the BPM versus wire/beam position can be

mapped. From this map, accurate calibrations of the BPM are established.

4.2 Simulating a relativistic beam

A schematic of the calibration apparatus is shown in Figure 4-1. A thin wire antenna
is threaded down the axis of the BPM. Short sections of beam pipe are attached to either
end of the BPM to maintain the proper boundary conditions. The wire is attached at one
end to the center conductor of a coaxial cable and at the other is soldered to a ball
bearing. The ball bearing is held in place by a magnet on the base of the apparatus,

stretching the wire taut. The position of the wire in the x, y plane is determined by two

74

75

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Coaxial cable -—">
RF Signal 8’ T x
Wire attached to a
center conductor r J
’
BPM W W I Short sections of
beam pipe

 

 

 

 

 

 

-"" Magnet

l/'

Stepper Motors

   

 

 

Figure 4-1: Schematic of pulsed wire apparatus for calibrating BPM.

76

two stepper motors, one on each axis, controlled by a MacintoshTM computer running
LabView© from National Instruments”. To simulate a highly relativistic beam, an rf sine

wave signal is generated on the wire.

4.3 Mapping the BPM

When mapping a BPM, the center of its aperture is found first. Moving the wire
along the x axis in the plus and minus directions until it just makes electrical contact with
the BPM wall locates the x center of the BPM. Likewise, the y center can be found.
Once the wire is positioned in the aperture center, the stepping motors are zeroed and the
response of each of the four electrodes is measured with the rf signal on versus wire
position within the BPM aperture. Typical maps for the electrodes are shown in Figures
4-2 and 4-3. These show the amplitude of the signal induced in the BPM electrodes

versus the position of the wire.

4.4 Fitting the map data
Once the map data has been taken, it can be used to find a calibration for the BPM.

By fitting the proper equations to the map data, calibrations for measuring the position of
the beam center and for measuring the quadrupole moment, (x2 >BPM — (y2 >an , are

found.

77

 

-10.0

a) Right electrode

 

E
a
E
x (m) 10.0
b) Left electrode

Figure 4-2: Maps of the a) right (0 = 0”) and b) left (0 = 180”) electrodes of a BPM.

78

flectrode Signal (mV)

 

a) Top electrode

. ."l\
I' I' Hum, 1.
.‘ \“unr‘t‘uc‘g.

\\ H .
0‘ 0“.‘|““‘uu. ‘

Electrode Signal (mV)

 

b) Bottom electrode

Figure 4—3: Maps of the a) top (0 = 90") and b) bottom (0 = 270“ ) electrodes of a BPM.

79
4.4.1 Centroid calibration

To calibrate the BPM to measure the beam center, the proper equation must be fit to
the map data. Na’r’vely, one might use equations like (1-14) and (1-15). However, this is
unwise. These equations assume an ideal BPM, but as was mentioned at the start of this
chapter, a real BPM will be flawed. The four electrodes will have slightly different
angular widths and each will have a slightly different capacitance, causing each to couple
to the beam differently.

It is simple to derive equations similar to (1-14) and (1-15) that take into account the
differences between the BPM electrodes. When this is done and the results are used to fit
the BPM map data the results are quite good. However, even better results are obtained if

the following equations are used:

x = x0 +s,,Rx +sx,R: +Sxy,RxR: (4-1)
and

y = y0 + SyRy + Sy,R: + Syx, RYR: (4-2)
where x0, Sx , Sx, , Sxy, , yo, Sy , Sy, and Syx, are all fitted constants. The values of RK

and Ry are defined as

AR
RX 5 20Log '11— (4-3)
L
and
Ar
Ry E 20L0g ? . (4-4)
B

This is a strange set of equations and it is not at all obvious why they should be

80

preferred. Their origin comes from the electronics used to process the BPM signals. In
larger machines there are often several hundred BPMs being monitored at all times. In
these situations it proves to be advantageous to do as much processing of the BPM
signals with analog electronics as possible before they are sent to the control system.
There are different schemes for accomplishing this task[24] and one of these naturally
measures the quantities in (4-3) and (4-4)[25], [26]. The expressions in (4-1) and (4-2)
were developed to take advantage of this fact and, as it turns out, give very good results
for the beam position[24], [27].
Fitting (4-1) and (4-2) to the BPM map data results in the following values for the

constants:

x0 = 0.122 mm,

Sx = 0.374 mm,

8‘, = —0.000072 mm,
Sxy2 = 0.00023 mm,
yo = —0.034 mm,

S3' = 0.373 mm,

Sy, = -0.000070 mm

and

SW, = 0.00022 mm.

Figures 4-4, 4-5 and 4-6 show how effective this fit is. Figure 4-4a shows a plot of Rx

versus the x position of the wire. Figure 4-4b shows the same plot but with the x position

81

RK vs. Actual x Position of Wire

 

 

 

-100 -s.0 -o.0 -4.0 ...0.. 00 2.0 4.0 6.0 8.0 1<.0

 

 

 

 

R, vs. x Position of Wire From Fit

mn

Ir Iv

 

l
1
20.0 i
l

I
l0.0 +

o“...

 

 

.100 -8.0 45.0 40 ..fi).. 0 2.0 4.0 6.0 8.0 it .0

' o

 

 

 

 

b)
Figure 4-4: a) R, versus actual x position of wire, b) Rx versus x position of wire as

calculated by fitted equation.

82

Ry vs. Actual y Position of Wire

 

vv-v

 

 

 

 

 

 

 

 

l 0
O
moi ....
l 0"
10.0 . ..
ML fl.
- o 0
31' e; fie i i . i —4r + —— —-——+——————
-100 -8.0 -6.0 -4.0 . .00. 0,0 2.0 4.0 6.0 3.0 l( 0
. . -10.0 1
.g... i
O -200 .1
. O
30.0
y (m)
a)
Ry vs. y Position of Wire From Fit
OE ‘ ‘fi t ——*—*—‘-— -————'
-1 1.0 -8.0 -6 0 -4 0 1c .0

 

 

 

 

2::

1b

5: .

o
r—-———r—— — i

 

y (mm)

M

Figure 4-5: a) Ryversus actual y position of wire, b) Ry versus y position of wire as

calculated by fitted equation.

83

Actual x Position — Computed x Position of Wire vs. Radial Position of Wire

0.4

 

O
0.3 1- O.
0.2 »
0.1 »

i; 00 «0 co.

-0.2 ,

0.0

~ Computed x [111111)

-0.1

 

At 111.11 x

-0.3 —

 

 

-o.4

 

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

Radius (mm)

3)

Actual y Position — Computed y Position of Wire vs. Radial Position of Wire

 

 

ltlunl 1‘ e Computed _1‘ (mm)

-0.3

 

 

-0.4

 

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
Radius (mm)
b)
Figure 4-6: a) Difference between actual x position of wire and that calculated from the
fitted equation versus radial position of the wire. b) Difference between actual y position

of wire and that calculated from the fitted equation versus the radial position of the wire.

0110111.!

101 lilt‘
Values

resolut:

4.4.: Q

T0

igill 5’
the be...

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be deri

 

and

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at 5.,a

 

84
calculated from (4-1) using the fitted constants. Figure 4-5 is the same as Figure 4-4, but

for the y direction. Figure 4-6 shows the difference between the calculated and actual
values of x and y versus the radial position of the wire, demonstrating sub-millimeter

resolution.

4. 4.2 Quadrupole calibration

To calibrate the BPM to measure the quadrupole moment, (x2>BPM -<y2)8m , one

might be tempted to fit equation (1-16) to the BPM map data. As when calibrating for
the beam centroid, however, this would give poor results because of the differences
between the BPM electrodes. An equation that takes into account their differences must
be derived.

The wire used in the apparatus depicted in Figure 4-1 is very thin. It represents a

beam that is essentially a delta function in x and y. Therefore, for a wire positioned at
(x,y) , equations (19), (1-10), (1-11) and (1-12) become

AR = a0R +amx+am(x2 —y2)+a3R(x3 —3xy2)+a4R(x4 —6x2y2 +y")+...+,

2

AL = a0L —ale+a2L(x —y2)—a3L(x3 —3xy2)+a4L(x’ —6x2y2 + y")+...+,

AT = a0T +alTy—a2T(x2 — y2)—a3T(3x2y—y3)+an(x’ —6x2y2 +y")+...+
and

2

AB = a0B —a,By—a23(x -y2)+a3B(3xzy—y3)+a48(x’ —6x2y2 +y4)+...+

where terms out to fourth order have been kept instead of just the first three. The aiR ’s,

aiL ’s, an ’s and aiB ’s are all constants. In (1-9), (1-10), (1-1 1) and (1-12) their values

 

 

were \

electro

There 1

One d0

Ms.

it canh

Ellen h

Where 1

 

85

were written explicitly in terms of the BPM radius and the angular width of the
electrodes. Now, however, it is assumed that the BPM electrodes are not identical.

Therefore
am at aiL at aiT at a,,,.

One does expect, though, that the constants are almost equal. That is

Making the definition

AR+AL—AT-AB

.=. 4-5
Q AR+AL+AT+AB ( )

 

it can be shown that, to fourth order, the quadrupole moment, x2 — y2 for the thin wire, is

given by
x2 —y2 = C0 +C,Q+C,(l—-Q)x+C,(l+Q)y+C,x’ +C,Qx‘ +C6y" +C7Qy"
+C8x2y2 +C9Qx2y2 +C,0(1—Q)x’ +C,,(1—Q)xy2

+ C,2(1+Q)x2y + C,,(1+Q)y3, (4-6)
where the C J. ’s are combinations of the am ’3, aiL ’s, aiT ’s and a“, ’s. Fitting the BPM

map data to (4-6) yields

Co =4.01mm2,
Cl =88.9 mmz,
C, =-0.181mm,
C, = 0.129 mm,

C4 = —0.00013 mm"2 ,

 

and

 

DTObI 31‘.
higher ..

aqua] 1'

 

86
C5 = 0.0029 mrn‘2 ,

C6 = —0.00086 mm‘2 ,
C7 = 0.0024 mrn‘2 ,
C8 = 0.00087 mm“2 ,
C9 = —0.0096 mrn‘2 ,
Clo = —0.0012 mrn’l ,
Cll = 0.0014 mm"1 ,
C12 = —0.0015 mrn‘I
and

C13 = 0.0012 mm".

Figure 4-7a shows the results of using this equation, plotting the difference between
the actual value of x2 — y2 and the value obtained with (4-6) and the fitted constants.
The results are quite good, being correct to within i 0.6 m2. (The wire was moved as
far away from the BPM axis as 9.0 mm. Therefore, the maximum value of x2 — y2 is
81.0 mm2 .) However, the obvious flaw with this equation is that, when making
measurements on a real beam, only x, y and Q, from (4-1), (4-2) and (4-5), will be

known. The higher order terms in (4-6), x4 , y’ etc., are unknown. Therefore, it would
seem that (4-6) is useless when making real beam measurements. We get around this
problem by recognizing that, when the beam is close to the axis of the beam pipe, the

higher order terms are very small. Figure 4-7b is a plot of the difference between the

actual value of x2 - y2 and the value obtained from the fit using the equation

Difference between Actual x2 — y2 and x2 — y2 From Fit vs. Radial Position of
Wire
0.8

 

0.6

Fr 0111

2
_\-

0.4

)

llllllUS V

F11. (1111111)

0.2

0.0 ,
4

-02 ,

7
A ,\._

7

\( [no] \‘

-o.4 .
-0.6

 

 

 

-0.8

 

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
Radius (mm)

a)

Difference between Actual x2 — y2 and x2 - y2 From Fit vs. Radial Position of
Wire
0.8

 

0.6

From

2
‘.\

0.4 .

7

minus .\‘

Fri (111111?)

0.2
0.0

2
’ .‘

-0.2 .

7

lcluol V

-0.4 .
-0.6 .,

 

 

-0.8

 

 

Radius (mm)

b)
Figure 4-7: a) Plot of the difference between actual value of x2 - y2 and its calculated

value using the fit of Equation (4-6) , and b) using Equation (4-7).

 

4 rrrrr

3CCUI

CODSL

4.5 Si

no nit

and [hi

 

 

 

 

88
x2 —y2 = c, +C,Q+C,(l —Q)x+c,(l+Q)y (4-7)

where the constants Co, C, , C2 and C3 are the same as those from (4-6). As can be
seen, the results from (4-7) are comparable to those from (4-6) out to a radius of about
4 mm. Therefore, the quadrupole moment of a real beam can be measured quite
accurately if the beam is kept within a 4 mm radius of the beam pipe center, a reasonable

constraint.

4.5 Simulating a diffuse beam
The are two problems with using a thin wire to calibrate a BPM. First, the wire has

no width. Therefore,

(X’ i - (fl = 0
and the quadrupole moment reduces to

(X’im -<y2>epn = (X’i-(YWY’ -7’ = X’ - 2’
where x and y are the positions of the wire. Second, the current signal on the wire is not
nearly as intense as an actual beam. The BPM maps in Figures 4-2 and 4-3 show that the
amplitude of the signals from the BPM electrodes peak at a little less than 6 mV. The

signal induced by our electron beam will easily be 20 to 100 times greater than this.

Based on the calibration done in the last section, we can expect to measure the
quadrupole moment to i 0.6 mm2 or better. However, when measuring the emittance,
we need to determine (x2 > — <y2> by subtracting 22 — yz away from the quadrupole

moment. How accurately this can be done is hard to judge with the pulsed wire

measl
uegc

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pnnci
scherr
i'OCt'lill

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also k
dismb

dUC to

Where

 

Weigh.

 

89

measurement because (x2 > — <y2> is zero and because the signal is much less than what

we get with an electron beam.

To simulate a diffirse beam where (x2>—<y2> is not zero, we can employ the
principle of superposition. In Figure 4-8 a square grid is superimposed upon the
schematic of a BPM. Using the apparatus in Figure 4-1, the wire is moved to each grid
location and the signals from the BPM electrodes are measured. The principle of
superposition tells us that the sum of the signals recorded for a particular electrode is the
signal one would get from that electrode if all the pencil beams were present
concurrently. By manipulating the size and shape of the grid, the value of (x2 > — <y2> is
changed. By shifting its position, different values for R and y result.

In addition to changing the size and shape of the grid in Figure 4—8, (x2 > — <y2> can
also be modified by assigning a weighting value based upon some superimposed

distribution function to each point in the grid. For example, the signal on the right lobe

due to the diffuse beam that is represented by all the wire positions would be
AR = ZciARi
where AR, is the signal on the right electrode for the wire at the ith position and ci is a

weighting value. The value of ci is determined by the distribution ftmction one chooses

to overlay on the grid of points.

Left ‘.

Figure

PUiSe-d

 

 

 

Left Lobe

  

  

90

Top Lobe

  

Bottom Lobe

  

Right Lobe
x

Figure 4-8: Schematic showing grid points for simulation of diffuse beam with a single

pulsed wire.

4.5.1 I

T0
for art
is sim;
Houel
COHIr'o
arbitra
in Fig

ment

 

 

 

9l

4. 5.1 Interpolation versus measurement

To simulate diffuse beams, the response of the BPM electrodes must be determined
for arbitrary wire positions. With the apparatus in Figure 4-1 this is quite easy. The wire
is simply moved to the desired location and the response of the electrodes is measured.
However, this is also very time consuming, even though the entire process is computer
controlled. A much faster method is to predict the response of the BPM electrodes at
arbitrary wire positions by interpolating between the grid points in the BPM maps shown
in Figures 4-2 and 4-3. To show that this is valid, an experiment was done to compare
the two methods.

In Figure 4-9, two sets of data are shown. Each point in the two plots represents a

single, simulated, diffuse beam. Each beam has for its center x = 0 and 'y' = 0. The

value of (x2)— <y2> is different for each beam and is calculated from the known wire

positions. The grid for each beam consists of 225 wire positions. A Gaussian
distribution is superimposed on each grid. In Figure 4-9a the response of the BPM for
each wire position is measured using the apparatus in Figure 4-1. In Figure 4-9b the
response of the BPM for each wire position is predict by interpolating between the points
of the grid of the BPM maps. (It should be noted that the BPM used for this experiment,
although of the same type, was not the same one whose maps are shown in Figures 4-2
and 4-3.)

Because all the beam centers are at zero, (4-6) says that plotting the value of

(x2 > — (y2 > versus the value of Q, from (4-5), should be a straight line. As can be seen

from the fitted lines in Figures 19a and 19b this is true. Using the fitted line, the

 

 

Figuh
115mg

eleclh

 

 

92

<x2> — <y2> vs. Q, Data From Pulsed Wire Apparatus

A

{D

 

   
 

 

 

 

 

 

 

 

 
   

 

 

2.0 ,
0: i r . 7‘ 9.8L
E -010 -0.08 -0.06 -0.04 -002 2 O0. 0. )8
i; -4.0 ..
i’ -6.0 _.
-8.0 a
19.9
Q (mm’)
a) Slope = 81.7 i 0.21, Intercept = -2.l65 :1: 0.0082 mm2
<x2> — <y2> vs. Q, Data From Interpolation
5.- i
4.0 ,L
l
.2 2.0 1
E: l
v _ 3. - as 1 .
°’ -0 06 oho 0.08 0 10

   
 

 

 

 

Q (m’)

b) Slope = 82.02 i 0.075 ,Intercept = —2.212 i 0.0028 mm2

Figure 4—9: a) Plot of (x2 > — <y2> versus Q for a number of simulated beams (i = y = 0)

using the apparatus in Figure 4-1. b) Identical to a) except that the responses of the BPM

electrodes for the individual grid points are interpolated from BPM maps.

predh

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93

predicted values of (x2 > - <y2> match the actual values to within i 0.1 mm2 for both sets
of data.
The slope of the line in Figure 4-9a is 81.7 $021 and its intercept is

—2.165i0.0082 mmz. In Figure 4-9b the line has slope 8202:0075 and intercept

— 2.212 i 0.0028 mmz. The properties of the two lines are very close, indicating the two
methods, although not identical, produce very similar results. These discrepancies can
largely be attributed to errors in the program that controlled the wire position in the
apparatus of Figure 4—1. This error was later corrected, but access to the equipment was

lost before this experiment could be repeated.

