w.
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07»

This is to certify that the
dissertation entitled

DESIGN, FABRICATION, AND TESTING OF
SUPERCONDUCTING RF CAVITIES FOR
HIGH AVERAGE
BEAM CURRENTS

presented by

David Joseph Meidlinger

has been accepted towards fulfillment
of the requirements for the

Ph.D. degree in Physics

 

 

Major Professor’s Signature

20 APRIL 1007-

 

Date

MSU is an affirmative-action, equal-opportunity employer

DESIGN, FABRICATION, AND TESTING OF
SUPERCONDUCTING RF CAVITIES FOR HIGH AVERAGE
BEAM CURRENTS

By

David Joseph Meidlinger

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Physics and Astronomy

2007

 

ABSTRACT

DESIGN, FABRICATION, AND TESTING OF SUPERCONDUCTING RF
CAVITIES FOR HIGH AVERAGE BEAM CURRENTS

By

David Joseph Meidlinger

For high current applications, it is desirable for the cavity shape to have a low
longitudinal loss factor and to have a high beam-breakup threshold current. This
dissertation describes three different cavities designed for this purpose: a six—cell el-
liptical cavity for particles traveling at the speed of light, a two—cell elliptical cavity
for subluminal particle speeds, and a single cell cavity which uses the TM012 mode
for acceleration. SUPERFISH simulations predict the peak fields in both of the ellip-
tical cavities will not exceed the TeSLA values by more than 10% but both will have
28.7% larger apertures. The elliptical designs assume the bunch frequency equals the
accelerating mode frequency. The beam pipe radius is chosen so that the cutoff fre-
quency is less than twice that of the accelerating mode. Hence all of the monopole and
dipole higher-order modes (HOMs) that can be driven by a Fourier component of the
beam have low loaded Q values. This simplifies the problem of HOM damping. The
TM012 cavity is predicted to have much higher peak fields than a 7r-mode elliptical
cavity, but offers potential advantages from its simplified shape; it is essentially a cir-
cular waveguide with curved end plates. This basic shape results in easier fabrication
and simplified tuning. Two prototype two-cell cavities were fabricated and tested at

cryogenic temperatures without beam.

..

 

1Fi-d—.‘L‘Ht. .k'

ACKNOWLEDGMENTS

I would like to thank my adviser, Walter Hartung, for his guidance, his professional-
ism, and his many helpful conversations throughout the years. I am the most grateful
for the unequivocal support, heartfelt encouragement, and selfless sacrifice of my wife;
Without her, none of this would be possible. This work was supported by the Na-
tional Science Foundation and the National Superconducting Cyclotron Laboratory

Fellowship.

iii

 

Contents

1 Introduction
1.1 Linear Accelerators ............................

1.1.1

Energy Recovery Linacs .....................

1.2 Radio-Frequency Cavities ........................

1.2.1
1.2.2
1.2.3
1.2.4

The Elliptical Cavity .......................
Superconductivity .........................
Issues Driving Cavity Design ...................
Outline ..............................

Superconducting Cavity Design

2.1 Waveguide Modes .............................
2.2 General Expansion of Electromagnetic Fields Inside a Hollow Cavity .
2.3 Applications ................................

2.3.1
2.3.2
2.3.3
2.3.4
2.3.5

Lossless Cavity ..........................
Cavity with Losses and Quality Factor .............
Wall Losses ............................
External Ports and External Q .................
Slater’s Theorem .........................

2.4 Cavity Figures of Merit ..........................

2.4.1
2.4.2
2.4.3
2.4.4
2.4.5
2.4.6
2.4.7

Accelerating Gradient and Transit Time Factor ........
Quality Factor and Geometry Factor ..............
Geometric Shunt Impedance ...................
Surface Electric Field Ratio ...................
Surface Magnetic Field Ratio ..................
Cell—to—cell Coupling .......................
Longitudinal Loss Factor .....................

Six-Cell Elliptical Cavity

3.1 The Approach to High Beam Currents .................
3.2 Six-Cell Design ..............................
3.3 Higher Order Modes of Interest .....................

Two-Cell Elliptical Cavity

4.1 Two-Cell Cavity Design .........................

4.1.1

SUPERFISH Simulations ....................

4.2 Higher Order Modes of Interest .....................

iv

 

5 Prototyping 73

 

5.1 Fabrication ................................ 73
5.1.1 Electron Beam Welding ..................... 74

5.1.2 Waveguide Coupling Measurements and Choice of Beam Tube
Length ............................... 77
5.1.3 Bead Pulls ............................. 84
5.2 Testing ................................... 86
5.2.1 Antenna Measurements ...................... 87
5.2.2 Cavity Preparation ........................ 90
5.2.3 Vertical Tests ........................... 96
6 Single-Cell HOM Cavity 112
6.1 Concept .................................. 112
6.1.1 Pillbox Trends .......................... 114
6.2 Cavity Design ............................... 124
6.2.1 Rounding the Comers ...................... 126
6.2.2 Adding the Beam Tubes ..................... 128
6.2.3 Loss Factor ............................ 132
6.2.4 Comparison with SOLEIL Cavity ................ 134
7 Conclusion 136
7.1 Summary ................................. 136
7.2 Implications ................................ 137
7.3 Possible Future Work ........................... 138
A Waveguide Modes 140
A.1 Maxwell’s Equations ........................... 140
A2 Circular Waveguide Modes ........................ 145
A.2.1 Transverse Electric Modes .................... 145
A.2.2 TE11 Mode ............................ 149
A.2.3 Transverse Magnetic Modes ................... 151
A24 TM01 Mode ............................ 152
B General Expansion of Electromagnetic Fields in a Cavity 154
3.1 The Complete Orthogonal Set ...................... 154
B.1.1 The Solenoidal Fields ....................... 155
3.1.2 The Irrotational Fields ...................... 159
B2 Time Evolution of the Expansion Coefficients ............. 162
C Analysis of CW RF Measurements 167
CI Direct Method ............................... 168
C2 Indirect Method Using 5:: ........................ 169
C3 Graphical Assessment of Systematic Errors ............... 172
Bibliography 174

v

List of Figures

1.1
1.2
1.3
1.4
1.5

3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11

4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8

5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9

TM01 field pattern ............................
Geometrical parameters used in designing the elliptical cell shape

The dispersion diagram for the TM010 passband. ...........
Multicell elliptical cavity cartoon ....................
The cell-to-cell variation in field intensity for a passband .......