4. 5.2 Check of BPM accuracy for measuring (x2 > — <y2>

As a check to verify that (x2>-<y2>, and the beam center, can be measured
accurately with a BPM, several simulated, diffuse beams were created. Ten of these,
arbitrarily chosen, are shown in Table 4-1. Each beam consists of approximately 121
wire positions. The table shows the actual values of i, y and (x2) —— <y2> , as calculated

from the known wire positions that make up the simulated beam, and the values as
calculated from the signals from the BPM electrodes. The BPM used for this experiment
is the one that was used to measure the emittance in the next Chapter and is the BPM
whose calibration constants are shown in this chapter.

Figure 4-10 is created by taking a single, simulated beam and moving its center radial

outward from the BPM’s center. The value of (x2 > — (yz) is held constant and 121 wire

Table 4-1: Table comparing the values of x, y and (x2 > — <y2> as measured by a BPM

to their actual values for several simulated, diffuse beams.

 

 

Actual x x Measured Actual y 7 Measured Actual (x2 ) .. (yz)

(mm) by BP M (m) by BPM (x2)- (yz) Measured by

(mm) (mm) (nine) BPM (mm’)
0.00 0.05 0.00 0.03 —3.74 —3.78
—l.9l —l.87 —1.29 —1.24 0.75 0.75
2.52 2.60 1.63 1.64 2.23 2.04
0.54 0.60 —3.86 -3.89 —1.27 —0.87
0.14 0.19 —0.99 —0.97 —1.31 —1.37
0.45 0.50 —0.23 -0.20 1.25 1.20
-0.29 —0.25 1.47 1.48 3 .00 3.09
—0.34 0.28 —O.37 —0.37 —12.5 —12.7
1.83 1.97 —0.81 —0.79 6.04 5.83
0.25 0.29 0.05 0.08 —O.94 —O.95

 

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95

positions were used to create each beam. Figure 4-10 is a plot of the difference between

the actual value of (x2) — <y2> and that calculated from the BPM signals. It shows the

expected decrease in accuracy as the radius increases.

Although only a few examples are shown here, numerous simulation of these diffuse

beams indicate that the BPM can measure the value of (x2 > — <y2> to i 0.2 mm2 reliably,

if the conditions are right. That means the beam must be stable, be well centered in the
BPM aperture and the data acquisition system must have sufficient accuracy. Typically
these conditions are not met with SPA, especially the beam stability requirement.

Therefore, when making error estimates, it was generally assumed that the BPM would

have an accuracy of i 0.5 mm2 when measuring (x2 > — <y2>.

4.6 Concerns with calibration method

Using a pulsed wire apparatus like that described here is a very common way to
calibrate BPMs. However, there are some concerns with its efficacy. The rf signal on the
wire is a single frequency sine wave. Our electron beam, on the other hand, consists of a
continuum of frequencies. Obviously, then, there is a possibility that the two signals are
not equivalent. In fact, it is not at all uncommon for the electrical center of the BPM to
change depending upon the frequency of the sine wave used to calibrate it[28]. This calls
into question the accuracy of the pulsed wire calibration.

An ideal calibration scheme would use the electron beam that the BPM is intended to
monitor instead of the pulsed wire. At the moment, it is not clear how this would be done

and in any case, as will be shown in Chapter 5, the SPA electron beam is currently too

expt

mic!

96
unstable for the task. Hopefully, future work will better this stability, allowing for clever

experiments to calibrate BPMs with the electron beam and eliminate any ambiguity that

might exist from the pulsed wire calibration.

~11 twirl liiflll".1|il"i

iii

1 .

ir‘tH' r» 11 I

97

Difference between Actual <xz> - <y2> and Measured <x2> — <y2> vs. Radial
Position of Beam, From Computer Simulated Data
0.45

 

0.40 i O.
0.35 h
0.30 i» O

”’1
o
o

0.25 ~~ .0

(mill

2. 0.20 .. .0

 

Difference. between amino] and measured

 

 

0.00 (196-... t #1 A. . a; a
0 1.0 2.0 . 4. .0 .0 7 0
.005 3 0 0 5 6

Radial Position of Beam (mm)

 

Figure 4-10: Plot of the difference between the actual value of (x2>—<y2> and that

calculated from the BPM signals in mm2 for a single simulated beam as it is move radial

outward from the BPM center.

Chapter 5

EXPERIMENTAL RESULTS

5.1 Introduction

In this chapter are presented the results of beam experiments intended to demonstrate
Miller’s technique for measuring the x and y rrns emittances using a BPM. The chapter
begins by describing the experimental apparatus and the data acquisition system. It then
moves on to describe three different experiments: the first is a check of the electron beam
stability, the second is a verification of the BPM calibration done in Chapter 4 and the

third demonstrates Miller’s emittance measurement.

5.2 Experimental apparatus and data acquisition

This section describes the experimental apparatus and the data acquisition system.

5. 2.1 Experimental apparatus

The experimental apparatus is merely the end section of the SPA beam line shown in
Figure 1-2. A schematic of this end section is shown in Figure 5-1. It consists of
quadrupoles 7 and 8, a steering coil, a BPM, the spectrometer and the drifis between the
magnets. (The fast deflector cavity, because it is not used, is left out of the figure.)

Quadrupoles 7 and 8 are identical. They are electromagnetic, have a pole length of

98

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2.75 inches and a gap radius of one inch. The fields of these quadrupoles have been

simulated and measured. From this it has been determined that their effective length is
86 mm and that, at a radius of 1 cm, the multipole components of the field are less than 1
percent of the quadrupole field. Attached to one pole of each magnet is a small Hall
probe. The Hall probe voltages have been correlated to the gradients of the quadrupole
fields. During beam operation, monitoring these voltages enables us to determine these
gradients to within a percent.

The steering coil is used for positioning the beam inside the BPM aperture. It is
electromagnetic, of a standard design and is capable of deflecting the 8 MeV electron
beam several mrad. Its field has not been characterized.

The BPM is the same BPM whose calibration is discussed in Chapter 4.

The spectrometer is an electromagnetic dipole magnet that bends the beam 90°. Its
edge angles are such that it focuses the beam on the screen shown in Figure 5-1. The
average energy of the beam is determined by adjusting the current of the magnet until the
beam spot is centered on the screen. Measuring the width of the beam spot determines
the energy spread of the beam. The spectrometer has been calibrated so that the average
energy can be determined to within 2 percent accuracy.

The three drifis in Figure 5-1 have the following lengths:

d, = 496 mm ,
d2 = 425 mm
and

d3 =237mm.

101
5. 2.2 Capturing BPM signals

A schematic of the data acquisition system for capturing the signals from the BPM
electrodes is shown in Figure 5-2. The signals from the four BPM electrodes travel down
50 Ohm, coaxial HeliaxTM cables of equal length to 300 MHz, low-pass filters. From
there they go to two, dual channel HPTM 54111D digitizing oscilloscopes. The
oscilloscope digitizers operate at 1 giga-sarnple per second with six bit accuracy. Linked
to the oscillosc0pes via GPIB is a PC running a control program written in LabView©
from National Instruments”. The PC stores the digitized BPM signals in binary files. It
can capture the signals from up to 99 individual beam shots. Each beam shot consists of

several (usually less than 10) beam bunches traveling one after the other.

5. 2.3 Analyzing digitized BPM signals

The digitized BPM signals are read into the IDL© data analysis software from
Research Systems, Inc. Here, each electrode signal is processed with a 250 MHz digital
filter. Utilizing the sampling theorem[29], the signals are filled in by interpolating ten
additional points for each data point taken to obtain a voltage versus time signal like that
shown in Figure 5-3. This trace shows two full beam bunches and part of a third from a
typical beam shot.

After the electrode signals are reconstructed, the peak-to-peak voltage of each beam
bunch is determined. This is defined as the voltage difference between the first negative

peak and the first positive peak of the bunch. The peak-to-peak voltage is only dependent

 

      
 

Hp 541 1 1D
Digitizing Oscilloscope

Hp 54] 1 1D
Digitizing Oscilloscope

    
   
   
     

GPIB Interface
to Computer

  

300 MHz
Low Pass Filter

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5-2: Schematic of data acquisition system for capturing BPM signals.

103

Voltage vs. Time for a Typical Beam Shot

100.0

 

80.0 -~

60.0 ~

40.0 «-

 

 

20.0 -»

 

0.0

 

 

Voltage (m\')

23:23
a 54 l
2604 l
-3.3 54 +
104 +

—2

-18 no -
—1605 -
-1355 a»
~1105 -
-855 L
.4’
-606 -
-3 56 T
~106
lAA
rim

_ 1)

-20.0 4-

-40.0 ..

 

 

450.0 ..

 

 

 

 

m. U u

 

-lO0.0

 

 

Time (as)

Figure 5-3: Voltage versus time signal from BPM electrode for a typical beam shot.

There are two full beam bunches and part of a third displayed.

104
upon the amplitude of the signal and has the property that any dc bias that might be

present is eliminated. Once the peak-to-peak voltages are determined, the values of x , 37

and (x2 > - <y2> for the desired number of beam bunches are determined according to

equations (4-1), (4-2) and (4-7) using the calibration constants determined in Chapter 4.

5.3 Stability experiment

The electron beam from SPA is not very stable beam—shot to beam-shot. This is due,
mainly, to fluctuations in the drive laser beam that strips the electrons from the photo-
cathode. This first experiment is meant as a check of this stability.

The drive laser is located a considerable distance from the accelerator vault. The laser
beam is transported several hundred feet from the laser location to its injection point in
the accelerator beam line via an evacuated pipe using multiple mirrors and lenses. There
are two stability issues associated with the technology used to create the drive laser beam:
amplitude fluctuations and pointing instabilities. The amplitude fluctuations are changes
in the amount of light energy contained in each laser pulse, causing corresponding
amplitude fluctuations in the electron beam. The pointing instabilities are small changes
in the location on the photo-cathode where the drive laser beam strikes. These are due to
small deflections of the laser beam caused by air currents and vibrations in the Optics.
Because of the great distance the laser beam travels, these small deflections are greatly
atnplified. This translates into position fluctuations in the electron beam and, because the
photo-cathode surface does not emit electrons uniformly, amplitude fluctuations as well.

To test how severe the electron beam instability is, two BPM experiments were done.

105
In the first, the stability of the beam shot-to-shot was investigated by measuring the

intensity of the beam (sum of all four BPM electrodes), 35, y and (x2>— <y2> for 99

successive beam shots. In the second experiment, the average stability of the beam was

observed by taking the average intensity, average beam center and average value of

(x2 > — <y2> over 99 successive beam shots. This was done for multiple sets of 99 beam
shots to determine if the average values of these quantities change over time scales on the
order of about one hour.

There are also long term (days) stability questions arising from changes to the photo-

cathode. The photo-cathode material used in SPA is CszTe, a fairly reactive substance.

Although the accelerator is maintained at a pressure on the order of 10’9 Torr, the photo-
cathode surface is slowly contaminated over time. Also, breakdown in the accelerator
cavities ofien results in electrical arcs that damage the photo-cathode surface. The end
result is a slow degradation of the photo-cathode performance. We attempt to neutralize
this effect by replacing the cathode every few months (when possible) and by increasing
the laser power to maintain constant charge levels. However, the effect of photo-cathode

aging on the electron beam has not been investigated.

5. 3.1 Shot-to-shot stability

In the first experiment, the electron beam was first transported to the position of the
BPM in Figure 5-1. To ensure that the majority of the beam that exits the accelerator
arrives at the BPM position, the intensity of the beam at the BPM at the beginning of the

SPA beam line (Figure I-2) was compared to the intensity of the beam at the BPM in

106
Figure 5-1. Using this diagnostic, and adjusting the strength of the focusing elements,

nearly 100 percent transmission of the electron beam from the accelerator exit to the
experimental BPM was achieved. This procedure was followed as a precursor to all the
experiments described in this chapter.

Once good beam transport was established, up to 99 successive beam shot signals
were captured using the data acquisition apparatus shown in Figure 5-2. This was done
without changing the focusing or steering of the beam. For each shot, the BPM signals

were processed to extract x, y, the intensity (sum of the four electrodes) and

(x2 > — <y2>. A typical result is shown in Figures 5-4 and 5-5.

As can be seen in the figures, the shot-to-shot stability of the beam is not very good.

This is especially true of the (x2 > — <y2> measurement, which is the most susceptible to

error. These fluctuations have two possible origins: actual changes to the electron beam
from shot-to-shot and the limited accuracy of the oscilloscope digitizers.

The digitizers operate with 6 bit accuracy and there was some concern that this
limited precision was responsible for the observed beam instability. The rrns scatter that
is introduced by the digitizers can be estimated and compared to the observed scatter.

In Figure 5-4a, the rrns scatter of the beam intensity versus shot number is calculated
to be

0‘ = il7.4 mV.
A simple calculation of the scatter due to the limited accuracy of the digitizers yields

Average Intensity

= 33'- 2Number ofbits = 353-] mV.

 

5 Digitizers

107

Beam Intensity vs. Shot Number

 

 

 

 

 

 

 

 

 

 

 

300.0
250.0.l~
0 o
r. O Q Q . O O .
E 200.00 ’9 . ‘0’. .9. ’0 ~00... \.O~.O 0’ o O”
t. '. $0. °¢.9..OOOOQ.
~; 0 ’ o 0 g «to. . 0 . o.
5 150.0” 9
E 100.0-L
5.5
50.0..
0.0 t . f . . . . .
o 10 20 30 40 50 60 70 80 90
ShotNumber
8)
<x2> - <y2> vs. Shot Number
20.0
15.0«» . O . . O
100 9 0 o
07‘ . .. . 9
E 5.0.. .0 ." o 0 o . ’
N, 0.0~lF §.——— % . . O ; +0; fit
. -50“: 10’ a)” 30 .40 .050 .60. 7.0 0 &.°. 90
'5. ' o
- . . O O O ’ O O .
40.0.- o
0 ~ 0 o
- - O . o
15.0 » , .
-2o.0 .
ShotNumbcr
b)

Figure 5-4: a) Intensity (sum of four BPM electrodes) versus successive beam shots. b)

(x2 > — <y2> versus successive beam shots.

108

x Position of Beam Center vs. Shot Number

 

 

 

 

 

 

 

 

 

 

 

 

 

2.00
1.50.»
1.00..
050+ . .
g 000 W—h—a—k 3 4 H. l .
7 lo°010~ 20 . 30 .40 .‘50 6% .73 ’30. 90
.050 ’0 o o co ’0
400‘... .o.o.0.0'o 0‘ .’ .. . ~ .0 . O o
-150 . . .9 O . ’. O O . O
o
-2.00
ShotNumbcr
a)
y Position of Beam Center vs. Shot Number
2.00
1.50..
1.00.. ° . °
2.. 050”. . ’ .’.... . O:O\O. ..o 0.0.0. .0 .I ’~OO
g 0.00 ~ fir $.13... .+ t t 4 $49 .4
>450}! o 1%. 20. 30 400 .3 ..600 70. q) 90
4.004» " . .
4.50..
.200
ShotNumber
19)

Figure 5-5: a) x position of beam center versus successive beam shots. b) y position of

beam center versus successive beam shots.

109
Assuming the scatter due to the beam and the scatter due to the 6 bit digitizers add in

quadrature, the scatter due to the beam is found to be
68m = $17.2 mV.
This indicates that beam fluctuations are the dominate source of noise for the intensity
measurement.
In Figure 5-5, the rms scatter for the x position of the beam versus shot number is
calculated to be
c = $0.41 mm
and for the y position
6 = $0.38 mm.

The scatter due to the digitizers is calculated to be

 

 

20 2J5
oDigitiRl’S = 11110 2Number ofbits Sx = i0'14 mm
in x and
20 2.5
GDigitizers = 1 mm 2Numbcrofbits Sy = 1130-14 mm

in y. SK and S y are the calibration constants found in Chapter 4. Therefore, the scatter

due to beam fluctuations is
one”, = $0.38 mm
in x and

0'Beam = $0.35 mm

in y.

110

In Figure S-4b, the rrns scatter of (x2) — <y2> versus shot number is calculated to be

o=$62 mmz.

An estimate of the scatter due to the accuracy of the oscilloscope digitizers yields

._ +__C_1____ _ +1 4 2
oDigitizcrs " - 2Numbcrofbits " — - mm a
where CI is the calibration constant determined in Chapter 4. Therefore, the scatter due

to the beam fluctuations is
(5ch = $6.0 mm2 .
The estimates of the scatter due to the 6 bit digitizers are accurate to about a factor of
2 for the four plots. Therefore, it is apparent that the dominate source of noise in these
measurements is beam related. Oscilloscopes with more accurate digitizers would have
been nice, if they had been available. But they are not necessary. The error introduce by

the oscillosc0pes is effectively swamped by beam noise.

5.3.2 Average stability

To check the average stability of the electron beam, a similar experiment was
performed. Again, good transport of the beam to the BPM in Figure 5-1 was achieved
first. In this experiment, sets of up 99 beam shots were captured using the apparatus in

Figure 5-2. Typically, each beam shot contained 5 beam bunches. For each set of 99

shots, the average values of 3:, y , the intensity and (x2 > - <y2> were calculated for each

beam bunch. This was done for successive sets of 99 beam shots without changing the

beam focusing or steering. The time lapse between measurements was 2 to 5 minutes.