ERL block diagram ............................
Cartoon of HOM and Low-frequency bunch spectrum .........
Cartoon of HOM and High-frequency bunch spectrum .........
Screenshot of the automated interface to SUPERFISH ........
6—cell cavity shape and electric field lines of the 7r-mode ........
The surface electric and magnetic fields for the 6-cell cavity ......
Field flatness plot for the 6-cell cavity ..................
Six-cell monopole HOM coupling strength ...............
The R/Qs for the monopole modes as a function of frequency.

Six-cell dipole HOM coupling strength .................
The R/Qs for the dipole modes as a function of frequency. ......

2-cell cavity shape and electric field lines ................
2-cell cavity surface electric and magnetic fields ............
2-cell cavity Ez along axis . . . .' ....................
2-cell and 6-cell cavity transit time factors ...............
Two-cell monopole HOM coupling strength ...............
The R/Qs for the monopole modes are plotted vs. frequency ......
Two-cell dipole HOM coupling strength .................
The R/Qs for the dipole modes of the two-cell cavity as a function of
frequency. .................................

Picture of 2-cell cavity half-cells .....................
Picture of electron beam welding ....................
Picture of a completed 2-cell niobium cavity ..............
Conceptual drawing of the 2-cell cavity .................
Cartoon of waveguide coupling field pattern ..............
Drawing of waveguide coupler test setup ................
Measured waveguide coupling Qext ...................
Dependence of waveguide Qext on the gap distance ..........
2-cell cavity beadpull ...........................

Images in this dissertation are presented in color.
vi

8
9
10
11
11

49
51
52
60
61
61
62
62
63
63
64

66
67
68
69
71
71
72

72

74
76
77
79
80
81
82
83
84

 

--

5.10 Diagram of the vertical test setup in the Dewar ............
5.11 Measured Qext for vertical test input coupler ..............
5.12 Measured Qext for vertical test pickup probe ..............
5.13 Picture of BCP in the acid room ....................
5.14 2-cell cavity ready for BCP ........................
5.15 High pressure rinse of the 2-cell cavity .................
5.16 The 2-cell cavity attached to the insert while in the clean room . . . .
5.17 RF system block diagram ........................
5.18 First round of Q0 measurements for the first 2—cell cavity .......
5.19 Second round of Q0 measurements for the first 2-cell cavity ......
5.20 Zero-mode and 7r—mode peak magnetic fields ..............
5.21 Q0 measurements of the second 2—cell cavity ..............
5.22 Residual resistance estimate .......................
5.23 Change in 7r—mode frequency with pressure ...............
5.24 Lorentz force detuning of the 2-cell cavity ...............
5.25 Second 2-cell cavity peak magnetic fields ................

6.1 A cartoon of the HOM cavity electric field pattern ...........
6.2 Pillbox TM012 transit time vs. particle speed .............
6.3 Pillbox TM01 transit time vs. length/ radius ..............

6.4 Pillbox trend for the maximum 6. ...................
6.5 Pillbox TM01p G increases with p ....................

6.6 Pillbox trend for 72%,); ..........................
6.7 Pillbox trend for the number of parasitic modes .............
6.8 A comparison of TM010 and TM01p magnetic field patterns .....

6.9 Flip— as a function of radius of curvature ................
800

6.10 G as a function of radius of curvature ...............

6.11 as a function of radius of curvature ................

6.12 (R G as a function of radius of curvature ...............

6.13 TM012 cavity electric field lines from SUPERFISH ..........
6.14 TM012 cavity axial electric field profile .................
6.15 TM012 cavity surface electric and magnetic fields ...........
6.16 Snapshots of the short range wake in the TM012 cavity ........
6.17 Scale drawing of the TM012 cavity and a SOLEIL cavity .......

agmfig
v

VG

C.1 The “Duality Triangle” ..........................

vii

87
89
90

92
93
95
98
103
104
105
106
107
109
110
1 11

113
117
118

120
121

122

123
125

127
128
129
130
133
133

134
135

173

 

 

"L It

1 I‘AA; ‘-

List of Tables

3.1
3.2

4.1
4.2

5.1

6.1
6.2

6.3

6.4
6.5

6—cell midcell parameter list and figures of merit ............ 56
6—cell endcell parameter list and cavity figures of merit ........ 57
2-cell parameter list and figures of merit ................ 66
2-cell figures of merit comparison .................... 70
Measured Q0 drop in 2-cell cavity tests ................. 111
B R .
Esst TM012 values of Bi: and Q for a pillbox ............ 126
EL and g for a pillbox(% = 0.425) with rounded corners ...... 127
5C0

Fit and 5 for a pillbox (-‘f = 0.350) with rounded corners ..... 129
Geometrical parameters of the TM012 cavity .............. 131
Figures of merit and field levels for the TM012 cavity as compared to

SOLEIL .................................. 135

viii

 

Chapter 1

Introduction

Since the late nineteenth century, advances in electrical and vacuum technology have
enabled scientists to probe the properties of matter by giving a beam of charged
particles a large kinetic energy and then allowing that beam to collide with a tar-
get material: Advances in electrical technology enabled high voltages to be produced,
which allowed particles to have high energy; Advances in vacuum technology removed
most of the air so that the beam only collided with the intended target. As an exam-
ple, consider an early method for producing a high voltage by creating a large direct
current with a parallel bank of voltaic cells and passing the current through a large
inductor. By opening the circuit, the magnetic field collapsed and, by Faraday’s law,
the energy stored in the magnetic field produced an electric field. Under the right
conditions, the electric field was large enough for electrons to overcome the attractive
forces of ions in the lattice of the metal electrode and leave the surface (a process
generally known as field emission), producing a spark. By encasing the electrodes in
an evacuated chamber containing a low-pressure gas, the electron beam collided with
the gas atoms and produced the characteristic line spectra of the gas atoms. Further
advances made even higher voltages possible, and x-rays were discovered experimen—
tally when an electron beam with thousands of electron volts of energy collided with

a solid metal electrode. These x-rays were quickly used to show by diffraction exper-

iments that the metal electrode did indeed consist of a lattice of individual atoms.
In the early twentieth century, electrostic voltages on the order of a million volts
could be produced with a Cockroft-Walton generator which allowed experimenters to
study nuclear reactions. The use of alternating electric fields eventually proved to be
superior for producing even higher beam energies corresponding to several millions of
volts. The advances in accelerator physics during the first half of the twentieth cen-
tury were primarily focussed on producing beams of higher energies, enabling further
study of subatomic particles. However, the size of the beam also increased with beam
energy, and the practical limitations on the size of a beam limited the final energy. The
development of strong focussing in the 1950’s separated the focussing requirements,
and therefore the beam size, from the beam energy, allowing even higher energies to
be achieved. The Tevatron, a modern high-energy accelerator at Fermilab, produces
a final interaction energy equivalent to one trillion volts, a remarkable increase by
twelve orders of magnitude in less than a century.

The goal of accelerator physics is generally to deliver a high flux of particle energy
to a chosen target. This may be done with high beam energy, high beam current,
or by decreasing the beam emittance (i.e. by decreasing the amount of phase space
occupied by the beam): The particular application determines the significance of
these three areas of energy, current, and emittance. For example, a hospital may
use a cyclotron to artificially produce radioactive isotopes for cancer therapy. Since
relatively small samples are needed, generating large beam currents at high energy is
not necessary. In contrast, a facility designed for transmutation of radioactive waste [1]
would ideally use the largest feasible beam current to convert waste at the highest
possible rate. Facilities which use a primary beam to produce a secondary beam often
need the highest possible primary beam current to produce the “brightest” secondary
beam. This is the case for numerous x-ray light sources [2—4] which produce x-rays
by accelerating electrons and for the Spallation Neutron Source [5] at Oak Ridge

National Laboratory which uses a proton beam to produce spalled neutrons. Electron

2

microscopy serves as an example where there is ample beam energy and current for
the application, but the beam optics limits the resolution, and therefore usefulness,
of the device.

The development of superconducting cavities for accelerating charged particles
has enabled new applications not previously possible with normal-conducting struc-
tures. The low dissipative power losses and high quality factor of superconducting
cavities permit continuous wave (CW) operation at high field. CW operation means
that energy is continuously delivered to the cavities without interruption, in contrast
to pulsed operation where the power is periodically switched off and back on. Also
because of the low losses, the aperture may be large, reducing interactions between
the beam and higher order modes (HOMs) of the cavity. This smaller interaction has
the advantage of reducing the heating of the cavity due to beam energy dissipating in
the HOMs, while also increasing the threshold current for beam instabilities caused
by HOMs. The advancement of superconducting technology has therefore resulted in
a proliferation of the number of high beam current accelerator projects. A number
of projects rely on superconducting cavities operating in an energy-recovery mode to
accelerate a low-emittance electron beam for the purpose of producing x-ray radia—
tion. The energy-recovery linear accelerator (ERL) concept has been demonstrated
at Thomas Jefferson National Accelerator Facility [6]. ERL projects such as the Cor-
nell ERL and 4GLS in Daresbury, United Kingdom are underway. Electron storage
rings have historically been used for high brightness x-ray sources. Because the elec-
trons continually circulate through the ring, a relatively small number of cavities is
needed to replenish the energy lost to collisions and to radiation. The x-rays produced,
whether by a linear accelerator or synchrotron, have found numerous uses in fields
such as condensed matter physics, material science, biology, and medicine. Scattering
experiments can reveal information about the onset of phase transitions, the atomic
lattice of new materials, and the molecular structure of proteins and viruses.

Interest in superconducting cavities has increased since the recent decision to use

3

 

l--v—x—.—v-

this technology for the proposed International Linear Collider (ILC), the highest en-
ergy linear accelerator conceived to date. This global commitment to superconducting
radio frequency (SRF) cavities has spurred research and development efforts for ex-
isting accelerators and new projects [7]: the B-factories at KEK and Cornell, the
CEBAF upgrade at JLAB, SNS at Oak Ridge, XFEL at DESY, ERL projects at
Cornell, 4GLS at Daresbury, ELBE at Forschungszentrum Rcssendorf, BESSY—ERL,
and the ERL/FEL at JLAB.

Since a large part of this thesis describes the design and prototyping of cavities
intended for use with an ERL, the following sections provides general background
information on the development of linear accelerators. The distinction of a linear

accelerator operating with energy recovery may then be better understood.

1. 1 Linear Accelerators

The first linear accelerator (linac) using alternating fields was constructed and tested
by Widerbe in 1927 [8, Sec.l.1]. A voltage of 25 kV at 1 MHz was applied to a single
drift tube with the entrance and exit tubes grounded. Singly charged ions would arrive
at the first accelerating gap at the peak of the radio frequency (RF) cycle, gaining 25
keV of energy. The ions then travelled through a drift tube which shielded them from
the electric fields for half a RF period. The ions then emerged from the drift tube
where the electric field at the next accelerating gap was at its peak value, gaining an
additional 25 keV of energy. There was no reason why more gaps could not be used
in the same arrangement to produce a higher final energy. Also, since the exterior
of the device was grounded, multiple sections could be combined. This experiment
demonstrated how alternating voltages can produce final beam energies greater than
that achieved in an electrostatic accelerator.

After World War II, high power microwave sources became available. Using a

higher frequency voltage source in the Widerbe design had the advantage of reducing

4

 

 

fin.;-|:‘— —;

the length of the drift tubes needed. The disadvantage was that there would be
significant radiation losses at high frequencies because of the unshielded capacative
gaps in between the drift tubes [9]. A solution proposed by Alvarez was to excite a
large cylindrical cavity containing drift tubes inside to shield the particle during half
of the RF cycle. The Alvarez structure may be viewed as several short cylindrical cells
coupled to each other and oscillating in phase. The currents flowing along the wall
shared between two neighboring cells will cancel each other completely, rendering the
walls superfluous.