111

Because the focusing remains constant, the average value of (x2 > — (y2> should not

change if the beam is stable.

The average stability of the beam was tested several times on different days using
different focusing and different amounts of charge per beam bunch. Figure 5-6 and 5-7
show the results of one of these experiments for a single beam bunch. Comparing these
to Figures 5-4 and 5-5, it is apparent that the average stability is better than the shot-to-
shot stability. However, it is still poor. Figure 5-8 shows the results of a similar
experiment done on a different day at different charge and different focusing. Obviously,
the beam was more erratic when this data was taken than it was for the data shown in
Figures 5-6 and 5-7.

These experiments demonstrate that the stability of the SPA electron beam is not very
good. This adversely impacts the precision with which one can expect to measure the

emittance using Miller’s technique.

5.4 Experiments to check BPM calibration

Using the pulsed wire technique described in Chapter 4, the BPM was calibrated to

measure 2, y and (x2 > — (yz). To check that this calibration is valid for a real electron

beam, the following experiment was performed.

This experiment is very similar to the one described in the previous section. Again,
good transport of the beam to the location of the BPM in Figure 5-1 was first established.
Then, without changing the upstream focusing, the electron beam was steered to different

transverse locations inside the BPM aperture using the steering coil shown in Figure 5-1.

 

112

Average Intensity vs. Measurement Number

700

 

600 -. . .
500 ll ‘ " .

400 ..
300 ~-

200 0

Average Intensity (ml)

100 Ji-

 

 

 

0 2 4 6 8 10

Measurement Number

8)

Average Value of <xz> — <y2> vs. Measurement Number

10.0
9.0 « -
8.0 « .

.7; 7.0 ..
E 6.0. I i

 

T

5.0 «»

9.12:: {ii} {iii

2.0 ..
1.0 0
0.0

 

 

 

0 2 4 6 8 10

Measurement Number

b)
Figure 5-6: a) Average intensity (sum of four BPM electrodes) of beam bunch versus
measurement number. b) Average value of (x2 > — <y2> versus measurement number.

Each measurement is the average over 99 consecutive beam shots.

113

Average x Position of Beam Center vs. Measurement Number

 

 

 

 

 

 

 

 

 

 

1.5 it

1.0 _
E 05 f
:3 00 J I 5 . i 4
:5" 0 I I i 6 8 g
3r. -O.5 »

-1.0 ..

-l.5

Measurement Number
a)
Average y Position of Beam Center vs. Measurement Number

1.5

1.0 ..
E 0.5 ~-
Z 00 a 1. a
g, h i 4 g I 8 § .
4?. -0.5 ~ I 1 I i

-1.0 —.

-1.5

Measurement Number
b)

Figure 5-7: a) Average value of x position of beam bunch center versus measurement
number. b) Average value of y position of beam bunch center versus measurement

number. Each measurement is the average over 99 consecutive beam shots.

114

Average Intensity vs. Measurement Number

 

 

 

 

300
250 ..
7‘ . . . . .
” o ‘ O a t o 0 o o
5 200” o ’ ‘ ‘ o .
E- 150 A
if 100 ..
3:
so .t
0 . . . .
0 5 10 15 20
Measurement Number
80
Average Value of <x2> — <y2> vs. Measurement Number
5.0

 

323% H i ,
this}. T i} . f-
r 4.01! 5 10 15 20
“:21: ll; H I}
4.04»
-5.0

’ )

’)

(mm

 

v‘

 

 

 

Measurement Number

b)
Figure 5-8: a) Another example of the average intensity (sum of four BPM electrodes)

of a beam bunch versus measurement number. b) Another example of the average value

of (x2 > -— (yz) versus measurement number.

115
At each location, the signals from 99 beam shots are captured with the apparatus of

Figure 5-2. Each beam shot consisted of 5 beam bunches. For each beam location, the

average values of SE2 — yz , the quadrupole moment and (x2 > — (y2> were calculated for
each beam bunch.
In Chapter 4, the BPM was calibrated (Equation 4-7) to measure
(lem - (flaw = 09- <y2>+ Y2 - Y2-
Recall from Chapter 1 that this is called the quadrupole moment of the BPM signal. In

this experiment, because the focusing is kept constant, the value of (x2 > — <y2> should
stay constant. (Even though the experiments fiom the last section show that this is not
necessarily true for the SPA electron beam, we will assume that it is for the present

circumstances.) Therefore, the only change in the quadrupole moment is due to the
change in the value of r2 — y? that occurs when the beam is moved to different locations
within the BPM aperture. The value of this term is known because the ability of the BPM
to measure 2 and y accurately has been verified on a similar electron accelerator[16].

Therefore, if the calibration constants in Equation 4-7 are correct, a plot of the quadrupole

moment versus x2 —72 should be a straight line with slope equal to 1.0. Figure 5-9
shows the results of one of these experiments.

The slope of the line in Figure 5-9a is 0.95. This is close to 1.0, suggesting that the
calibration is fairly accurate. Repetitions of this experiment indicate that a slope of 0.95
is typical.

It is tempting to try to correct the calibration so that the small error that is indicated is

eliminated. However, because of the instability of the electron beam, illustrated by Figure

116

2

Quadmpole Moment vs. T — y

 

 

 

-30.0

, ‘)
Quadrupole Moment (mm‘)

 

 

 

 

2 2
<x > — <y > vs. Measurement Number

5.0
4.0 T

213% l l i l i
:35} 3ij 7.} {in}
an. {i} i T

 

(mm?)

2
.\ \

 

 

 

 

 

-3.0 «-
-4.0 ..
-S.0
Measurement Number
b)

Figure 5-9: a) Quadrupole moment versus x2 -y2. The slope of the fit line is

0.95 $ 0.021 . b) (x2 > — <y2> versus measurement number for the same data points shown

in a). Each point is the average of 99 beam shots.

 

117

5-8b and the previous section, it is uncertain how much faith one can put into this
calibration check. Also, this experiment says nothing about the constant term in Equation
4-7. Therefore, no change to the calibration found in Chapter 4 was made.

One other note. This experiment was performed several times and in each case the
data yielded plots very much like that shown in Figure 5-9, even thought the amount of
charge per bunch varied by as much as a factor of 20. This demonstrates the BPM’s
expected insensitivity to charge when measuring the beam position and quadrupole

moment.

5.5 Emittance measurements

The final experiments presented in this chapter are intended to verify that the
emittance of the beam can be measured by Miller’s technique utilizing the BPM and the
beam line components shown in Figure 5-1. The main difficulty is that there is no
independent verification that the emittance numbers obtained this way are, in fact,
correct. As was discussed in the Introduction, one of the main reasons this method is
being pursued is that there are no other satisfactory approaches to measuring the rrns
emittance of photo-injector electron beams. The validity of Miller’s technique must be
inferred by looking at the numerical error estimates and the behavior of the emittance as
beam parameters are changed.

The first part of this section goes through the approach to a single emittance
measurement. This will serve as an example and it can be assumed that all other
emittance measurements are done in the same manner. The second part of this section

shows the results of measuring the emittance of the beam with different amounts of

118

charge per bunch. It is expected that the emittance will increase as the charge is
increased. The third part of this section shows the results of measuring the emittance
with different amounts of magnetic flux on the photo-cathode. It can be shown that the
emittance changes in a predictable way as the strength of the magnetic field in the photo-
cathode region is changed [6], [30]. All of the emittance measurements presented here
were done at an average beam energy of 7.77 MeV with an energy spread of less than 1

percent.

5. 5.1 Single emittance measurement

To measure the emittance with Miller’s technique, the first thing that needed to be
done was to find focusing settings for quadrupoles 7 and 8 in Figure 5-1 that resulted in a
stable matrix equation. This subject was discussed at length in Chapter 2 where it was
concluded that a quadrupole triplet was a very good optics configuration for performing
this measurement. As can be seen in Figure 5-1, however, only two quadrupoles were
used in this experiment. Initially, three quadrupoles were used. However, it became
apparent early on that the emittance of the SPA electron beam was not as good as was
predicted by simulation. (This will be discussed later.) Because of this, the beam size
tended to be bigger than anticipated and there was a substantial problem with the beam
intercepting the beam pipe walls when the triplet configuration was used. To minimize
this problem, just the two quadrupoles were used. Although this does not provide as
good a resolution as the triplet configuration, it proved to be adequate.

Utilizing the ideas presented in Chapter 2, 18 settings for quadrupoles 7 and 8 that

produced a stable matrix equation with sufficient resolution were determined. Added to

119
these 18 settings is one more that is known to transport 100 percent of the electron beam

from the accelerator exit to the experimental BPM. This 19th quadrupole setting allows

one to determine if a significant amount of the beam is intercepting the beam pipe wall

for any of the other 18. The matrix A that results is shown in Figure 5-10. The figure of
merit (FOM), as defined by Equation 2-9, for this matrix is

FOM = 0.30
This shows that these quadrupoles settings result in a matrix equation that is sufficiently
stable.

After choosing the quadrupole settings that produce the matrix in Figure 5-10,
considerable time was spent focusing the beam as it entered the section of beam line
where the emittance measurements occurred (Figure 5-1). The goal was to adjust the
properties of the entering electron beam so no beam was lost in the diagnostic section of
beam line (Figure 5-1) for any of the 19 quadrupole settings chosen for the emittance
measurement. Again, this was done by comparing the intensity of the beam at the BPM at
the start of the SPA beam line to the intensity of the beam at the BPM in Figure 5-1. This
task was complicated by the unstable nature of the beam intensity (Figure 5-6a and Figure
5-8a) and, in the end, was not accomplished with complete success. Very often
measurements were discarded when too much scraping of the beam occurred.

When making an emittance measurement, the quadrupoles in Figure 5-1 were set to
each of the 19 settings in turn and the beam was approximately centered in the BPM
aperture using the steering coil. For each setting, 99 beam shots were captured and stored

on the computer. Each beam shot typically contained 5 beam bunches. Each of these 5

120

- 1.1450 2.9264 1.8697 -0.8651 -2.4034 -1.6693_
8.1 73 5 -0.3 626 0.0040 -0.0073 0.0060 -0.001 3
5.0447 -1.4027 0.0975 -0.0478 -0.1 145 -0.0686
2.8484 -1.7965 02833 02306 — 0.4483 — 02179
0.0033 0.1852 25660 - 0.0260 0.0067 — 0.0004
0.0578 0.7794 2.6251 - 0.0014 — 0.0140 — 0.0360
0.1694 1.3450 2.6699 - 0.0461 — 0.1495 - 0.1212
0.0127 - 0.0032 0.0002 — 9.1505 — 0.1079 — 0.0003
0.0048 0.0208 0.0227 — 6.0285 1.1742 - 0.0572

A = 02427 0.4643 02221 — 3.3388 1.6799 — 0.21 13
0.0289 — 0.0036 0.0001 — 0.0092 02934 — 2.3290
0.0016 0.0150 0.0350 - 0.0078 — 02723 - 2.3836
0.1237 0.3197 02067 — 0.0596 - 0.7340 - 22605
0.0643 — 0.0039 0.0001 — 5.3022 — 12.7231 - 7.6325
0.0184 - 0.0350 0.0167 — 4.6875 — 1 1.3255 — 6.8409
0.0001 - 0.0050 0.0668 — 4.0891 — 9.9595 — 6.0643
6.8308 15.4086 8.6894 — 0.0934 — 0.0209 - 0.0012
6.0439 13.7133 7.7786 — 0.03 81 0.0330 0.0071

_52439 1 1.9836 6.8464 — 0.0052 0.0313 — 0.0469_

 

 

Figure 5-10: A matrix associated with settings for quadrupoles in Figure 5-1 used to

measure the electron beam emittance.

121

bunches were analyzed to find the intensity, 3?, y, and (x2 ) - <y2>. Any beam outside a
4 mm radius of the BPM center was then discarded because the BPM calibration for the
quadrupole moment is not valid in this range (Chapter 4). Of the remaining beam shots,
the average value of the intensity and (x2 > — <y2> was calculated for each quadrupole
setting.

A typical result of such a measurement is shown in Figure 5-11 and Table 5-1. The

charge per bunch was approximately 0.2 nC'. Figure 5-11 is a plot of the average beam

intensity for each of the 19 quadrupole settings. The second column of Table 5-1
presents the average measured value of (x2 > — <y2> for each of the 19 quadrupole settings
for one of the beam bunches. Using this data and the matrix A shown in Figure 5-10,
one can solve for the rms beam parameters as described in Chapter 2, yielding:

(x2) = 3.46 mm2 $ 0.408 mmz,

(xx’) = -351 mm mrad $ 0582 mm mrad ,

<x'2> = 426 mradz $1.004 mradz,
<y2> = 0.95 mm2 $ 0.333 mmz,

(yy') = -0.22 mm mrad :t 0532 mm mrad,

and

<y’2> = 1.42 rnradz $ 0.933 mradz.

The error estimates are obtained from the well known covariance matrix as derived in

Appendix F and assume the measurement errors are normally distributed.

122

Intensity vs. Measurement Number

80

 

70 4.

50 ..

40 ..

Intensity (m\')

30 4»

20 at

 

 

 

0 t : 4 : t : 7 . .
0 2 4 6 8 10 12 14 16 18 20

Measurement Number

Figure 5-11: Average beam intensity (sum of four BPM electrodes) at BPM versus
measurement number for emittance measurement a charge equal to approximately

0.2 nC/bunch'.

123

Table 5-1: Average measured value of (x2 > — <y2> , predicted value of (x2 > — <y2> (from

fit) and the difference between them for each quadrupole setting in emittance

measurement for charge equal to approximately 0.2 nC/bunch'.

 

Measurement Average value Error estimate, Predicted value Difference
Number 2 _ 2 o, for average 2 _ 2 between
0f ()(‘m>m2)< y > value of Of ()(‘m>mz)< y > predicted value
(x2 > _ (Y2) and measured
(m2) value (m2)
1 -2.6 $0.90 -1.0 -1.6
2 23.1 $0.95 29.6 -6.4
3 26.3 $0.82 22.7 3.6
4 23.3 $0.91 16.9 6.3
5 6.6 $0.87 10.3 -3.7
6 8.3 $0.81 8.6 -O.3
7 7.5 $0.89 7.1 0.5
8 -8.6 $0.82 -8.6 0.1
9 -7.1 $0.86 -6.1 -1.0
10 -2.7 $0.77 -3.7 1.0
11 -2.9 $0.76 -3.3 0.3
12 -3.7 $0.83 ~32 -O.5
13 -2.6 $0.81 -2.9 0.3
14 -12.5 $0.77 -12.8 0.3
15 -11.8 $0.85 -11.4 -O.4
16 -10.0 $0.80 -10.0 0.0
17 11.3 $0.82 6.5 4.8
18 11.2 $0.86 5.9 5.3
19 -6.0 $0.85 5.2 -11.1

 

124
Using (2-1) and (2-2), the x and y nns emittances can be calculated, yielding

ex 21.6 mm mrad $0.49 mm mrad
and

2y = 1.1 mm mrad $ 0.36 mm mrad.

Multiplying these number by the relativistic factor By to obtain the normalized emittances
(Equations 2-3 and 24) gives

em = 25.2 mm mrad $ 7.94 mm mrad
and

syn = 18.5 mmmrad$5.89 mmmrad.

The formulas for the error estimates are derived in Appendix F. Again, normally
distributed measurement errors are assumed.

After performing the fit to estimate the rrns beam parameters, one can then use those

estimates to calculate the predicted values of (x2>— <y2> , shown in column 4 of Table

5-1. In column 5 of Table 5-1 is shown the difference between the average measured
value of (x2 > — <y2> and the predicted value. From this column, the value of o, the error

in the measurements, is estimated to be $4.71 mm2 (See Appendix F). This is much
different from the values of o in column 2 of Table 5-1. It is apparent that there is often
poor agreement between the measured and predicted values.

To improve the fit to the data, data points where excessive beam scraping occurred
need to be eliminated. For instance, in Figure 5-11, the intensity for measurement 19 is
obviously much lower than the rest. The electron beam, for whatever reason, changed,

or, more likely, was intercepting the beam pipe walls during this measurement. So, this

125

data point is thrown away. Although less obvious, the same is true for measurements 2, 3

and 5. Now, doing a fit without these data points yields
(x2) = 5.34 mm2 $ 0.250 mm2 ,
(xx') = —4.07 mm mrad $ 0.169 mm mrad ,

<x'2> = 4.37 rnradz $ 0.270 mradz,
<y2> = 0.97 mm2 $ 0.069 mmz,

<yy’) = —0.20 mm mrad $ 0.121mm mrad ,
and
(ya) = 1.41 mrad2 $ 0.194 mradz.

The emittances are now calculated to be:

a, = 2.60 mm mrad $ 0.079 mm mrad

and

8,, = 1.15 mm mrad $ 0.067 mm mrad.
Or,

em = 42.1 mm mrad $ 1.29 mm mrad
and

syn = 18.6 mm mrad $1.08 mm mrad.

The estimated error in these beam parameters is much lower than it was before.

Taking the new estimated rrns beam parameters, one can again predicted the value of
(x2 > — <y2> for each of the measurements kept. The result is Table 5-2. From column 5

the value of o, the error in the measurements, is now estimated to be $ 0.98 mm2 (See

126

Table 5-2: Same as Table 5-1, but measurements 2, 3, 5 and 19 are all discarded because

of beam scraping.

 

Measurement Average value Error estimate, Predicted value Difference
Number of (x2 > _ (Y2) o, for average of (x2 > _ <y2> between
(mmz) v Inc of (mmz) predicted value
(x2 ) _ <y2> and measured
(mmz) value (mm )

1 -2.6 $0.90 -0.3 -2.3

4 23.3 $0.91 23.3 0.0

6 8.3 $0.81 8.7 -0.4

7 7.5 $0.89 7.1 0.4

8 -8.6 $0.82 -9.3 0.7

9 -7.1 $0.86 -5.7 -1.4

10 -2.7 $0.77 -4.0 1.3

11 -2.9 $0.76 -3 .4 0.5

12 -3.7 $0.83 -3.3 -0.4

13 -2.6 $0.81 -2.7 0.1

14 -12.5 $0.77 -12.8 0.3

15 -11.8 $0.85 -11.3 -0.5

16 -10.0 $0.80 -10.3 0.3

17 11.3 $0.82 10.8 0.5

18 11.2 $0.86 10.2 1.0

 

127

Appendix F). This is much closer to the values of 0' given in column 3 than the is

previous value of $ 4.71 mm2 .

It might be argued that there is no reason to expect that the estimated value of o from
the fit be anywhere near the values of o in column 3. Based on the stability experiments
presented in the previous sections it should be much bigger. However, obtaining a
smaller estimated value of c was not the reason that data points were discarded. That was
based solely on concerns that the beam was scraping. The fact that the estimated value of
0' did approach the values of o in column 3 is quite remarkable. In fact, this was a typical

behavior. Based on observation, whatever beam fluctuations that caused the large

changes in the average value of (x2 > — <y2> shown in the stability experiments seemed to

be damped when quadrupoles 7 and 8 were set to their measurement values.
Unfortunately, no experiments were done during this run cycle to verify this

quantitatively

5. 5.2 On the question of beam scraping

The last section presented an example of a fairly typical emittance measurement using
Miller’s method on SPA. However, when analyzing the data certain data points were
discarded to improve the fit to the model. As it turns out, this was also typical.
Sometimes, as was the case with measurement 19 in the example, discarding a data point
was easily justifiable. Obviously, some kind of beam scraping had occurred. However,
as is the case with at least one of the other three data points that were thrown away,

arguing that the beam was intercepting the beam pipe is not always so easy. As can be

128

seen by comparing Figures 5-8a and 5-11, the fluctuations in intensity seen in the
emittance measurement (Figure 5-11) are not so different from the fluctuations in
intensity when no focusing changes are made (Figure 5-8a).

The risk one takes when throwing away “bad” data points is that the fault may not be
in the data, but in the model used to fit it. By throwing out data to make the fit better, one
could be ignoring a physical effect that the model does not take into account.

To verify that this is not true here, Miller’s emittance measurement was simulated
using the particle code PARMELA[21]. First, a beam bunch containing 20,000 particles

and having a Gaussian distribution was generated. Its rms beam parameters were

(x2) = 526 mm2 ,

(xx') = —3.97 mm mrad ,

<x'2> = 4.31 mradz,

<y2> = 1.08 mm2 ,

<yy'> = —0.25 mm mrad ,
and

<y'2> = 1.39 mradz.
The beam bunch has an energy spread of 1 percent and a F WHM length of 3 mm
longitudinally. This is about what is expected of the SPA beam. The rrns beam
parameters were chosen to be close to those measured in the example from the last

section. The charge was assumed to be 1 nC in order to exaggerate the space charge

effect, if any. A Gaussian distribution was used because the point of this exercise is to

129
simulate a diffuse beam, not because that is the expected distribution of the SPA beam.

The distribution of the SPA beam is generally unknown.

PARMELA takes this beam bunch and moves the particles down a simulation of the
beam line shown in Figure 5-1 and includes non-linear effects as it does so. The
simulation was run twice, one with a boundary where the real beam pipe should be and
one without. When a particle in the first simulation “hit” the beam pipe, it was discarded.

After running both simulations for each of the 19 quadrupole settings, the value of

(x2>— <y2> was calculated for each. Also, for the first simulation, the number of

particles left at the BPM position was recorded. A plot of this number versus
measurement looks like Figure 5-12. As can be seen, the way in which the beam
intensity increases from measurement 2 to 4 is very similar to that seen in Figure 5—11 for
the real data.

Fitting the data from the first simulation and keeping all the data points yields Table
5-3 and the rrns beam parameters in Table 5-4. Comparing the predicted values of
(x2 > - <y2> for measurements 2 through 4 to the values calculated from the simulation,

there is a similarity to the relationship between the predicted and measured values of

(x2 > — (yz) in Table 5-1: the predicted value of (x2 > — <y2> for measurement 2 is larger

than the measured value, the predicted value for measurement 3 is about right and the

predicted value for measurement 4 is too small.

When the values of (x2) — <y2> from the second simulation, where there is no beam

pipe, were fit, the predicted values matched the simulated values. Also, if measurements

130

Intensity vs. Measurement Number

25000

 

20000»O..OOOOOOOOOOOOOOOO

15000 «»

10000 4»

lnlt-nsilt (Number of Particles)

5000 4»

 

 

0 t * : * : 1 % . .
0 2 4 6 8 10 12 14 16 18 20

 

Measurement Number

Figure 5-12: Intensity vs. measurement number for simulated emittance measurement.

131

Table 5-3: Value of (x2>- (y2> from simulation, predicted value of (x2)— (yZT (from

fit) and the difference between them for each quadrupole setting in simulated emittance

measurement with beam pipe.

 

 

Measurement Calculated value of Predicted value of Difference between
Number (x2) _ (Y2) (m2) (x2) _ (Y2) (m2) predicted value and
from simulation simulated value
(m2)

1 -0.3 -0.1 -0.2

2 32.7 34.8 -2.1

3 27.3 25.9 1.4

4 21.2 18.8 2.4

5 10.4 10.8 -0.4

6 8.6 9.2 -0.6

7 7.0 7.8 -0.8

8 ~95 -9.5 0.0

9 -6.7 -6.6 -0.1
10 -3.8 -3.9 0.1
11 -3.6 -3.6 0.0
12 -3.6 -3.6 0.0
13 -3.1 -3.1 0.0
14 -14.1 -14.1 0.0
15 -12.6 -12.6 0.0
16 -11.1 -11.0 -0.1
17 13.1 12.9 0.2
18 11.8 11.6 0.2
19 10.2 10.1 0.1

132

Table 5-4: Values of the RMS beam parameters for different fits in simulated emittance

 

measurements.
RMS Beam Actual Predicted Predicted Predicted Predicted
parameters values values from values from values from values from
fit using data fit using data fit using data fit using data
from from from from
simulation simulation simulation simulation
with beam with beam with beam without
pipe pipe, but pipe, but beam pipe
discarding discarding
data points 2 data points
and 3 2, 3 and 4
(x2) 5.26 4.10 4.84 5.28 5.26
(mmz)
(xx') -3.97 -3.51 -3.71 -3.93 -3.92
(mm mrad)
(x’2> 4.31 4.49 4.29 4.34 4.34
(mradz)
(Y2) 1.08 1.05 1.05 1.05 1.06
(M)
<yy') -0.25 -0.25 -0.25 -0.24 -0.24
(mm mrad)
(y’2> 1.39 1.58 1.57 1.57 1.57
(mradz)

 

133
2 through 4 are discarded from the first simulation, then the fit to that simulated data is

also very good. Table 5-4 shows the results for the rrns beam parameters for several
scenarios.

What the simulation without beam pipe boundaries shows is that there are no
unexpected discrepancies between the model and what is really going on in the beam
dynamics. The linear model is quite good, even at a relatively high charge of 1 nC per
beam bunch.

What is apparent from the first simulation, where the beam pipe was present, is that
the beam intercepting the pipe wall is a problem. From this one can conclude that

throwing away data points because it is suspected that beam scraping has taken place is
acceptable. In fact, from simulation, a good rule of thumb is that, when (x2 > — <y2> gets

bigger than about $ 20 mm2 , beam scraping should be looked for carefully.

What this simulation does not address is the difficulty of distinguishing beam
scraping from the inherent instability of the electron beam. As has already been
established, changes in beam parameters and intensity are to be expected as a normal
property of the beam. Determining which changes are due to the beam intercepting the
wall is often arbitrary. This is not a very satisfactory solution, but until improvements to

the stability of the electron beam are made it is unavoidable.

5.5.3 Emittance versus bunch charge
To verify that this method for measuring the emittance actually works is difficult. We

would like to compare the numbers from this technique to others. However, as has

134

already been discussed, there are no other methods that we trust because of the unique
qualities of a photoinjector beam. This limits us to more a more indirect approach.

One way to verify Miller’s technique is to vary the emittance in a predictable way.
For instance, as the charge per beam bunch is increased, we expect the emittance to
increase. By measuring the emittance at different charge levels, a definite trend should
emerge. In this experiment, the emittance of the beam is measured versus the charge per
bunch of the beam. The charge per bunch is the only variable changed. The focusing
elements upstream from the experimental section are kept constant. The emittance at
each charge level was measured using the method described in section 5.5.1. Each beam
shot contained 5 bunches. The emittance for each bunch was measured at each charge
level. The average of these 5 emittances at each charge level is plotted in Figure 5-13.
The x emittance shows a definite upward trend, as expected. The trend for the y

emittance is less clear, although it does seem to increase as well.

5. 5.4 Emittance versus magnetic field

A better check of the efficacy of Miller’s emittance measurement is to measure the
emittance as the magnetic field on the cathode is changed. In Figure I-l, three solenoids
are shown around the front end of the SPA photoinjector. The two large solenoids are
used to focus the beam as it is accelerated. The smaller bucking coil is used to cancel the
magnetic fields of the two larger solenoids in the region of the photo-cathode. What will
be shown is that, as the current through the bucking coil is moved away from its proper

value, the emittance of the electron beam will increase.

135

Average Normalized Emittance vs. Bunch Charge

 

70.0

LA

5:

o
”-1

8
o
1.1

30.0 4- I i
I If

B
0

Normalized Emittance (mm mrad)

10.0 '1'

 

 

 

0.0 : 5 4 4 5 : 4 4
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

Bunch Charge (nC)‘

Figure 5-13: A plot of normalized emittance versus bunch charge.

0.45

 

 

Ox Emittance
I y Emittance

 

 

136

The front end of the SPA photoinjector is designed so that the electric and magnetic

fields are cylindrically symmetric about the axis that runs through the center of the

cathode. Therefore,

% a

w'm"

(5-1)

where d) is the scalar potential and A is the vector potential. The equations of motion

for a charged particle in cylindrical coordinates are

d . .

31-(ymi) —')(mr02 = q(Er +r0Bz —2Be),
Et-(ymrzél) = q(1~:6 +213r 413,),

and

d .
$(ymZ) = q(Ez + m, — r013,)[6].

Using (5-1), we have

 

 

6A6
E9 =— at ,
6A9
Br — - aZ
and
1 a
BZ =;“a—r-(I'A9)

Substituting these into Equation 5-2 gives

 

 

 

__( 2(3)— (6A9 .6A6 .6A9 iAej
rdtymr —q 6t+r6r+zaz+ r

(5-2)

137

or
d .
a:(ymr20 + qrAe) = 0.
Therefore,
ymr20+qrAe = constant. (5-3)

This is equivalent to saying that the canonical angular momentum in the Hamiltonian
formalism is a conserved quantity when the system has cylindrical symmetry.

The magnetic flux through the circle defined by the radial position of the particle is
w = [fr-as = [(w A)-dS = (TA-di = 2mA, [6].

Therefore (5-3) can be rewritten as
2 ‘ q
ymr 9+5“, = constant. (5-4)

This is known as Busch’s theorem [6].

The bucking coil on the SPA photoinjector was incorporated in the design in order to
nullify the magnetic fields from the other two solenoids on the surface of the cathode so
that w in (5-4) is zero. When its current is not properly set, however, there will be a
magnetic field perpendicular to the photocathode surface that is nearly constant with
radius. Then 1); will be given by

w=nfiBw
where Bc is the magnetic field on the cathode and rc is the radial position of the electron

at the cathode. Equation (5-4) now becomes

erzB

C C

 

ymrcz0- = constant.

 

138

This is significant because, when the electrons are emitted from the cathode surface, their

angular velocity will be very small. So, we can say that

 

2' q ercch

where r is the radial position of the electron at some point downstream.

As the beam is accelerated away from the cathode, the magnetic fields, and therefore
the magnetic flux, from the solenoids decrease to zero. According to (5-5) the angular
velocities of the electrons grow to compensate. This leads to an increase in the in x and y
momenta of the electrons as they pass into the field free region downstream from the
cathode, enlarging the area of the beam in x and y phase space. Since the x and y rrns
emittances are proportional to this area, they also increase.

An analyses of the beam envelope equation [6], [30], shows that the emittance

increases with the magnetic field on the cathode according to

RZB 2
en=\/s§+(e ° °) (5-6)
4mc

where an is the normalized rrns emittance, t:i is some intrinsic emittance and Rc is the

 

 

radius of the cathode. This equation holds for both the x and y directions although the
intrinsic emittance will be different for each. By moving the current of the bucking coil
away from its proper value and using Miller’s technique to measure the emittance, a
distinct curve that follows (5-6) should emerge. The results of such an experiment are
shown in Figure 5-14.

The fitted curves in Figure 5-14 are slightly different. The magnetic field dependent

term was slightly bigger (20%) for the x direction than it was in the y direction for a

139

given magnetic field. However, the fits to (5~6) are quite good considering the simple
model used to obtain the dependence of the emittance on the magnetic field.

According to Figure 5-14, the x emittance is determined with better accuracy than the
y emittance. This is not a failure of the technique, but a problem with the signal to noise
ratio. The poor stability of the electron beam limits the resolution of Miller’s emittance
measurement. It would appear that oftentimes the value of the y emittance falls beneath
this resolution. With improvements to the beam, better values with smaller error bars for

the y emittance will result.

'The charge per bunch was measured using a wall current monitor. Recent experiments
have call into question its calibration. Therefore all charge measurements are not

accurate to better than a factor of two.

140

Normalized rrns Emittance vs. Magnetic Field on Cathode

-AA 2

 

 

 

 

 

 

 

“LU
45.0
g 40.0
a . I
v 30.0 Fit to x Emittance
g 9 x Emittance
E 25'0 —Fit to y Emittance
20.0 I 1 Emittance T
Y .
E 15.0 i
5.0
: 4 : 3;, . 4 : t
-80 -60 40 -20 O 20 40 60 80

Magnetic Field on Cathode (Gauss)

Figure 5-14: Emittance versus magnetic field on photo-cathode.

CONCLUSION

The goal of this thesis was to present a sound theoretical basis for measuring both the
rrns emittance and the rms pulse length using BPMs and to demonstrate Miller’s
emittance measurement technique experimentally. This has been achieved. However, in
the process some questions were raised and problems encountered.

The poor stability of the SPA electron beam presented the greatest obstacle to
implementing Miller’s emittance measurement successfully. The hope we have is to use
Miller’s method as a routine emittance measurement. To do this, however, the beam
stability must be improved. At the time of this writing, changes are being made that
should improve the stability considerably.

As was mentioned briefly in Chapter 5, the measured x and y emittances of the SPA
beam were greater than anticipated. Simulations of the machine show that, at 1 nC per
beam bunch, the normalized emittances should be about 10 mm mrad. At 0.2 nC per
beam bunch, the normalized x emittance was three to four times this value. The
normalized y emittance was, perhaps, more reasonable. It is not yet known why this
discrepancy exists. Possible reasons are that the simulations are wrong, something in the
beam line is causing the emittance to grow, the emittance measurement is flawed or, quite
simply, we are not running the accelerator in an optimal fashion.

In its previous incarnation as the injector for the HIBAF facility[14], PARMELA

141

. 142
simulations of the injector and experiments agreed quite well. For this reason, we have

great faith in PARMELA to accurately predict the performance of SPA. For the
simulations to be incorrect, a substantial physical effect would have to be neglected.

In our PARMELA simulations of SPA, it is assumed by the code that the photo-
cathode emits electrons uniformly when struck by the laser beam. However, we know
that this is not true in the real accelerator. When fabricated, a photo-cathode’s surface
does not have a uniform quantum efficiency. This is a result of the fabrication process.
These discrepancies are small, and simulations done in the past indicate that they do not
degrade the beam quality significantly. However, it is not clear what happens to the
photo—cathode surface over time after it is placed inside the accelerator.

Because of limited access to the photo-cathode preparation equipment, the photo-
cathode that was in place when the emittance measurements presented here were
performed was well over one year old and had been inside the accelerator for nine
months. This is longer than any cathode of this type has been used. During this time the
cathode was damaged by arcing in the accelerator cavities, it was poisoned on one
occasion when a vacuum pump failed and its average quantum efficiency degraded by a
factor of 20. It is unknown what this abuse did to the cathode surface. It is possible that
the initial shape of the beam, because of the damage to the cathode, is the cause of the
larger emittance.

There is also a possibility that some part of the SPA beam line is producing nonlinear
forces that are causing substantial emittance growth. This is unlikely. However, the

possibility should also be investigated.

143

There is also a chance that the emittance measurement is flawed somehow. However,
I consider this to be the least likely explanation. Presented in this thesis is ample
evidence that Miller’s emittance method is sound. Also, based on the poor behavior of
the beam, there is no reason to think that the emittance is any better than what was
measured. The difficulties in transporting the beam and beam spot size on the two
screens in the beam line tend to support the higher numbers.

The simplest explanation for the higher numbers is that the accelerator is not being
run optimally. That is, the currents in the solenoids are such that the emittance of the
beam is not minimized. The simulations are run such that the magnetic fields from the
solenoids are very close to their optimum value. Hence, if the currents in the real
solenoids are not set properly, simulation and experiment will not agree. Since the
magnetic fields from the solenoids have never been mapped, setting their currents

properly is not automatic. With experience, this condition should improve.