Another major advancement in linear particle acceleration was the method of
launching a travelling wave with a velocity equal to that of the particle [8, Sec. 1. 1]. The
longitudinal electric field of the travelling wave would continually push the particle,
transferring energy to it. A single cylindrical tube cannot accomplish this, since the
phase velocities of the waveguide modes are greater than the speed of light. A periodic
placement of disks with holes cut in the center for the beam are used to reduce the
phase velocity to the speed of light. Such a structure may be viewed as a series
of cylindrical cells, with each cell coupled to its nearest neighbors. This method of
particle acceleration was eventually used at the Stanford Linear Accelerator Center.

An alternative method for accelerating a beam with coupled cylindrical cells is
to form a series of independent multicell cavities. Each cavity is independent in the
sense that it is powered separately with its own adjustable amplitude and phase. A
standing wave is established in each cavity, with the electric field in each cell 180° out
of phase with respect to both of its neighboring cells. The length of each cell is set to
g} where 6 is the particle velocity divided by the speed of light and A is the speed
of light divided by the RF frequency. This cell length ensures that the particle is in
a given cell for half of the RF period and can therefore be accelerated by the electric
field in every cell. This is the method used for superconducting cavities and has the
operational advantage of greater flexibility: each individual cavity may be adjusted

as needed or even shut down if necessary.

5

One of the primary characteristics of all linacs is the ability to produce high-
energy, low-emittance beams. In circular machines such as electron storage rings, the
emittance of the beam is determined by the synchrotron radiation emitted in the
bending magnets. The emittance of the beam in a linac is essentially determined by

the source, not the machine lattice. Some other advantages of linacs [8, Sec.l.1]:
1. Strong focusing can easily be provided to confine a high-intensity beam.

2. The beam experiences a single pass, avoiding resonances which can occur when

beam is fed back in a circular machine.

3. The final energy of the beam is not limited by synchrotron radiation losses since

there are no bending magnets.
4. Injection and extraction are simpler.

5. A linac can operate at any duty factor, including CW.

1.1.1 Energy Recovery Linacs

Superconducting cavities allow large electron beam currents which can be used for
applications such x-ray free electron lasers (FELs) and electron cooling of a heavy ion
beam. The low emittance electron beam of a linear accelerator is capable of producing
very bright, coherent x-rays. However, FELs only convert approximately 1% of the
available beam energy to x-ray energy. For example, a 10 kW laser would require
1 MW of beam power, with 990 kW of beam power left over after lasing. Simply
disposing this high-energy, high-power beam would be economically unattractive and
would produce enormous amounts of radioactivity at the beam dump. Energy recovery
is a method to generate a high—energy, high-current beam using only a small fraction
of the available beam power.

The concept is to first start with a high-energy, low-emittance source of electrons,

such as an RF or DC photo-injector. Electron bunches leave the source with speeds
6

 

 

near that of light. They are then further accelerated, providing the power that the
application, such as lasing or cooling, will later extract. The bunches are then acceler-
ated to high energy through a main linac consisting of several independently phased
cavities. After reaching high energy, the bunches are used for the intended applica-
tion. Bending magnets then recirculate the beam back to the main linac. The bunches
enter the main linac 180° out of phase with the RF, so the bunches decelerate, giving
their energy back to the cavity mode. When the bunches leave the main linac, they
have the same relatively low energy they started with when they left the source, and
can be safely dumped. The cavities in the main linac act as energy storage devices.
The energy stored in the accelerating mode will decrease during the first half of an
RF period when a bunch is being accelerated, but the energy will be restored dur-
ing the second half of the RF cycle as a subsequent bunch is decelerated. Averaged
over an RF period, the net beam loading of the cavity is zero, and power must only
be supplied to compensate for the dissipative losses of the cavities. The situation is
analogous to reactive elements in a lumped circuit, where the instantaneous power
going into or out of the element can be large but there is no net power consumed by

the lossless element.

1.2 Radio—Frequency Cavities

An RF cavity is essentially a closed metal structure which uses one of its electromag-
netic modes of oscillation to accelerate beam. A standing wave is established in which
the field pattern has an electric field pointing primarily in the direction that the beam
travels. By driving the cavity at the mode’s resonant frequency, the relatively small
field of the forward travelling wave from the power supply can result in a large field
in the cavity. The only power which must be externally supplied is to compensate for
the dissipative losses of the metal walls of the cavity and the power transferred to the

beam. The basic shape used for superconducting cavities and the salient features of

 

t

 

 

 

 

I—-——-’-d
~-—-.--1
-——>-1

—_.’_

_-=—_?__
r..-—
l..__..?
i-——->—I
E

-——-,-

 

 

 

 

Figure 1.1: The TM010 field pattern has a large electric field along the beam axis [8,
Sec.l.2].

the accelerating mode field pattern will be described in the following section.

1.2.1 The Elliptical Cavity

As the name implies, the pillbox cavity is a cylinder with the diameter larger than the
radius. The lowest frequency resonant mode for such a structure is the TM010 mode.
As can be seen in Figure 1.1, the electric field along the beam axis is very high. The
electric field is highest along the beam axis if there is no hole to allow the beam to
pass through the cavity. When an opening is made for the beam pipe, the peak electric
field will occur along the surface of this opening. The peak magnetic field occurs at
a radius that is approximately 73% of the cavity’s outer radius. For superconducting
cavitiae, all corners are rounded with an elliptical profile. The purpose of rounding the
corner at the iris is to reduce the peak electric field and therefore reduce the chance
of field emission ocurring at high field. The purpose of rounding the outer equatorial
edge is to avoid hard multipacting barriers and reduce the peak magnetic field on the
surface of the cavity. Large surface magnetic fields are maintained by large electrical
currents on the surface and therefore locally dissipate large amounts of power. If this
heating occurs near a defect in the material, the increased temperature could initiate

a thermal instability which destroys the superconducting state. Figure 1.2 shows the

 

 

 

 

 

 

 

 

Riris

 

 

 

 

e— —u

L

 

Figure 1.2: An elliptical profile is used for the equator and iris of the cell with a
straight section connecting the two ellipses.

geometrical parameters which characterize the cell shape:

D = equatorial radius of the cell. It is adjusted primarily to tune the resonant fre-

quency to the design value.