Appendices

Appendix A

Appendix A
CHARGE DISTRIBUTION INDUCED ON INNER SURFACE OF BEAM PIPE

DUE TO RELATIVISTIC ELECTRON BEAM

A.1 Introduction

Consider Figure A-1. An electron bunch that is described in the lab frame by the
density distribution pL(r'i) traveling with a relativistic velocity, v, inside a metal pipe
with radius a. The distribution function pL(x‘) is referenced to the coordinate system
whose origin corresponds to the pipe center. The electric potential, (I), is defined to be
zero on the pipe. In the metal walls of the pipe, the electron bunch induces an image
charge. A relationship between the bunch density distribution and this image charge will
be derived.

The problem will broken down into two parts. First, the problem is solved in the

beam frame where v = 0. Then, to find the solution in the lab frame, a relativistic

transformation is made.

A.2 Solution in beam frame
In the beam bunch’s rest fiame it has a density distribution that will be denoted by
p364). To find the image charge distribution that is induced by the bunch, Poisson’s

equation must be solved,

144

*‘<

radius = a pL(i’t)

k .

 

 

 

 

 

 

 

(I) = 0
b)
Figure A-l: Relativistic electron beam pulse in metal pipe: a) cross sectional view, b)

longitudinal view.

146

VZMR) = 4313365),
for the potential (12(x) using the proper boundary conditions. This problem was first

done by Smythe[31]. The image charge distribution on the pipe walls is then given by

l

 

 

 

 

 

 

o(0,z) = —4_7t.fi. E m, fr =f
1 1 aq>(r,o,z)
‘5 ‘2; a. 0"")
In cylindrical coordinates, Poisson’s equation becomes
62:55:) + -} at?) + :1,- 523‘," + 6:32)?) = 4an0?) (A-2)
The boundary conditions on (12(x‘) are
(142))”, = 0 (A-3)
and
<1>(s)|zm = 0. (A-4)
To solve for the potential, the Green’s function, (3(a)?) , is found. Then
cr>(2) = [p,(x')o(x,sz')d3x'— ‘4— cj ¢(x )————— ad” ——),xda' [32]. (A-5)
Volume 11: Surface

By evaluating the surface integral over the boundary where the potential is zero, (A-3)

and (A-4), (A-S) becomes

<1>(i) = IpB(X')G(x,x')d3x'. (A-6)

Volume

By definition, the Green’s function has the following properties:

G(x,x') = G(x',x), (A-7)

147

17206152): —47r6()‘t — x') (A-8)

and G(x,X') has the same boundary conditions as (12(2). Therefore, from (A-3) and

(A-4)

 

= o (A-9)

and

 

= 0. (A-lO)
In cylindrical coordinates, the Dirac delta function is given by
_ _ 1
5(x — x') = ;5(r — r')5(0 — 0')5(z — 2’).

Therefore, from (A-8),

1
Woe-1,2): —47t;8(r—r')8(0—0’)6(z—z'). (A-ll)
It is known that
1 2n
; Icos[n(0 — 0')]cos[n'(0 — 0')]d0 = 5m , (A-12)
0

% [5.4.46 .. 6414.14.49 —e')]de = 5......

and

 

2 8
32.12 (Km) Irlv(xvm Slim,“ 22de = 5m“. (A-13)

v+1
where v 2 —1 and xum is the Bessel function zero[33]. Therefore, we can expand the

function

148

Tao—mew)

in the following series[32]:

—6(r—r)8(0— 0): 2a ZJHXU 3{Amcos[n(o-o')]

m=1

 

 

+ B"m sin[n(0 —0')]} (A-14)
where
l - 0
an = 2 ’ n— ,
1 , n at 0
2 211: a 1 1'
Am- — 43213.50 m) ojdoojdn-r-ao—r')6(0—0')Jn(x,,m 2) cos[n(0—0')]
( ‘3
2J0 xnm _
= a
naZJrzr+l(x rim),
and

 

[do ]dn-:-8(r — r')8(0 — 001(me 3314140 -0')]

00
D
B
l
:1
m
N
t...
A
X
3
V

Expanding the Green’s function in a similar fashion yields

66:30:22: “ZZZ um(z, z')J M( 3AM, cos[n(0 -0')]. (A-15)

The constants are the same as in (A-l2) and an(z,z') contains the z dependence of
G(x,x’). (The sin term is dropped because G(i‘t,)‘i') = G(x',x) and

sin[n(0-0')] 9’: sin[n(0' —0)]. Therefore, it is expected that the sin term coefficients

149
should, in fact, be zero.) Substituting (A-15) into (A-l 1) yields

62G 16G 162G 62G

 

 

 

 

 

_. _., 1 r r r
V G(X, X=) 51‘2 -1-'-"a-r—+‘lja92 + 622 =—47t-r-6(r—r)8(0—0)8(z—z),
a’Jnhm-il 164%.?)
2a cos[n(0— 0012A“, an(z,z at, +; (it
L _

 

 

 

2 622 , ' 1
4.1.0-4 ”(4.) “92> z4......-r.).(.-.t...-..).
r a a 62 r

Making the definition

this expression becomes

62J (x) 16] (x) 112

23a“ COSID(9“9')];Anm Tan(z,z 'X"( a —) T—ax2+§_:K__x—21“(x)j

 

azz , ' 1
+Jn(xnm 1) “(22 z )} = —47t-5(r — r')8(0 —0')8(z— z').
a 62 r

By using Bessel’s equation,

this becomes

 

Zan cos[n(0- 9' ”2:4,“,an ma)T62 Z'g‘zgz z)-(:fl)22nm(z,z’)]

m=l a

= _4n%5(r—r')8(0—0')8(z—z’).

150
Multiplying both sides of the equation by

1 r
;c03[n (0 — 0 )]rJ n. (xwm. E)

and integrating over r and 0 from 0 to a and 0 to 27: respectively gives

gan izj‘coinm — 0')]cos[n'(0 — 00kt); Am 1d" (xnm BJ n, (xm' 3.) dr

@Tazzm(z,z') —(xfl)22nm(z,z')T

 

022 a
1 271a r 1
= —4n — j Icos{n'(0 - 0')]rJn. xm, — -5(r — r')8(0 — 0')8(z — z')drd0.
1t 0 o a r

Using (A-12) and (A-13) yields

2J2, ' ' 2 , ' , . ' 2
a a n+l(xnm )An.m,[:a an (zgz)_(in_fl_) Zn'm'(z,zr J

"' 2

 

 

From (A-14)

2 2 I
an’ a Jn'+1(xn'm')An'm' = J"! (x ’ l L) ,
2 "m a

leaving the equation

 

 

2 r 2

where the primes on n and m have been dropped for convenience.

To solve this differential equation, first simplify the notation by making the definition

151

*0

III
S
3

Then,

622nm (2’ 2') [(2
522 _

 

an (2,2') = —47t5(z — z').

62mn (z, z')

Multiplying both sides of the equation by 62

and integrating, gives

.1.
2

 

rummage-.1

(Z (z,z'))2
"m = 47:ng (z', z’) + constant.

(Zlm(2)) — k2 2 '9.

Equation (A-10) requires that, as 2 —-) $00 , Z goes to zero, and, as a consequence, so does
its derivative. Therefore,
47tZ;m (z',z') + constant 2 0 ,

leaving

(Z:1m (z,z'))2 _ k2 (Zmn(z,z'))2 = 0
2

2

 

 

Therefore
an (2,2') = ejtkz+C ,
where C is some constant.
To determine the final form of the function an (z,z') , first rewrite the constant C in
the following way:
C -—) AeC.
This gives

an (2,2,) : Aeikz+C .

152
From (A-10) and (A-7)

 

an(z,z') —> 0 (A-17)

2.41..
and
an(z,z') = an(z',z). (A-18)
Now, consider the special case of a point charge located at 5%.

(9(2): [p,(x')G(x,x')d3x' = [q5(i'—x,)6(rr,x')d3x'=qG(x,ro).
V V

Therefore, 2' can be considered the longitudinal position of the point charge. Given the

symmetry of this problem, it is expected that the function an(z,z') will be symmetric
about that position. This, combined with (A-17) and (A-1 8) leads to the solution
an (2,2') = Ae'k'z‘z'l.
To find the amplitude, A, first take into account the discontinuity at z = z' . To do

this, recall (A-16) and do the integration

 

z'+e 2 ' 2"”:
l [d 2.3.2ng ) — 182.42.29sz = ’4“ [5(2 " 2')

z'—e z'-e

where e —> 0. This becomes

A(— Ute-““2“" — A(+ k)e'k(z""+‘) = —4n .

When e—>0

Therefore

153

2 -'—‘Z‘Z
Z...(zz)=—”3e ., ' we

run

Combining (A-19) and (A-lS) results in

G(3i, i')=-Za cos{n(0— 0')]ZJn Hm(x —)Jn(xm g) ’Eafl'Z‘z'l

a"'° "“1 xxnerzrrn( nm) e

for the Green’s function. Substituting this into (A-6) to find the potential and then

 

putting the potential into (A-l) gives

o(0,z) = E1; jpg(i')gan 004149 " 99]

®Zxa 'd’x'.

2
m=1nxnmJn+l(xnm)

 

Using

J;(xm)=—J,,,(x, m)[33],

and the final expression for the image charge in the beam rest frame is

Jnxn( 2M)
-x —"—"'|z-z'|

0(0, (-z)1t — -a—12Jpa(x' )Za cos[n(0— 0' ”20:1— Jnx( )e " d3x' (A-20)

where

154

A.3 Surface charge distribution in lab frame

Now that the problem has been solved in the beam frame, a relativistic transformation
is made to the lab frame. The situation is that depicted in Figure A-2. K is the lab frame,
where one wishes to know the surface charge distribution and K' is the beam frame,
where the surface charge distribution is already known. The frame K' moves with

velocity, v, relative to the K frame in the z direction. In K' , the surface charge, from

(A-20), is

-1 0°
o'(0',z') = E2— [(3801022) an cos[n(0' —0")].
v n=

1,"

0° Jn(xnm :) —-x—"fllz'—z"l 3

692 e . dx" (A-21)
.(x...)

ms]

 

To get the charge distribution in the lab frame, K, make the following transformations:
E, =yE; =>o=yo'
r' —> r ,
0' —-) 0 ,
r" —> r" ,
0" —> 0" ,
z’ —> y(z— Bct) ,
z" —) y(z" — Bct) ,
d3x" —> r"dr"d¢"ydz" = yd’x" ,

and

155

 

 

 

 

 

 

Figure A-2: Lab and beam frame for relativistic electron beam bunch traveling down

beam pipe.

156

p302") ——) pB(r",0",y(z" — Bet»

where

o|<

and

 

Substituting these into (A-21) gives

6(9424) = 72 lelr'xemaz" —Bct)) .. a. cos[n(e—9")]

7:3 V
J( L)
n xnm a -x"m|z-z“|
J

a 3 n _
.(X...) e dx . (A 22)

 

 

a)”:

m=1

 

In the lab frame, the bunch density distribution will be different from the bunch
density distribution in the beam frame in two ways: the lab density distribution will be
shorter because of length contraction, and, because of conservation of charge, it will also

be more peaked. Therefore
yp.(r'.e',y(z'-Bct))=p.(r'.e',z',t) (A—23)
where pL(r',0',z’,t) is just the bunch density distribution in the lab frame, K.

Substituting this into (A-22) and making the double primes single primes for convenience

yields the final expression for the surface charge distribution in the lab frame,

157

o(0,z,t) = % IpL(r',0',z',t)ian cos[n(0 - 0')]
n=0

2
V

where

A.4 Relativistic approximation

d3x' (A-24)

As a final note, a useful approximation is derived that is valid when the electron beam

is highly relativistic.
One definition of the Dirac delta function is

lx-x'l

 

 

 

 

 

 

 

e 8
6(x — x') 5 lim
8—90 28
Making the definition
a
8 E
Yxnm
gives
_|z-2'|
3‘ _|z-ZI
-yxT"m|z-z'| _ a 6 Wm _ 2 a C E
— 7X... 3 _ 7X... 22
Yxnm

When 7 is large, then, one can make the approximation

158

X
~17 —'“‘l(z-2’|
3

5(2— 2').

 

11m

Substituting this into (A-24) gives

o,(0 z,)t =--- :(‘TpL r' ,0', z' ,t)za cos[n(0- (9’)].

.2 (.4

pipe
”‘1 xnmJ run (xn

 

From (A-13) any function of r can be expanded in the series

f(r) = gAva(xm i)

where

am a2 2
aXJn+1

——(——2X.,m )Jrf(r)Jn (xnm 3dr.

Expanding the function

we".

the relation

 

xnmJn+1 (xnm)

results. Therefore, (A-25) can be approximated by

6(0, 2, t): ~ at; I pL(r’, 0', z, t ){1+2Z(— 4") cos[n(0—0')]}r'dr'd0’.

area of
PIPe

r'dr'd0' .

(A-25)

159

The term inside the parentheses is the distribution function of an infinite line charge. (See
Appendix B) This is a result of the well known “pan-caking” effect when charges become
highly relativistic. The electric field lines become almost perpendicular to the direction

of motion.

 

 

Appendix B

Appendix B
SURFACE CHARGE DISTRIBUTION ON INNER SURFACE OF
CYLINDRICAL METAL PIPE DUE TO AN INFINITE LINE CHARGE INSIDE

THE PIPE AND PARALLEL TO ITS AXIS

B.l Introduction
In this appendix, the distribution function for the image charge induced on the inner
surface of an infinite metal pipe due to an infinite line charge is calculated. The line

charge is inside the pipe and parallel to its axis.

B2 Constant potential surfaces due to two infinite line charges in free space

To begin, the constant potential surfaces that occur in the presence of two infinite line
charges in free space are calculated. The line charges are parallel to each other.

Figure B-l shows two infinite line charges perpendicular to the page and parallel to

the z axis. One has a charge per unit length of it and the other —7\.. They are positioned at

(0,0) and (0,d) respectively. The electric potential due to these line charges at some
point (x, y) is given by
<1>(x, y) = -27.1nr, + 2;. 1m, (B-l)

where

160

161

 

 

 

 

7» 4»

4 d >

 

 

Figure B-l: Two infinite line charges parallel to the z axis with charge per unit length it

and 4., respectively.

162

r,2 = x2 «I-y2
and
r22 =(x—d)2 +y2.
The constant potential surfaces occur when

LI-

1.2

where k is some constant. Squaring this expression leads to
r,2 = kzrzz.

Using (B-2) and (B-3) yields
x2 +y‘2 = k2(x—d)2 +k2 2,
x2 +y2 = kzx2 —2k2dx+k2d2 + k2 2,
x2(1—k2)+2k2dx+ y2(1—k2) = 1862,

kzd kzd2
x2 +2x1_k2 +y2 =1—k2 ,

 

 

k’d 1&12 ,_k2d2 k‘d’
1_k2+(1_k2)2+y -1_k2+(1_k2)29

 

 

x2 +2x

 

 

T kzd )2 + , 1842-1642 +k‘d2
y =

 

x+1_k2 (1—k2)2 9
kzd 2 , 1(de
X+1_k2 +y =‘('i:—kz—)'2-.

This is the equation for a circle with its center at

03-2)

(B-3)

163

 

 

kzd
x0 = _1_ k2 (3'4)
and
Yo = 0
The radius is
kd
R = 1- k, . (B-S)

8.3 Electric potential from infinite line charge inside an infinite metal pipe

It proves necessary to calculate the electric potential for an infinite line charge inside
an infinite metal pipe before the image charge distribution can be determined. The line
charge is parallel to the pipe axis. In this section, using the results just derived, the
electric potential for an arbitrary point is determined for the given situation.

In Figure B-2 an infinite line charge with a charge per unit length 7» is placed in a

metal pipe of radius a. The line charge is parallel to the pipe axis and is located at
(r',0'). Because the potential on the pipe must be a constant, an image line charge of
opposite sign appears at (R,0').

In the primed coordinate system, the potential on the pipe is constant if r' is set equal

to the result in (B-4),

X3 = r. = _ (B-6)

 

and the radius of the pipe is set equal to the result in (B-5)

164

 

constant
potential

 

 

Figure B-2: Infinite line charge inside an infinite metal pipe.

165

 

 

 

a _ kd B 7
“ 1— k2 ' ( ‘ )
Using (B-6) and (B-7) to solve for k and (1 yields
1.!

k — ; (B-8)

and
2 '2
d = a ,r (B-9)
r
Using (B-9) the radial position of the line charge 4. is given by
2 _ ,2 2
R=r'+d=r’+a ,r =a—,. (13-10)

The distances from the line charges it and 4. to some point (r,0) can be determined
by the law of cosines. Identifying these distances as r, and r_A , we have
r: =r'2 +r2 —2rr'cos(0-0') (B-11)
and
r3, = R2 + r2 — 2chos(0 —0').

Substituting (B-10) into the last expression gives

4 2

r3, = 37+ r2 —2r§r—,-cos(o—0'). (13-12)

The electric potential at (r,0) can be calculated using (B-l), (B-1 1) and (B-12).

<1>(r,0) = —22.1r(:—:T

166

y a. a. a
= —2}, lnlr'2 +r2 -2rr'cos(0—0')] 2 —lnT:;,—2"+I'2 —2r?c08(0-9')]
q)(r,9) = —}.Tln[r'2 + r2 — 2rr’cos(0 -0’)]

a’ a2
—1nTr7+ r2 —2r7cos(0 -0')TT (B-13)

B.4 Surface charge distribution from infinite line charge inside infinite metal pipe

B. 4.1 Image charge distribution

Now that the electric potential is known, (B-13), the induced surface charge, 0(0) , on

the inside of the pipe is given by

 

 

 

 

 

 

l _.
O-(9)="‘———fi E , fizf'
47: m
_ '1—E _L6<D(r,0)
41! r [=3 _ 411; fit ,._.a
From (13-13),

V a2 ‘

M_ x, 2r-2r'cos(9-6') Zf-ZficOdG—e) }

al- _— 12 2 _2 , 9_9' — 4 2
1‘ +1' IT COS( ) :2 +1.2 _2rar_'cos(9_er)

 

 

 

Therefore,

 

167

 

 

 

 

 

 

 

 

a I
0(9) — _fi a-r'COiG—G') a—7C040-9) ?
_ —27: r'2 +a2 —2ar'cos(0—0') _ a2
. Fl 2:2 —2ar'cos(0—0')]J
A a—r'cos(0—0')——+r'cos(0—0')
Cy(e) =__T a >
27: r'2 +a2 —2ar'cos(0—0')
l 2 _ '2
6(0) = — a r (B-14)

27:a r’2 +a2 —2ar'cos(0—0')°

B. 4.2 Expansion of image charge distribution in Fourier series

It will prove useful to expand (B-14) in a Fourier series. Making the definitions

 

x 5 Ir: S 1
and

aEO—W
yields

f(x,a)=1+ x21:::cosa = 2139+;(an coso: +bn cosa) (B-15)
where

a“27:1_f=5[ f(x,o:)coino:)d0t

and

bzfisz(x,a)81n(na)da.

. .

l 68
Evaluating the first integral,

 

 

(1 - x2) 2} cos(no:)do:

 

 

 

a" = 7: 01+x2—2xc050t
(1- x2) “ cos(no:)do:
= 2 I 2
7: 0 1+ x - 2xcoso:
(1— x2) 7:x“
an = 2 2 (From Tables).
7: I1 —x T [T
Since x is less than or equal to 1, E
1—x2=|1—x2T. .
Therefore
(1 - x2) 7:xn

 

= 2x". (B-16)

The second integral is

(1 - X2) 2]- sin(no:)do:

7: 1+x2 —2xcoso:

 

 

0

b = - (1 2:2) [ln(1 + x2 - 2x cosa)]

27:

 

= 0. (13-17)

 

0

Substituting the results in (B-16) and (B-17) and the definitions of x and or into (B-15)

yields

 

0(9)- l a2_rv2 —--L 1+2i(r—')ncosln(0—0')l
— 27:a r'2 +a2 —2ar'cos(0-0') — 27:a "=1 a °

Appendix C

- 'I

9""

Appendix C
VOLTAGE SIGNAL GENERATED ON OSCILLOSCOPE FROM BEAM
POSITION MONITOR SIGNAL TRAN SMITTED THROUGH TRANSMISSION

LINE

C.l Introduction

As an electron beam bunch passes through a BPM, it generates a signal on the BPM’s
electrodes. This signal is transmitted by a transmission line, passed through a low-pass
filter and displayed on an oscilloscope. In this appendix, an expression for the voltage
signal on the oscilloscope is derived. The coupling between the electron beam and the

BPM electrodes is modeled with a simple electrical circuit.

C.2 General Solution

In this section, the voltage signal seen by the oscilloscope is derived. This is done

using a simple circuit model and the results from Appendix A.

C. 2. 1 Solution to circuit model

The coupling of the BPM electrodes to the beam image charge can be modeled by the
circuit in Figure C-1 [17]. The image charge is represented by the current, ib. Zc is the
characteristic impedance of the transmission line and Cp is the capacitance of the BPM

169

 

170

Transmission Line

 

 

 

/\

K)

K/
o

 

 

 

Figure 01: Circuit model of BPM electrode coupling to electron beam. Cp is the

capacitance of the BPM electrode. Zc is the characteristic impedance of the transmission

line. ib represents the image induced image charge from the passing electron beam.

171

electrodes. In the frequency domain, this model is accurate to approximately 2 GHz[l6].
Beyond this, it breaks down.

The relationship between voltage signal into the transmission line, VT, and the

current, ib , is

Taking the Fourier transform of both sides of this equation, yields

VT(co)
Z

C

 

Ib((o) = + ijwVT(o))

where the well known Fourier transform of a function is given by
12(0)) = If(t)e”"°'dt.

Solving for VT(m) gives

Zch(0))

v = —.
Aw) 1+ ijch

(C-l)

Because the beam bunches generated by SPA are very short, a large part of the signal
content is at very high frequencies. Since the model is inaccurate above 2 GHz, low-pass
filters are placed on the oscillosc0pe inputs in order to ensure that the we are far from that
regime. The end result is that the original signal is modified significantly by the time it is
displayed on the oscilloscope. Therefore, transfer functions are assigned in the frequency
domain: 0(0)) for the transmission line and A(w) for the combination low pass filter
and oscilloscope. The resulting voltage signal is given by simply multiplying equation

(C-l) by these transfer fimctions, yielding

 

172

Zch(m)

WORMAQD) .

V...(@) =

Taking the inverse Fourier transform of this function gives

1H,*°°21(m)

_,,1+j(D (DC ch o(co)A(w)ej°"d0). (C-2)

v,..(t)=—

C. 2.2 Expression for beam current
To find an expression for Ib(a)) , one first finds the current generated by the image

charge, ib(t) . It is given by the time derivative of the total amount of charge on the BPM

electrode, or,

1 (t)-— [0(9, z,td

tArea of
Electrode

The surface charge distribution for a highly relativistic beam is known from Appendix A,

area of
PH”

o(9,,=zt) 2? I pL(r', 9',,)zt{1+2n2(r:—)n cos[n(0—9')]}l'dr'd0'.

The BPMs used on SPA have square electrodes. The electrodes have longitudinal
length L, longitudinal location 20 , angular width 20: and angular location 90. Integrating
the expression for the surface charge distribution over the surface area of the electrodes

gives

“(0:21—59 :Eldezoidz I pL(r', 9' ,z,t){1+22(r: _)" cos[n(O—9')]}da'

n=1

The integral over 9 is simple and results in

.1

173

_1 zo+L co , n .
ib(t)=;§: Idz I pL(r',6',z,t){2a+4Z(%) smnacoin(9—9')]}da'.

zo area of “=1 n
we

 

Taking the Fourier transform yields

_1+oo ' (1 10+].
Ib((0) = Elem m a JdZmI§L(T',9',Z,t)
pipe

 

® {20: + 4% (—) .. Sinnn“ cos[n(9 — 9')]}da' . (03)

r

n-l 3

Inside equation (O3) is the expression
d zo+L

E lpL(r',9',z,t)dz (04)

1.,

Recall from equation (A-23) that z and t occur in the charge distribution together as

2 —- Bct. In other words, the density distribution function can be rewritten as

pL(r',9',z,t) = pL(r’,9',z- Bct).

Taking the time derivative of the distribution gives

(1 d
EpL(r',9',z,t) = —BcEpL(r',9',z— Bct).

Therefore, if the time derivative is moved inside the integral given in (04) then

+L
20d

I d—tPL(r':9',Zst)dZ = ‘BC[PL(r',O',zo + L — Bct) - pL(r',G',z0 - [3a)].
20

Then (C-3) becomes

 

174

Ib((1))=—§:Zlodtem"t flpL( r' ,9' ,—zo+L Bct)- pL(r', 9’ ,—z0 —Bct)]

area of
pipe

®{2a + 42(31131" “‘1 cos[n(9 - 9')]}da' . (C-S)

 

As was just done for the z integral in equation (C-3), the time integral can be isolated

from (OS) leaving

M

I = Ie‘j“‘[pL(r',9',zo + L— [3ct)—pL(r',9',z0 —Bct)}lt

.20 .31

 

 

-‘3( +L)+°° '—“’—( +L-Bct)
= e 1"“ 2° J‘eJ‘3c 2° pL(r',9',zo + L— Bct)dt

-®

J10 [Jae-(2031:) (r 0, lo Bet)dt

Making the definitions:

z'azo+L—Bct

and

220 —Bct,

gives

no ml
{Ezojejflcz "pL(r',9',Z" d2".

:—_—e-i::(lo+I-)MD j z,
:fe 3° ('r ,,z9' ')'dz +6

 

I=

Recognizing that the two integrals are identical yields

+00 ,~ “’20”. _jwz.,+1. Jm_ _Jw
Je "m e 5° ‘3‘ pL(r',9’,zo+L—Bct)—e BCe BCpL(r',9',z0 —Bct) dt

"l

175

‘j—To
e B“

BC [l—e-EL) IejEc-z'pL(r'ae'aZ')dZ'-

I:

 

Substituting this back into (C-5) gives

 

6%?“ --9_ +°° 3,.
I b((1)):e 27t[l—e 19*) I da' Idz'eJ‘“ pL(r',9',z’)

 

® {20: + 4; ( r—Tn Sin ”0‘ cos[n(9— e H}. (06)

C. 2.3 Solution

Now that an expression for the current has been found, one can write the complete,

general solution for the voltage signal seen by the oscilloscope. Substituting (06) into

(C-2) yields

 

1 M"Z o(co)A(co) ~1320 43: *°° 131'
_ c Be [So I I Be 7 I I
vosc(t) _ 47:2 _£1+jo)CpZ, e [l-e )Iifda idz e pL(r ,9 ,z)
" pipe

 

®{2a +4Z(a —)n Si" no” cos[n(e— 9 fined“). (C-7)
n=l
C.3 Approximate solution for short beam pulse
In general, the analytical forms for the charge distribution and the transfer functions
0(a)) and A((o) will be unknown. What is known, however, is that the electron beam
bunches from SPA are very short. The maximum expected full width at half maximum

length (FWHM) is 6 mm, or 20 ps. Because of the low-pass filter that signal passes

176
through, the beam pulse is essentially a delta function longitudinally for all frequencies of

interest:

(.0

[dz'e”jl’3"pL(r',e',z'); [dz'pL(r',e',z') (C-8)

to high accuracy. Substituting this into (07) gives

 

vosc(t) = 211:,— ! da'sz'pL(r',9',z'){2a + 42 (13:) " sinnno: cos[n(9 — 90]}

area of ~00 n--l
PIPe

 

m 4320 .0,
(8 [e 9° [1_e-7EL]Zco(w)A(m)

cjm‘d . C-9
27: 1+ ijch m ( )

-®

CA Gaussian beam

To demonstrate that (C-8) is a reasonable approximation, we will demonstrate it using
a Gaussian longitudinal distribution.

If the beam bunches have a Gaussian distribution longitudinally, the total beam pulse
distribution is

2:2

‘_2
e 20

 

pL(r',e',z')—>p,.(r'.e') 2....-

From (C-8) two integrals can be defined:
I. = ldz'pdrzezz'): p:(r',e')

and

177

2.2
,0) . " 2
-1—2 e 2"

1.: Jaye-Wmoaegz'):p.029»1m ..

 

27:6 '

Therefore, to show that (C-8) is valid, it is enough to show that a third integral,

2:2

_j£D_.z' e 202

+oo
I3 = sz'e B“
—cn

 

27:0 ’
is very close to one for all frequencies of interest.

To do I3 , first complete the square:

 

 

1 19m

I =
3 2no_w
Making the definition
_1_ L21

2 27::3'e-2 -- —o
dxfioe‘x = =e 2 9°
I JZno

-0

 

 

“1%)! .. ' £0). ‘[‘° )’

27:0

1 a) 2
To see how close I 3 is to one, the function e 2(3" ) versus frequency is plotted in
Figure C-2 for a beam bunch with a FWHM length of 6 mm, or 20 ps. This is the
approximate maximum length beam bunch expected. The relationship between a, the

rms width of the pulse, and the FWHM for a Gaussian pulse is

FWHM FWHM
o = 5
241n4 2.35

 

As can be seen, even at 1 GHz this function is very close to one.

 

1.0000 ,
0.9998 «»
0.9996 +—
0.9994 1

0.9992 1-

Function Value

0.9990 «-

0.9988 « p

0.9986 ‘

 

178

1

 

 

-fl, _

A

 

 

0.9984
0.0

1.0

r

2.0

A
1

A
f

3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

Frequency in MHz

1 co
Figure C-2: The function e 49° ] vs. frequency for a 6 mm, or 20 ps, FWHM long

electron pulse.

Appendix D

Appendix D

EXPANSION OF BPM SIGNAL AMPLITUDE IN CARTESIAN COORDINATES

D.l Introduction

In Appendix C an expression, equation (C-9), was derived for the voltage signal seen
by an oscilloscope due to the signal generated in a BPM electrode by an electron beam
bunch. In fact, it is the amplitude of this voltage time signal we are interested in.

In this appendix, the first few terms of the amplitude in (C-9) are converted to
Cartesian coordinates. These few terms are then shown to be important beam parameters.
In Chapter 1, it is demonstrated how the BPM electrode signals can be exploited to

measure them.

D.2 Expansion of amplitude in Cartesian coordinates

The amplitude in equation (O9) is

 

A=-4—1— I da' Idz'pL( r ,,9'z ){2a+42(3a— 4..) Sinnnac04n(9— 9)]} (D-l)

area of -co
P'Dc

in cylindrical coordinates. To change to Cartesian coordinates, start by making the

substitution

pL(r',9’,z') —> pL(x',y',z’).

179

180
Then, for simplicity, change the bounds on the transverse integral from being just over

the beam pipe area to i 00 in x and y. This is justified because the beam is, by necessity,
confined to the interior of the pipe. Including areas outside the pipe by going to :1: 00 will

contribute nothing to the integral. Therefore, (D-l) becomes

{em-MOM

A=4'l_(x2”lpn ",,y z)

-¢D-€D—®

 

®{2a + 42C _:..n) 8'“ "a cos[n(9— 9 )]}dx dy'dz (D-2)

D.2.1 Expansion of summation term in Cartesian coordinates

Looking at the summation in (D-2) out to n = 3 gives

 

2a+42(a -)n Sinnnacoinw— 9')]= 2a+4sinar—'cos(9— 9')

+ 2 sin 20:(%) 2 cos[2(9— 9' )] + -sin 3o:(a —) cos[3(9— 9’ )] + O(: —).

Exploiting the relationships
x' = r' cos9'
and
y' = r' sin9' ,
these terms can be written in Cartesian coordinates.
Term 1: 2o:

4 sino:

 

Term 2: 4 sino: i—cos(9 — 9') = r'[cos9' cos9 + sin9' sin9]

 

l8l

= 432nm (x' cos9 + y' sin9)

 

I

2 .
2 2
Term 3: 2 sin 2a(£-) cos[2(9 — 9')] = 5:12 o: r'2 [cosZ9' 90529 + sin 29' sin 29]
a

 

 

 

 

2 ' 2
= 8:: 0Lr'2[(cosz 9' —sin2 9')cosZ9+25in9'cos9’sin29]
= 28:1220t[(x,2 -y'2)cos29 +2x'y'sin29]
4 ' 3 4 ' 3a , ,
Term 4: 33in3a(r:) cos[3(9—9') =38"; r"[cosB9'cos39+s1n39's1n39]

= % sizga r'3[(cos3 9' — 3cos9' sin2 9')cos39 + (3cos2 9' sin9' — sin3 9')sin39]

 

= 1;- sizga [(x'3 — 3x'y'2)cos39 + (3x'2y' — y'3)sin39]

 

Therefore, (D-2) becomes

 

1 MM‘HD 4 - .
A = In? jjij(x’,y',z'){2a+ 8:"! (x'cos9+ y'sm9)

-ao-oo—ao

 

 

2 $211220: [(x'2 — y'2)c0829 + 2x'y' sin29]
4Sin3a I3 I I2 I2 I I3 '
+5 a3 [(x —3xy )cos39+(3x y —y )sm39]
+ 0(ai‘) }dx'dy'dz'. (D3).

D. 2.2 Expressing amplitude as a sum of moments
In Chapter 1, two averages where defined. The first is called an ensemble average.

This is the average of a quantity over the beam distribution in the coordinate system

 

182

whose origin corresponds to the center of the beam. An ensemble average is denoted by

angled brackets:

j j j quantity x p(x)dV

Origin corresponds to beam center

llldildv ’

Origin corresponds to beam center

 

(quantity) (D—4)

where p(x) describes the electron beam’s spatial distribution. By definition

<X>=(y)=<2)=0.

The second average defines the transverse, or xy, origin to correspond with the center of
the beam pipe. The longitudinal, or z, origin corresponds to the longitudinal center of the
beam bunch. This average will be referred to as a BPM average and is denoted in the

following way:

”Iquantity x p(3‘:)dV

xy origin corresponds to beam pipe center. 2 origin to beam bunch center

llldildv

xy origin corresponds to beam pipe center, 2 origin to beam bunch ccnte

 

(quantity) am a (D'S)

The average values of x and y with this definition yield the x and y position of the center
of the beam with respect to the beam pipe and
(z) = 0.
The first integral in (D-3) is

j I ij(x',y',z')dx'dy'dz' = q,

where q is the total charge in the pulse. This is by definition of the beam bunch

distribution function. The other integrals in (D-3) are BPM averages defined by (D-5)

and multiplied by q. That is

 

183

 

l 4sino: ,
A = 47:2 {20: + a Q<X>Bm cos9 + q<y>BPM s1n9)

23i
a

 

2
+ 2 0‘ [q(<x2 >BPM _ <y2 >BPM )c0529 + 2q<xy>BPM sin29]

4 ' 3
+ 3 81:3 a [q(<x3>BPM — 3<xy2 >BPM )COS39 + q(3<x2y> BPM — <y3>BPM)Sin 39]

+091

The primes have been dropped for convenience. Typically, the x and y center positions

 

of the beam, given by (x)BPM and <y>9m are denoted by x and y, respectively.

Therefore, this becomes

4 .
A = %{2a + L?E(2cos9 + Vsin9)

2 srr122a [((x2 >9m — (y2 >9m )c0529 + 2<xy> BPM sin 29]

 

a

4 ' 3 -
+ 3 81:3 o: [((x3 >9m - 3<xy2 >9m)°°S3e + (3(x2y)BPM — <y3 >BPM)s1n 39]

+o(;1—,-)}. (D-6)

The term (x2)BPM — <y2>9m in (D-6) is given special treatment. Let us look at the

 

integral of x2 over the beam bunch density distribution.