L = half-length of the cell. It is set equal to 3}, where 6 is the particle velocity
divided by the velocity of light and A is the speed of light divided by the resonant

frequency.
Riris = beam tube radius.

a = angle that the straight section connecting the two ellipses makes with the ver-

tical.

d = location where the straight section would intersect the beam tube if both were

continued indefinitely.
a = minor semi-axis of the iris ellipse.

A = minor semi-axis of the equator ellipse.

9

 

(Frequency)

 

 

 

 

1.... ' 2.... ............................
o 02 0.4 0'6 0L8 1

Figure 1.3: The dispersion diagram for the T M01 passband [10, p.134].

b = major semi—axis of the iris ellipse.

B = major semi-axis of the equator ellipse.

Several elliptical cells are welded together to form a single multicell cavity. The
TM010 mode of a cell couples to those of its neighboring cells via the electric field.
This coupling causes the TM010 mode to split into a passband of closely spaced
modes equal in number to the number of cells. The dispersion diagram for the TM010
passband of a hypothetical 5—cell cavity is shown in Figure 1.3.

The width of the passband (range of frequencies of the modes in the passband) is
determined by the strength of the cell-to—cell coupling, which is primarily determined
by the iris radius. A longer cavity with more cells would have more modes in the
same frequency range, therefore increasing the number of cells reduces the difference
in frequency between the adjacent modes of the passband.

Figure 1.4 shows an example of a 5-cell cavity, illustrating the electric field lines
for the accelerating mode. The accelerating mode has the highest frequency in the
passband and is referred to as the 7r-mode since the phase advances by 180° from
cell to cell. The field pattern in each cell is TM010 for the accelerating mode. The

higher-order modes have their own passbands, and the field pattern in each cell is

10

 

.Ll‘ v’ ‘ ‘
Fl gig/L “EELS-:53: h
”J l Uflkf \jn‘x xv Jig .j—j

/

Figure 1.4: Conceptual schematic showing the alternating electric field lines of the
7r-mode [10, p.4].

 

 

1 I r r 1 I

 

 

 

 

 

 

 

 

 

 

r I _ I 1 I
fr/Slmode ‘ . ‘Il/S mode
0 I) ll ll n 4 0 [l
vj U ' U
-1 u 1 1 l 4 -l 1 4 1 1
1 2 3 4 5 1 2 3 4 5
Cell # Cell #
1 T I I 7 I l I I I I r
. 27/5 mode I "IOdC

 

 

 

 

 

 

 

 

3 2 3
Cell # Cull #

 

 

 

 

 

3
Cell #

Figure 1.5: The modes within a passband are characterized by the sinusoidal variation
in electric field amplitude from cell to cell [10, p.132].

that of the HOM. All passbands will contain as many modes as there are cells. For

the mth passband mode, the amplitude in the jth cell is proportional to

sin [m1r (2302131)] (1.1)

where N is the total number of cells in the cavity. This variation in cell amplitude is

 

graphically depicted in Figure 1.5 for the case of a 5-cell cavity.

11

I-‘- :- fin.

1.2.2 Superconductivity

There are several advantages to using superconducting (SC) cavities made from bulk
niobium instead of normal-conducting (NC), copper cavities for particle accelera-
tors [10, Sec.l.2]. High gradients may be achieved with both technologies, but the
mechanical properties of copper and the quality of the cavity vacuum degrade if a
copper cavity’s temperature exceed approximately 100°C. To prevent excessive heat—
ing at high gradients, the duty factor is typically lower than 1%. Because the Q of
3 SC cavity is typically 105 times greater than a copper structure, high gradients
are possible at high duty cycles and up to CW operation. The high Q of SC cavities
reduces the power dissipated, in the range of 1-10 W. This power must be dissi-
pated at cryogenic temperatures however, which is inherently inefficient due to the
low Carnot efficiency of refrigeration at or below 4 K. When the Carnot efficiency
is combined with the mechanical efficiency of the refrigeration system, the overall
efficiency is approximately 0.3%. However, when the efficiency of the klystron used to
' power a NC cavity is taken into account, a SC structure would require approximately
200 times less AC wall power to operate than a NC structure at the same duty cy-
cle. The high Q also enables larger cavity apertures to be used, which significantly
reduces any interactions between the beam and the cavity HOMs. The beam will
deposit less energy in the HOMs; The HOMs will degrade the beam emittance less
and higher beam currents can be accelerated before HOM-induced beam instabilities
arise. Larger apertures thus allow for higher average beam currents.

SC cavities bring not only improved performance but also additional challenges.
To attain high Q’s, cleanliness during fabrication and processing of bulk niobium SC
cavities is crucial. The power per unit area dissipated locally on the inner surface of

a cavity is given by the expression
1 2
P = ERSH , (1.2)

12

 

where H is the local magnetic field on the surface and R3 is the surface resistance.
The BCS theory of superconductivity predicts that at temperatures below half of the
critical temperature, the surface resistance has the approximate form [10, Sec.3.4],

A 0
a (1.3)

RBCS = Aw2 e-

Here w is the angular frequency of the RF, A(0) is the energy gap at zero tempera-
ture, k3 is Boltzmann’s constant, T is the temperature, and A is a temperature and
frequency independent constant whose value is determined by the properties of the
material. The surface resistance measured in experiments is consistent with the BCS
prediction, therefore lowering the temperature of the helium bath can dramatically
reduce the surface resistance. However, the decrease in surface resistance with tem-
perature does not continue indefinitely. Eventually the measured surface resistance
reaches a constant value independent of temperature. This residual resistance (R0) is
essentially due to impurities and foreign material on the niobium surface or trapped

magnetic flux. The surface resistance of a SC cavity therefore has the form
Ago:
R3 = Aw2e B +120. (1.4)

The value of R0 typically attained is in the range of 10 to 20 n0. For cavities oper—
ating at frequencies around 1.5 GHz, the temperature must be less than or equal to
approximately 2 K for the BCS resistance to drop below this level [10, Sec.9.1]. To
achieve these low limiting values of surface resistance and to ensure adequate rates
of heat transfer to the helium bath, careful precautions must be taken.