M44344) WWW

LHXZPJXJ, z)dxdydz = Lulfi — 2x2 + 2x2 + 22 — x2)pL(x,y,z)dxdydz

= [flux—XV +2xx—x2]pL(x,y,z)dxdydz

"Q—Q—Q

r‘,-*1

 

184
= q[<(x — 302 >am + x2 ].
Because of the definitions of the ensemble and BPM averages given in (D-4) and (D-S),

subtracting the term 2 from x inside the angle brackets changes the BPM average to an

ensemble average. That is

92>...aka-92>...921992922).

Similarly

q(y2 lam = q[<(y - imam + Y2] = q((y2)+ Y2 )

Therefore,

 

 

Appendix E

Appendix E
FIRST ORDER CALCULATION OF THE TRAJECTORY OF AN ELECTRON

DUE TO IDEAL FAST DEFLECTOR FIELDS

E.1 Introduction

 

A fast deflector is a cylindrical cavity operating in a TMno mode. In this mode the
dominant field is a time varying dipole field transverse to the z direction. This is shown
to be true in the first part of this appendix. In the second part the trajectory of a

relativistic electron through this dipole field is calculated to first order.

E.2 Electric and magnetic fields of fast deflector
The fast deflector is a cylindrical cavity that Operates in a TM“0 mode. A schematic

is shown in Figure E-l. The ideal electric and magnetic fields are

E, = EOJ,(k,,r)cos9cos(mt+¢), (E-l)
a2 . .
Br = a) 2217c,- E0J1(k“r) sm9 s1n((ot + (b) (E-2)
and
B, = (o x—ac—,E,J;(knr)cos9sin(wt + 4)). (13-3)

All other field components are zero. The constant kH is defined as

185

 

538 .880:ch “mam mo oumaosom ”Tm 2:me

 

 

186

 

 

 

 

 

 

 

187

x
a
where xll , equal to 3.8317, is the first zero of the first Bessel fimction and a is the radius
of the cavity. The angular frequency, (9, is given by

a) = 27:f
where the frequency, f, is 1300 MHz for our cavity. The relationship between the angular
frequency and the geometrical properties of the cavity is

a) = knc.
Therefore, the cavity radius is

a-Efi-flfi-OMm
- a) —27:f-' '

The length of the cavity, L, is independent of frequency and on SPA measures 14.48 cm.
The maximum amplitude of the electric field, E0, is 24MV/m, from measurement. The
kinetic energy of the electrons as they enter the cavity will be approximately 8 MeV. The
aperture of the cavity, the opening that the beam travels through, is one inch in diameter,

or 1.27 cm in radius.

E.3 Transformation of fields to Cartesian coordinates
The magnetic field vector is given by
1‘3 = 133+ B99+Bzi = 13,? + 13,9
where B2 = 0. The relationships between the cylindrical coordinate unit vectors and the

Cartesian unit vectors are

188

f: 710089 + ysin9

and

9 = —xsin9 + ycos9.
Therefore,

B = B,(xcos9 + ysin9)+ Be(— xsin9 + ycos9)

= 52(BIr cos9 — B6 sin9) + 9(13, sin9 + B6 c059).

This gives

Bx = B, cos9 — B9 sin9 (E4)
and

By = B, sin9 + B9 cos9. (E-S)

Substituting (E-2) and (E-3) into (E-4) gives

2
13K = [to —5a-—2-J,(knr)E0 sin9cos9 - a) ——a-7J;(k”r)Eo sin9cos9]sin((ot + (b)
xnrc xnc

80)

= WED sin29[x—:;J,(k”r)— J{(k”r)]sin(rot + (1)).

A property of Bessel functions is
xJ{(x) = J,(x) — xJ2(x) [33].
Therefore,

11(X)
X

 

J;(x)= —-Jz(x).

Remembering that

189

 

xnr knr knr
- 12(knr)
Therefore
13, = £7J2(knr)Eo sin29sin(mt+¢). (E-6)
11

Substituting (E-2) and (E-3) into (E-S) gives

 

2
B, = [9;2a—m7EoJi(knr)sin2 9 +9 x11. E.J:(k..r)cosz ejsintnt + 9)

ma

 

Eo[;3—J,(knr)sin2 9 + J{(k”r)cos2 9] sin((ot + 11>).

2
xllc 111.

Using

J{(k”r) = ;a—rJ,(kllr)—J2(knr)

ll

 

gives
By = xi; Eo[fI-;J,(knr)sin2 9
+;:—IJ,(knr)cosz 9 -J,(k,,r)cos2 9) sin(o)t + ¢)
13y = $Eo(fi;ll(knr)—J,(k,lr)cosz 9] sin((ot + 4)). (E-7)

The Bessel functions J , (knr) and J 2(knr) can be written as infinite series. The first

190

terms of these series are

a:
2

and
l 2
§(k”r) 7
respectively. The value of kll is
x11 -1
kll = —— = 0.2737 cm .
a

The electrons are limited to the region defined by the aperture through which they enter
the cavity. If these two functions are plotted versus radius as in Figure E-2, it is apparent
that the first term is a very good approximation of the full Bessel fimction in each case
when the maximum value of the radius is limited to the value of the aperture radius, 1.27

cm. Therefore, (E-l), (E-6) and (E-7) become

 

k
E, E Eo-—2]-'-xcos(mt+¢) , (E-8)
an) 1 . COX .
Bx 5 :1? E0 Zkflxy srn(cot + o) = 4an ony srn(0)t + 4)) (E-9)
and
B = —‘°a—E (1— ixzj sin(mt + M (13-10)
V _ 2xnc2 0 4a2 '

The y magnetic field, given in (E-lO), has a small quadratic term. However, as can be

seen in Figure E-3, this results in, perhaps, a three percent variation in field strength along

 

191

 

0150‘”

0.040"

0.020 "

 

 

P
4P
qt-

 

'11-
y.
~41-
b
1..

0.(X)0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Radius (cm)

a)

0.8 0.9 1.0 l . 1 1.2

 

0.019
0.014 1
0.012 1
0.010 7

0.(X)8 ‘

Function Values

0.“)6 i-

0W4 '1‘

 

 

 

 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Radius (cm)

b)

0.8 0.9 1.0 1.1 1.2

1:
Figure E-2: a) J,(k“r) (blue) and 421: (red) versus radius. b) J 2(knr) (blue) and

1
-8-(l:nr)2 (red) versus radius.

 

 

192

1.000 ,

 

0.995 ._
0.990 ._
0.935 -.
0.930 4.
0.975 ,

0.970 4 ~

 

Function Value

0.965 -»
0.960 ..

0.955 T

 

 

0.950 : : : 1e : : : : : : t t
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

 

x (cm)

2
_ ll 2
4a2

Figure E3: 1 vs. x (cm).

 

193

the x axis. Therefore, this term will also be dropped, leaving

B ~—2a—E '
,zzxnc, osrn(0)t+¢). (13-11)

E.4 Demonstration of dominant field

Of the three field components present inside the fast deflector cavity, it is the y
magnetic field given in (E-l 1) that has the dominant effect. This can be seen if we first
calculate the magnitude of the ratio of the x magnetic field to the y magnetic field. Using

(E-9) and (E-l 1) this is

 

 

 

 

 

 

 

 

(0X11
B, < gacz onmaxymx _ xflrjm _ (3.8317)’(1.27cm)2 _ 0 03
lByl- C03 E _ 2a2 _ 4(14.Ocm)2 _ ° °
2xnc2 0

Therefore, the x magnetic field magnitude will be, at most, three percent of the y
magnetic field. Therefore, the x magnetic field can be ignored.

To demonstrate that the electric field can also be ignored, an estimate of an electron’s
energy change as it passes through the cavity is necessary. Since the fast deflector cavity
is not intended to give, or take, energy from the electrons, it is expected that this change
will be slight.

The equation of motion for the longitudinal direction is

d , _ __
5(sz)-qEz — eEz

Since the electron are already highly relativistic, the longitudinal velocity will not change

substantially. Therefore,

 

 

 

194

d'z
-—=0
dt

leaving

and substituting (E-8) for E2 gives

dy _eE ok,___,_x
dt — 200m ——cos(0ot+¢).

To estimate the energy change, the time dependence of x will be assumed to be

constant. This makes integrating this equation trivial:

eEoknx sin(0)t + ‘1’)
213cm 0)

 

AYE-

 

Gigi: Yout —Yin
in

tin
The time that an electron exits the cavity, tom, is equal to the time that it enters the cavity,

tin, plus the time it takes for it to traverse the fast deflector. Therefore

 

1,, at, +£
_ BC
and
A ~_eEok11X . [I +_L_)+ _ . ( t + ) (E12)
y: 20cm00 srnro in [30 ¢ srn00in :1) . -

For a given electric field amplitude, it is easy to find the maximum energy change from
(E-12). Figure E—4 shows the maximum relative change in gamma versus x for an 8 MeV

electron when E0 is set to its maximum value of 24 MV/m. The largest relative energy

195

1.000 «

 

0.995 ~

0.990 . »

0.985 «»

0.980 -

0.975 « »

0.970 -~

0.965 . »

l 0 - Fractional Energy Change

0.960 < -

0.955 ~ »

 

0.950 ‘. : 4 .T

 

 

 

0.0 0.1 0.2 0.3 0.4 0.5 0.6

x (cm)

A
Figure E-4: 77- vs. x (cm) for an 8 MeV electron.

0.7

0.8

0.9

1.0

a". '

 

196
change is about four percent near the edge of the aperture. Since the energy change will

generally be much smaller than this, the electric field can also be neglected.

E.5 Solution to the first. order equations of motion

If only the y magnetic field in (E-l 1) is kept, we are left with a simple equation of

motion: ! .
'.....- l-
q

(1 .~.
a—t-(me) = —q'sz = esz. '

 

Substituting (El 1) into this equation and setting

ymii = $5213;on sin(o)t + ¢). (E-13)

To solve (E-13), we will first change to more useful coordinates. First,

 

a‘aa'za‘ma-
Therefore
2 dzx ma ,
ym(Bc) y = eBCmEO sm((ot + 4)) ,
or
e cm
H = E ' t .
x Bymc 2x110, 0 sm((n + ¢)

The value of t is given by

197
t _ a
_ [30 ,
where z is the electron’s longitudinal location and Be its longitudinal velocity. It will be

defined that 2 equal to zero is the entrance to the fast deflector cavity. Therefore

e ma (oz
" :: E o —+ . E-14
x Bymc 2xnc2 0 sm[ BC (1)) ( )

 

It is easy to integrate (E-l4). Doing this yields

6 3 (DZ
X'(Z) = _‘Y_1;-2-:CZ— E0 005(13:+ 4)] + C1, (E-IS)

where cl is a constant. Integrating again gives

x(z) = _i a

Be, 032
mmEogsm —+¢ +ClZ+C2, (E-16)

[3c
where c2 is also a constant.

The position and divergence of the electron at the fast deflector cavity entrance are

defined as

xi 2 x(0)
and

x; 5 x'(0) .

Substituting these values into (E-15) and (E-16) and solving for the constants c1 and c2

results in

 

a
el = x.' +-— x c2 EO cos¢ (E-17)

and

 

 

198

—sin¢. (5'13)

E.6 Electron position and divergence at exit of fast deflector cavity

In practice, we are not concerned very much with the trajectory of the electrons
through the fast deflector. It is the final trajectory, at the cavity exit, that is more
important. Also, since the electrons will always be part of a beam bunch, the notation
used to this point needs to be modified slightly.

At the exit to the fast deflector cavity, 2 is equal to L, the cavity length. Using (E-l 5)

and (E-16), the position and divergence of the electron at the cavity exit are given by

 

e a Bc _ 03L
XL = X(L) = ——'YI;2X”CZ E0 ES"{E;+¢) +CIL+C2 (E-19)
and
x' - x'(L) - -—-e——a—-E c049£+¢j + c (E-ZO)
L ym anc2 0 [3c "

The phase angle, ‘1’: can be rewritten as
<l> = in) + M + 4)..
The angle (ho + A4) is defined as the phase of the beam bunch center with respect to the
cavity fields. The angle A4) is included for calibration purposes. Its magnitude is defined
to be much less than one. The angle 4), is defined as the phase of a particular electron in
the bunch with respect to (to + Ad).

As mentioned in Chapter 1, the beam bunches on SPA are 6 mm F WHM in length or

less. This is a time duration of 20 ps. At the frequency 1300 MHz, then, the magnitude

 

 

199

of (b, will be

[4),] _<_ (10 ps)(21tf) = 0.082 .

Therefore,

<<1

 

IA¢+¢.

and one can make the following approximations:

and

wL (0L
COS(E+¢O +A¢ +¢zj = COS(E+¢O) 005(A¢+¢z)
_S'm(CE—:’+¢0) sin(A¢+¢z),
=cos(9£+¢ )[l—(A¢+¢ )2]—sin(gli+¢ )[(A¢+¢ )‘(M’Hl’ )3]
_ BC 0 2 BC 0 z z
cos(c;—:+¢O +134; 4.4),) z cos(%+¢0) —(A¢+¢,)sm(%+¢,], (5-21)
, mL . (0L
smlgwo + 13¢ W.) = smlgwo) COSW’ +¢zl
+cos(m—L-+¢ )sin(A¢+¢ )
BC ° ‘ ’

sin(%cli+¢o + Ad) +45) 5 sin((;—:'+¢O) +(A¢ +¢z)cos(%:;+ (be) , (E-22)

cos¢ = cos(d>o + M) + (bl) = coscb0 cos(A¢ + 92)‘ sim1>0 sin(A¢ + (1),),

cos¢ s 005% — (mp + ¢z)sin¢0 (13-23)

200

sin¢ = sin(<t)o + A4) + (1)) = Sin¢o cos(A¢ + (1)2) + sin(A¢ + (b1)cos¢l)0 ,
sin¢ s sin<t>0 +(A¢+¢z)cos¢0. (E-24)
Substituting (E-Zl), (E-22), (E-23) and (E-24) into (E-17), (E-18), (E-19) and (E-20)

yields

. L L
x, = _%fiao%[sm[%;+¢ol +(A¢+¢.)cos(-“§;+¢ol]

l .40..-.wmwl

 

fifi‘aofiglsiwo+<A¢+¢zl°°sl>ol

, e a BC . (0L . (0L
xL = Xi + LXi +RmEo ;{Sm¢o +Ecos¢o “Sln(—B;+¢o)

+ (A4) + ¢Z)[cos¢o —:—:sin¢o - COS[% + 90):” ,

and

 

e a (0L
xi, = X: +;m—2x”c2 Eo{cos¢o “003(B_c+¢o]

+(A¢ +¢z)[8in((g_:+¢°l — studio}.

 

Appendix F

 

Appendix F
COVARIANCE MATRIX AND THE STANDARD ERROR IN THE

CALCULATION OF THE EMITTANCE

F.1 Introduction

As demonstrated in Chapter 2, the rms beam parameters, (x2), <x'2>, (xx') , <y2>,

<y'2) and <yy'), are determined by inverting a matrix equation. The x and y rms

emittances are then calculated according to

 

a. J<x2><x'2>-<xx'>2 (F-n

and

 

8y J<y2><y'2)-<yy'>2 . (F-2)

In this appendix, expressions for the standard errors for the rms beam parameters and the
x and y rms emittances are derived.

The standard errors in rrns beam parameters are determined from the well known
covariance matrix[34]. The expressions for the standard errors for the x and y rrns

emittances were first demonstrated by Miller et al.[12].

201

 

 

202

F.2 Estimating measurement errors
In this section, an estimate for the measurement errors in a linear set of equations is
derived. It is assumed that these errors are normally distributed.

Assume the general matrix equation,
Ax = b ,
where A has m rows and n columns, i has dimension n and B has dimension m. The

vector 5 is determined by measurement. Therefore, there will be some standard error, 0',

associated with the individual entries of 5. Although it might be true that the value of
this error is well known from the measurement process, it can also be estimated from the

data.
Assume that m > n and A is of rank n. To solve for 'x’ , first multiply both sides by
the transpose of A , leaving

TB.

>u

TX:—

>1:

AT; is a positive definite square matrix and can be inverted. Multiplying both sides of

the equation by its inverse gives

.4

The matrix AT; has a set of n eigenvectors v, defined by

1:: _.
ATb.

>1:

1;)

and

Al:

F. '

 

203

where A, is the corresponding eigenvalue. The set of v, ’s form an orthonormal basis in

the space spanned by it. Therefore, one can write x and the vector ATE as linear

combinations of the eigenvectors:
n
i = Z civ, (F-3)
i=1

where the constants, ci , are defined by the dot product

‘

i-vi,

Ci

and

where

Dotting '9, into both sides,
v, (2mm) = cjkj = Vi (Edith) = dj,
i=1 i=1
substituting in the expression for d j and solving for c j gives

J

Using (F -3) gives

 

204

v. . (F-4)

.l

.. (3TB) - v,

H 7L]-

The vector 5 can be rewritten as
B = 50 + 5,.