The BCS resistance has a strong dependence on temperature, so it is desirable
to increase the rate of heat transfer through the bulk niobium as much as possi-
ble to ensure that the temperature rise from the outer to inner surface is as small

as possible. Electron-phonon scattering essentially determines the value of both the

13

 

electrical and thermal conductivities of a metal at room temperature. These con-
ductivities are higher at cryogenic temperatures since the number of phonons is re—
duced [10, Sec.3.2.1]. The conductivities eventually saturate at limiting values where
electron scattering with impurities dominates. Since it is relatively easy to make
precise measurements of electrical conductivity, the ratio of the saturated electrical
conductivity to the room—temperature electrical conductivity is used as a measure of
the purity of bulk niobium. High values of this ratio, called the residual resistance
ratio (RRR), are therefore desirable. High purity niobium is expensive, with the cost
roughly proportional to the RRR: A RRR of 250 is typical. During the stamping of
elliptical half-cells and machining of weld preps, care must be taken to avoid using
steel. This prevents any iron from contaminating the surface and providing magnetic
flux to be trapped during cooldown. Also, high permeability metal must be used to
shield the cavity from stray magnetic fields (including the Earth’s magnetic field)
during cooldown. Convential welding techniques such as tungsten-inert-gas (TIC)
welding would contaminate the niobium surface, so all welds are typically done with
an electron beam under high vacuum. If hydrogen concentrations are above 2 ppm
by weight, then there is a risk of a lossy hydride forming on the Nb surface at tem-
peratures between 60 K and 150 K [10, Sec.9.4]. The acid used for etching the cavity
surface must be chilled below 20°C to avoid excessive hydrogen to be released on the
surface. Baking at 700°C to 900°C under vacuum is necessary to ensure that hydrogen
concentrations are low enough to preclude precipitaion of hydrides. An alternative ap—
proach to baking, which was adopted here at MSU, mitigates the formation of the
hydride by cooling down quickly (approximately 1 hour) while in the temperature
range of 60 K to 150 K.

1.2.3 Issues Driving Cavity Design

Superconducting cavities are an ideal choice when large beam currents and high duty

cycles are desired. These characteristics combined with the efficient use of beam power

14

in an ERL have lead to a number of projects striving for 100’s of mA of electron beam
current for use in a high power FEL or for electron cooling. The primary challenges
for the cavity at these high beam currents are to reduce the amount of beam-induced
HOM power as much as possible and to effectively damp the dipole HOMs that
are primarily responsible for the multipass beam breakup (BBU) instabilities that
ultimately limit the maximum available beam current.

If a particle enters a cavity on the central axis when a dipole HOM has been
excited, then the particle will leave with a deflection in the horizontal or vertical
direction. The optics of the recirculation line will cause the tranverse momentum
imparted to the particle by the HOM to result in the particle entering the cavity with
a transverse displacement when it returns for deceleration. The transverse offset can
cause the particle to further excite the HOM and this process can continue until the
particle collides with the cavity wall [6]. The threshold current at which a multipass

BBU occurs is predicted to the correct order of magnitude by the expression [11]

-2pc

1,h = 6(5)QkM12. (1.5)

 

Here 1) is the momentum of the particle, c is the speed of light, 6 is the charge of
the electron, g is the shunt impedance of the HOM, Q is the quality factor of the
HOM, k = % is the wavenumber of the HOM, and M12 is the transfer matrix element
relating the transverse momentum at the cavity exit to the transverse displacement
of the particle at the entrance of the same cavity during the next pass. The HOM
of concern is the one which corresponds to the lowest threshold current. The beam—

induced power dissipated by a longitudinal HOM is given by

P = iii—"’1 (3), (1.6)

where w is the angular frequency of the HOM, q is the bunch charge, I is the average

15

 

beam current, and 5 is the shunt impedance of the HOM. The goal is then to minimize
both the 8 and loaded Q values for all HOMs which interact with the beam. In
contrast, strong interactions between the cavity accelerating mode and the beam are

desirable for efficient acceleration, so the general aim for this case is to maximize both

5am Q.

1.2.4 Outline

This dissertation will present the design of three superconducting cavities intended for
acceleration of beams with high average current. Elliptical cavities with two cells and
six cells have been designed for use in an ERL. The two-cell cavity is designed for use
as an injector cavity, operating without energy recovery. The six—cell cavity is suitable
for acceleration in the main linac. The third cavity shape is a single cell suitable for use
in an electron storage ring. All three cavities have large apertures, which reduce the
HOM heat load and increase the threshold currents for beam instabilitias. Chapter 2
describes the solution for general cavity fields and the key figures of merit used to
describe these field patterns. The figures of merit allow one to compare different cavity
shapes and judge which shape is better for an intended application. The figures of
merit are therefore important for understanding the design goals of the three different
cavities. Chapter 3 describes the design criteria for the shape of the six-cell cavity
and the method used to meet these criteria. Here the basic approach to avoiding
HOM-induced beam instabilities is described and will constitute one of the primary
design criteria. Chapter 4 describes the design and shape of the two-cell cavity. Two
copies of the two-cell cavity shape were prototyped and tested. The details of this
fabrication process and the results of the tests are presented. Chapter 6 describes the
basic concept of using a HOM in a single cell cavity as the accelerating mode. The
purpose of using the HOM is that it allows the aperture of the cavity to increase
significantly without significantly increasing the heat dissipated in the accelerating

mode. The increased aperture has the advantage of diminishing interactions between

16

 

the undesirable HOMs and the beam and therefore enabling larger beam currents to

be used.

17

 

Chapter 2

Superconducting Cavity Design

The main design considerations for superconducting radio-frequency cavities will be
described in this chapter. We begin with the waveguide modes in Section 2.1 and
then proceed to cavities in Section 2.2; the standing wave fields in a cavity can be
thought of as being the superposition of travelling wave fields of a waveguide. The
effects of wall losses and coupling ports are discussed in Sections 2.3.3 and 2.3.4. The
main figures of merit that guided the design of the 2-cell, 6—cell, and HOM cavity are

then described in Section 2.4.

2. 1 Waveguide Modes

A brief review of the electromagnetic field patterns of waveguides will be useful for
describing closed cavities, which may be viewed as a waveguide with its ends shorted,
and for understanding how the energy stored in cavity modes may couple out through
a beam pipe, which may be viewed as a waveguide with a circular cross section.
The modes of propagation in a waveguide can be derived from Maxwell’s equa-
tions, as done in several texts [12-14], as well as in Appendix A. The starting point is
a hollow tube of constant cross-section with vacuum inside and conducting walls. A

solution of Maxwell’s equations is sought with the fields varying sinusoidally in time

18

as ejwt. In the vacuum region with no source terms and perfectly conducting walls,
the electric and magnetic field both satisfy the wave equation. Solutions are sought
which vary sinusoidally in the axial direction (2) as e’ jfiz. One can then separate
out the axial component of the fields from the transverse components.

In order to satisfy the boundary conditions imposed by the conducting walls of the
waveguide, longitudinal field components must be introduced. This is in contrast to
plane travelling waves, for which the components are purely transverse. Likewise, the
simple proportionality between the frequency and the wavenumber of a plane wave
becomes more complicated for a waveguide.

An expression relating the frequency of a waveguide mode to its wavenumber

generally has the form

k3 = 1.3+ 52 (2.1)

This Pythagorean relation between the critical wavenumber kc, the free space
wavenumber kg, and the axial wavenumber H in Equation (2.1) provides a simple
physical interpretation of a waveguide mode. Below a critical frequency, we = also, a
mode may only be excited evanescently. Evanescent is a general term used to describe
modes which diminish rapidly beyond the point of excitation and therefore do not
propagate electromagnetic energy. Mathematically, these evanescent solutions arise
in the same way that exponentially diminishing state functions appear in a classically
forbidden region as solutions to the Schrodinger equation. Physically, the mode re-
sponds as a combination of a lumped inductor and lumped capacitor: The mode can
only store energy during one half of an RF cycle before delivering it back to the RF
source, similar to the behavior of a lumped capacitor or inductor in low frequency
circuits. At we, a standing wave is established in the transverse direction. The wave is
a superposition of a wave travelling away from the axis and a reflected wave travelling

toward the axis. The type of mode and the shape of the waveguide cross section will

19

 

determine the value of kc when the appropriate solution to the wave equation is found.
For frequencies above we, a wave travelling in the axial direction is established with
the wavenumber B. The field pattern can therefore be viewed as the superposition of
two travelling waves which propagate in equal but opposite directions slightly off axis
and continually reflect off of the walls of the waveguide. The two waves travel with
the free-space wavenumber k0 which has he as a transverse component and fl as an
axial component.

The notion of impedance is a useful but potentially confusing one for RF ap-
plications, since the term impedance is used to describe four different, but related,

quantities [13, page 188].

Intrinsic Impedance

There is the intrinsic impedance, 77, of the medium which is the ratio g- for a plane

wave in the medium.

Wave Impedance

The wave impedance, Z, is the ratio i- of the transverse fields for a travelling wave
of a given mode and its value is a function of the waveguide geometry, the medium

of the guide, and the mode number.

Characteristic Impedance

The characteristic impedance, Z0, is uniquely defined only for transmission lines as
the ratio of the voltage to current in a travelling wave. The product of voltage and
current is the instantaneous power passing through the transverse plane and may
be calculated by integrating the Poynting vector over the area of the plane. The
characteristic impedance is therefore a function of the transmission line geometry
and the wave impedance. Since voltage and current may not be uniquely defined for
a waveguide mode, the characteristic impedance is not uniquely defined either. In

20

 

this thesis, the characteristic impedance for waveguide modes is chosen to be equal
to the wave impedance. Once voltage or current is defined in terms of the transverse
fields of the mode, the remaining quantity must be defined such that the product of
voltage and current is equal to the integrated energy flux at the plane. A suitable set
of reference transverse fields, Et and Ht, may be normalized such that Et is real and

satisfies the expression

/ Et2da =1. (2.2)

Let the transverse fields of the mode then be expressed as,

e = vEt (2.3)

h = iZHt (2.4)
with the coefficients calculated by evaluating the following integrals,

’U = /(e Et) da (2.5)
2' = Z2 / (h - Ht) da. (2.6)
These coefficients serve as an equivalent voltage and current since the ratio of v to i

is equal to the wave impedance and the product of v and i is equal to the total power

passing through the plane,

 

vE't _ v
iZHt — 2

-;-Re (vi*/Et2da) = $Re(vi*). (2.8)

I___i;~|m
ll

-;—Re[/(exh)-da

Load Impedance

The impedance of a load, Z1, attached to a transmission line or waveguide is generally

not equal to the characteristic impedance. In this case, a travelling wave will be par-

21

 

tially reflected at the load and a standing wave will be established in the transmission
line or waveguide. By measuring the amplitude and phase of the reflected wave, the
impedance of the load may be established. The load impedance is defined as the ratio
of voltage to current for the superposition of the forward and reverse travelling waves,

 

Here Vf, If and Vr, Ir are the voltage and current of a wave travelling in the forward

and reverse direction respectively.

Circular Waveguide Modes

The case of a circular waveguide is of particular interest for the present work. In
the case of a conducting wall of circular cross-section, separation of variables may be
done in cylindrical coordinates. The solutions to the wave equation are trigonometric
functions of the azimuthal coordinate, while the radial dependence takes the form of
Bessel functions.