The vector 50 is the ideal value of [3. The vector 6,, contains the errors associated with

5. Substituting this into (F -4) gives

 

n 3TB ‘ l.‘ a TB -‘. n K" -‘.
i=x0+éx=z[ (28") vjvj=Z—( 1°) v’v,+Z———( 2’) ’vj.
i=1 1' i=1 j i=1 j

So, the ideal value of x is

 

II 1:15 -‘.
j-r( X0.) vjvj

J

X0:

and its error is given by

n =T‘ -"
a, = 1:21ij . (F-S)

J

Because a, is unknown, 'e‘ll is also unknown. However, ‘e‘, can be estimated. To do

this, start by recognizing that
(ital-(git) = Sallie),
= ViiT=vj = HIV.
(Xvi).(.ivj) = it,
NOW, we make the definition that there is a set of m orthonormal vectors, 2, , that span

the space of 5. The first 11 of these vectors are defined by

 

 

205

 

gi E lAv:| = J7: . (F-6)

6.. = Zatz. (F-7)

where

 

'é,= in. (F-8)
i=1 7% ’
Bydefinition
-T=T=‘
=_. _ _=_ viAAv.
(ATgi)-v12g;r VlzTL,

 

when is n and
(Talia, =0

when i > 11. Substituting these results into (F -8) gives

 

t=ziaf—,—v '1 =2f (M)

j=l i-l jsl
Each element of 6,, is the error in the measurement of the corresponding element of

5. Assume that these errors are normal and, on average, equal. Then the magnitude of

 

ZCNS
the vector 'éb is

|'é,|=,/é,-'é, st 2 =05

where o is the average measurement error. Using (F -9) one gets

=lzagl [Easy-l iaj Em

i=|

Therefore, on average,
2
2 — ‘ ‘ ~ 2
aj =(eb -g,) :0.

The vector 5 can also be expanded as a sum of g, ’s.

b: i“ gi)gi

=1

ZG'EE = filo-50 +éb)'§a]§t ,

i=1 I=l

2(5.gi)g =Z[(b +6, )g,]gi+ 210% +éb).gi]§i.

i=l i=1 i-n+

Because 50 is perpendicular to g, when i > m,
2;.(5 °§i)§i = 20; ii); + iéltéb 'gi)§i a
or

b- (52.)E= 266.. 202:

is al i-n+l

Taking the square of both sides and using (F-IO) yields

[B-i(fi-g,)§,]z = m, -g,)2 Etna—w.

i=1 i=n+l

(F-lO)

 

 

207

Therefore

62; 1 [B—Z(B-g.)g,]2. (F-lO)

 

m—n i=1

By the definition in (F-6),

 

 

 

(5 ' gi)gi = [iTvi Kim)-

Since 31%, is a constant,

(B - g); = (3v,)iTv, = A9,??? = 3(v, -v,)i,

-.
—.

(Bale. -- Ai-
Putting this into (F -10) yields

2 1
m-n

 

III

0' [B— ‘]2 (F-ll)

F.3 Covariance matrix
In this section the Covariance matrix is defined and its relationship with the estimated
errors in the components of i is established

To begin, define the matrix

208

i

E

: = -|
(ATA) .
The n eigenvectors of the matrix ATA can be written as

(V0
Viz

<l
II

 

 

\Vin)

This allows E to be written as

”v“ v2| . . . vulqu/l.1 O . . . 0 "v” v12 . . . v,“1
v12 . v“2 0 1/2»2 . v21 . V2“
is: [20].
. . . . 0 .
_vln v2n . . . vm‘L O . . . 01/).n__vnl vn2 vnn_

 

 

 

 

Multiplying this out, one gets the symmetric matrix

B 1'21. 0 Viiviz vilvin 7
:3x 2 k . . . z—x

i=1 i is] i i=1 i
n 2

n vilviz Xi
2'? Z t

i-l i i=1 i

 

 

i: = (F-12)
: vi(n-l)vin
i=1 2'i
: vilvin ' i vi(n-l)vin " _\i
L is] M . . . i=1 xi i=1 7‘1 _

The covariance matrix is defined as

pan

(:2 o2 (F-13)

 

 

209

where 0'2 is given by (F-l 1).
Equation (F-9) is an expression for the error vector ‘e‘x :

‘

8ij

$221.5” H 2‘]—

 

 

j=l i=1
The estimated value of a j is given in (F -10). Therefore, the errors in the individual

components of i is given by

(e x.) =622— " (F-14)

F1 Aj

where exi is the error in the i‘h element of a, and vji is the ith element of the jth

eigenvector of AT; . The expression in (F -14) is just the expression for the diagonal
entries of the covariance matrix defined in (F -12) and (F-13). Therefore, the magnitude

of the error in the ith element of i is

 

 

e as C.. . (F-lS)

xi u

F.4 Estimated error in calculation of the emittance

For the particular set of linear equations dealt with in Chapter 1, n = 6 and

{<x2>\

(XX")
<6: 3 . (F-16)
(w’)
<

yr2>)

MI
W
N

 

 

K

From the elements of i , the x and y rms emittances are calculated according to (F-l) and

 

210

(F -2). In this section the errors in the emittances are estimated.
To begin, from (F -1), the x emittance squared is
82 = (x2><x'2>— (xx’)2 = x1x3 -— x;
where x1, x2 and x3 are the first three elements of the vector i given in (F-l6). Recall

that we can write

 

i = i0 + "éll ,
then
2 2
3x = (x0: + ex] Xxos + 6x3)“ (x02 + 6x2) -
Multiplying this out,

8i = me03 _ X32 + XOlex3 + xoaexi + exlex3 - 2x026x2 _ 6:2 - (F’17)
The first two terms in give the actual, or ideal, emittance
8:0 = x017%3 " X32-
The remaining terms give the error in the emittance estimate.
To find a value for the emittance error, first make the substitutions
x0l —> xl ,
x02 —> x2
and
x03 —> x3.
Since x01, x02 and x03 are unknown, they are replaced with x1 , x2 and x3 , the best

estimate of their values. Then, from (F -17), the error in the emittance squared is

~ 2
errei = x1e“3 + x3e“ +ex|ex3 —2x2ex2 —ex2.

 

211
This can be divided into two terms:

err82 -:- term1+ term2 ,

where

_ 2
terrnl — exle,‘3 — ex2

and
term2 = xlex3 + x3e“ -2x2ex2.
From (F-9),
{exl
ex2
. 6 a v
1". = _Z i i
. j l A)
le6)

 

 

Substituting this into (F -l 9) and (F -20) gives

‘:“““l.-1J"‘ll§J-’-’l léi—fil

 

and

 

 

For terrnl , we have

tam-l.l—’fll27-l list—l

 

 

 

(F-18)

(F-19)

(F-ZO)

 

 

 

 

Since one can only estimate the magnitude of aJ. and ai using (F-10), terrnl cannot be

reduced any further. However, its magnitude can be estimated by adding in quadrature.

That is

6 6

termlzz 22

j=l i=1

0...“

a. 2
jX"(Vj1Vi3 _ ijViZ)

:ylfa

4

6

G 2
22th (VJJsz3 -J.2v 1iv 3vj sz 2+vJ.2vJ22)
i=1 j '

 

Ill

6
.31

J

6 2 6 6
~ 2 vi! 2 V_i3)_ 2[ 2 V__jlvj2][ 2 Vi__2___Vi3]
= 0' 0' 0' 0'
[ g— :]( isl A’i j=l A" iszlfl A.
6 v2
_J_ 622%(0 22 _i_V2)
i=1 j is] A'i

The terms in parentheses can be identified as elements of the covariance matrix, (F-13).

 

Therefore
terrnl2 E CJJC33 —2CJ2C23 +C22C22 = CJJC33 +C§2 —2C12C23. (F-21)
For term2 we have
tenn2=x1[26:a’:-—i]+x3[za ’ —:—:l]— 2x2[:ajvj2:l
J— ,. . J?

a

6

J
=Z—(x vJ.3 +x 3VJJ— —2x2vJ.2).
J=l A'j

213
This can also be estimated by adding in quadrature:

2

° 2

—-J —2 v
(xlvj3 +x3vjl x2 jz)

trm22_==.
e X,-

6 a
PI
0,2
2 2 2 2 2 2
(xlvj3 +2xlvj3x3vj, —4x,vj3x2vj2 +x3vjl —4x3vj,x2vj2 +4x2vj2)
J

III
EM]
>’

Ill

6 v? 6 v.v. 6 v. v.
X26322?) +2x1x3[022_1 ’3] —4x1x2[622—: J3]
i=1 1' j

i=1 ,- i=1

6 v? 6 v. v. 6 v?

+ xiozzl—Jj -4x2x3[622-3;:f’3-] +4x§(GZZf-].

1“ J H J J=1 J

The summation terms can again be identified as elements of the covariance matrix, 63 ,
yielding

term22 5 xfC33 + x§C11+4x§C22 +2xlx2C13 —4x,x2C23 —4x2x3C,2. (F-22)
Since (F -21) and (F -22) are estimates of the magnitude of terml and term2, they
cannot be substituting into (F -1 8) to get the error in the emittance squared. However, by

again adding in quadrature, the magnitude of err£2 can be estimated:
err:2 s terml2 + term22
5 xfC33 + ngH +4x§C22 +2xlx2CI3 —4x,x2C23 -4x2x3C12

+ C11C33 + C22 - 2C12(323

Therefore,

2 _ 2 2
ex —x,x3—x2 j: errei ,

 

 

214

a: = <x2 ><x'2>—<XX'>2 i(<x2 >2C33 +<Xfl>zcn +4<xx'>2C22 +2<x2 ><x'2>C13

l

—4<X2><XX'>C23 —4<XX'><X'2>C12 +C11C33 + ng _2C12C23 )3

The emittance is then

 

8x = J<x2><x'2>-(xx')ierrei ,

 

811',

8x = \sz)<x'2>-(xx')\/1:t (XZXX'ZSx-(xx’) '

 

 

When the emittance is significantly bigger than the error,

 

 

|' err8i
8): = J<x2)<x'2>-(xx'>lli 2(<X2><x.2>_(xx'))i,

 

 

 

2 ’2-xx’2i 1 x22 +x'22
3x=\/<X XX > < > 2J<x2><x'2)-(XX')2 (< >C33 < >Cu

+4<XX'>2C22 +2<X2 ><XIZ>C13 —4<x2>(xx')C23 _4<XX'><X,2>C‘2

1

+ C11C33 + C22 — 2C12(323 )5-

Through a similar procedure, the y emittance can be found to be

_ 2 ,2 _ , 2 + 1 2 2 + '2 2 44
e,_J<y ><y > <yy> -2J<y2><yfl>_<yy,>,(<y > c, <y >c

+4<yy'>2C« +2<y2><y'2>C46 -4<y2><yy')Css - 4<yy'><y'2>C4s

 

 

 

l

+C,,C66 +C§, —2c,,c,,)5.

 

REFERENCES

”'4 '. shiny-1

REFERENCES

[1] R. Warren, “Star Wars and The FEL,” (private communication).
[2] C. A. Brau, Free-Electron Lasers, (Academic Press, Sand Diego, CA, 1990).

[3] J. M. J. Madey, “Stimulated Emission of Bremsstrahlung in a Periodic Magnetic
Field,” .1. Appl. Phys. 42, 1906 (1971).

[4] J. S. Fraser, R. L. Sheffield, E. R. Gray, P. M. Giles, R. W. Springer and V. A. Loebs,
“Photocathodes in Accelerator Applications,” Proceedings of the 1987 Particle
Accelerator Conference, IEEE catalog No. 87CH2387-9, 1705 (IEEE, New York,
1987)

[5] J. S. Fraser and Richard L. Sheffield, “High-Brightness Injectors for RF-Driven Free-
Electron Lasers,” IEEE J. Quantum Electron. 23, 1489 (1987).

[6] M. Reiser, Theory and Design of Charged Particle Beams, (John Wiley & Sons, New
York, NY, 1994).

[7] P. J. Tallerico and Jean-Pierre Coulon, “Computer Simulation of the Lasertron With a
Ring Model,” Proceedings of the 1987 Particle Accelerator Conference, IEEE
catalog No. 87CH2387-9, 1806 (IEEE, New York, 1987).

[8] B. E. Carlsten and S. J. Russell, “Subpicosecond compression of 0.1-1 nC electron
bunches with a magnetic chicane at 8 MeV,” Phys. Rev. E 53, 2072 (1996).

[9] B. E. Carlsten, J. C. Goldstein, P. G. O’Shea and E. J. Pitcher, “Measuring Emittance
of Non-thermalizcd Electron Beams From Photoinj ectors,” Nucl. Instrum. Phys. Res.
A 331, 791 (1993).

[10] K. Kim, “RF and Space Charge Effects in Laser-driven RF Electron Guns, “ Nucl.
Instrum. Phys. Res. A 275, 201 (1989).

[11] R. T. Avery, A. Faltens and E. C. Hartwig, “Non-intercepting Monitor for Beam
Current and Position,” IEEE Tans. Nucl. Sci. 18, 920 (1971).

215

 

216

[12] R. H. Miller, J. E. Clendenin, M. B. James, J. C. Sheppard, “Non-intercepting
Emittance Monitor,” Proceedings of the 12”? International Conference on High
Energy Accelerators, 602 (Fermilab, 1983).

[13] B. E. Carlsten, S. J. Russell, J. M. Kinross-Wright, M. Weber, J. G. Plato, M. L.
Milder, R. Cooper and R. Sturges, “Subpicosecond Compression Experiments at Los
Alamos National Laboratory,” Proceedings of the Micro Bunches Workshop, AIP
Conf. Proc. 367, 21 (AIP, Upton, NY, 1995).

[14] W. D. Cornelius, S. Bender, K. Meir, L. E. Thode and J. M. Watson, “The Los
Alamos High Brightness Accelerator FEL (HIBAF) Facility,” Nucl. Instum. Methods
Phys. Res. A 296, 251 (1990).

[15] B. E. Carlsten, “Calculation of the noninertial space-charge force and the coherent
synchrotron radiation force for short electron bunches in circular motion using the
retarded Green’s fimction technique,” Phys. Rev. E 54, 838 (1996).

[16] J. D. Gilpatrick, J. F. Power, R. E. Meyer and C. R. Rose, “Design and Operation of
Button-Probe, Beam-Position Measurements,” Proceedings of the 1993 Particle
Accelerator Conference, IEEE catalog No. 93CH3279-7, 2334 (IEEE, New York,
1993).

[17] J. D. Gilpatrick, H. Marquez, J. Power and V. Yuan, “Design and Operation of a
Bunched-Beam, Phase-Spread Measurement”, Proceedings of the 1992 Linear
Accelerator Conference, AECL catalog No. 10728, 359 (AECL Research, Chalk
River, Ontario, Canada, 1992).

[18] K. L. Brown and R. V. Servranckx, “First- and Second-Order Charged Particle
Optics,” Presented at the 3"‘1 Summer School on High Energy Particle Accelerators
July 6-16, (Brookhaven National Laboratory, Upton, NY, 1983).

[19] S. J. Russell and B. E. Carlsten, “Measuring Emittance Using Beam Position
Monitors,” Proceedings of the 1993 Particle Accelerator Conference, IEEE catalog
No. 93CH3279-7, 2537 (IEEE, New York, 1993).

[20] G. Strang, Linear Algebra and its Applications, Second Addition, (Academic Press,
Orlando, FL, 1980).

[21] L. M. Young, (private communication).

[22] K. R. Crandall and D. P. Rusthoi, “Trace3-D Documentation,” Los Alamos National
Laboratory report, LA-UR-90—4146 (1990).

217

[23] J. F. Power, J. D. Gilpatrick, F. Neri and R. B. Shurter, “Characterization of Beam
Position Monitors in Two-Dimensions,” Proceedings of the 1992 Linear Accelerator
Conference, AECL catalog No. 10728, 362 (AECL Research, Chalk River, Ontario,
Canada, 1992).

[24] R. E. Shafer, “Beam Position Monitoring,” Proceedings of the 1989 Accelerator
Instrumentation Workshop (Brookhaven National Laboratory), AIP Conference
Proceedings 212, (AIP, Upton, NY, 1989).

[25] J. D. Gilpatrick, J. Power, F. D. Wells and R. B. Shurter, “Microstrip Probe
Electronics,” Los Alamos National Laboratory document, LA-CP-89-488 (1989).

[26] J. D. Gilpatrick and F. D. Wells, “Double Balanced Mixer Operation used as Phase
and Synchronous Detector as Applied to Beam Position and Intensity Measurements,”
Los Alamos National Laboratory document, LA-UR-93-l622 (1992).

[27] J. D. Gilpatrick, “Microstrip Measurement Algorithms,” Los Alamos National
Laboratory document, LA-UR-93-1639 (1992).

[28] P. R. Cameron, M. C. Grau, M. Morvillo, T. J. Shea and R. E. Sikora, “RHIC Beam
Position Monitor Characterization,” Proceedings of the 1995 Particle Accelerator

Conference and International Conference on High-Energy Accelerators, IEEE
catalog No. 95CH35843, 2458 (IEEE, New York, 1996).

[29] J. A. Cadzow, Discrete-Time Systems, (Prentice-Hall, Inc., Englewood Cliffs, NJ,
1973)

[30] D. W. Feldman, S. C. Bender, B. E. Carlsten, J. Early, R. B. Feldman, W. J. D.
Johnson, A. H. Lumpkin, P. G. O’Shea, W. E. Stein, R. L. Sheffield and L. M.
Young, “Experimental Results from the Los Alamos FEL Photoinjector,” IEEE J.
Quantum Electron. 27, 2636 (1991).

[31] W. R. Smythe, Static and Dynamic Electricity, (McGraw-Hill Book Company, Inc.,
New York, NY, 1939).

[32] J. D. Jackson, Classical Electroaynamics, 2nd Edition, (John Wiley & Sons, Inc.,
New York, NY, 1975).

[3 3] W. H. Beyer, Editor, CRC Standard Mathematical Tables and Formulae, 29’h
Edition, (CRC Press, Inc., Boca Raton, FL, 1991).

[34] W. H. Press, S. A. Teukolsky, W. T. Vettering and B. P. Flannery, Numerical
Recipes in FORTRAN, The Art of Scientific Computing, Second Edition, (Cambridge
University Press, New York, NY, 1992).

 

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