As in all waveguides, the modes of propagation in the circular waveguide can be
classified as either TE (transverse electric, with zero longitudinal electric field) or
TM (transverse magnetic, with zero longitudinal magnetic field). The lowest modes
of each classification are referred to as TE11 and TM01.

For the TM01 mode in a circular waveguide of radius a, we obtain kc = 2.405/ a,
and the fields in a cylindrical coordinate system (radial, axial, and azimuthal coordi-

nate = p,z, and ()5, respectively) are

e2 = .10 (2.4055) . (2.10)
e = —Jk—5J1 (kcp) up (2.11)
C
_ _ _fl2
h — h¢n¢ — flkc J1 (kcp) 11¢. (2.12)

22

 

2.2 General Expansion of Electromagnetic Fields
Inside a Hollow Cavity

Any electric or magnetic field pattern in a cavity may be expressed as a superposition
of the corresponding field of the eigenmodes. This representation of the cavity fields
is useful for determining the cavity figures of merit and for modelling interactions
between the cavity and the beam. The derivation of the modal expansion of the
fields can be found in the text by Slater [l4], and in Appendix B. The results are
summarized briefly in this section.

Any vector field is determined uniquely once the values of the curl and divergence
are known at all points. A general vector field may then be described as a combination
of two different sets of orthogonal functions. One set contains solenoidal functions for
which the divergence is zero. The other set contains irrotational functions for which
the curl is zero.

Considering a hollow cavity with boundary conditions imposed on its bounding
surface, one can solve for a set of orthonormal functions (solenoidal fields Ea and Ha
and an irrotational field Fa). The electric field, magnetic field, current density, and
charge density in the cavity can then be expressed as linear combinations of these

functions:

E = Z<eaEa+faFa) (2.13)
H = ihaHa (2.14)
J = i<jaEa+gan (2.15)
p = 2::Tal/Ja- (2-16)

In order for the functions to be orthonormal, they cannot have the same dimen-

sions as the actual electric and magnetic fields. The properly dimensioned electric

23

 

fields Ea and fig are related to the orthonormal functions via

= __ 2.17

E. 203E: < >

Ha = iii; (2.18)
2Ua

With the proper mathematical operations, Maxwell’s equations can be written in

terms of the coefficients and the appropriate surface integrals:

dra
= _ 2.19
Raga dt ( )
kaha = ja+ {(1531,, x H) da + ed—d—t (2.20)
kaea = —f(E X Ha)‘ da- [151—Sta. (2.21)

The time evolution of the coeflicients ea and ha can be described in terms of

ordinary differential equations:

d283,

_d_t2 +waea = __d_[ja+j[(EaxH) d3] ——[jf(ExHa) da}2.22)
Wdlwaha = 2%][3 +]{(EaxH).da]; ———[jf(ExHa) d8] (2-23)

When the terms on the right hand side are zero, the fields oscillate sinusoidally
in time with angular frequency wa, as one would expect. The first term on the right
hand side, ja allows for energy to be transferred into or out of the cavity via the
beam. The other terms allow for power to be lost in the walls of the cavity, or for the

power to arrive into or to leak out of the cavity through an external port.

24

 

2.3 Applications

2.3. 1 Lossless Cavity

If the cavity walls and dielectric material are both lossless, then Equations (2.22) and

(2.23) reduce to

d211,, 2 d263, 2
a

Eff-HUG 3=W+w

ea = 0. (2.24)
An excited cavity mode will therefore freely oscillate sinusoidally with angular fre-
quency, wa = 7kg; These frequencies and field patterns correspond to the eigenvalues

found in numerical simulations.

2.3.2 Cavity with Losses and Quality Factor

There are various energy-loss mechanisms which are proportional to the square of the
cavity fields and contribute to the overall Q of the cavity. Of these “quadratic” losses,
heating of the dielectric medium filling the cavity is not directly relevant to the work
of this thesis since the cavities addressed are all operated under vacuum. However, a
brief description of the effects of dielectric losses is useful for clarifying the treatment
of resistive wall losses, which will be addressed in the next section. A cavity with a
lossy dielectric has finite conductivity and will therefore have an associated volume

current, J = 0E, which has expansion coefficients
ja = 06a- (2.25)

Substituting Equation (2.25) into Equation (2.22) results in

d268, 0 deg

2
dt2 + E 712' + waea = 0. (2.26)

 

25

 

A similar equation is produced by combining Equations (2.21), (2.23), and (2.25):

dzha + a dha

W EE- + wgha = 0. (2.27)

Either equation may be solved by assuming a sinusoidal solution with complex angular

frequency. The equation for the angular frequency of free, damped oscillations is then

1.2% — w2 + J—f-w = 0, (2.28)
which has the solution
. ma 1 2 _ €LUa

The electromagnetic field pattern of the modes with dielectric losses are identical
to those of the loss-free case, but the damping produces a frequency shift which is
of order 612. This is the familiar shift in natural frequency of a damped harmonic
oscillator.

The general definition of the Q of a mode with angular frequency w, stored energy

U, and power loss P is
_ Q

Q P . (2.30)

Dissipative losses are not the only mechanism that causes stored energy to leave a
cavity mode. There must be some means of coupling power to the cavity, and this port
therefore also provides a means for energy to leak out of the cavity. Also, a charge
bunch passing through the cavity can be accelerated and therefore take energy from
the mode. If the cavity mode initially starts with some stored energy and is allowed
to oscillate freely without being driven by an external RF generator, then the power

leaving the cavity through the various loss mechanisms will cause the cavity fields to

26

 

 

 

decay. The rate of decay is faster when the lost power is larger,

dU wU
_ = _p= __
dt Q

wt

U(t) = U0 e‘U . (2.32)

(2.31)

The rate of decay of the cavity fields may be found from the fact that the stored

energy is proportional to the square of the cavity fields:

\/U(t) = (1065053. (2.33)

The rate of decay is a function of what is referred to as the loaded Q (denoted Q L),
which is the Q calculated using the total power leaving through all mechanisms. It also
has significance when the cavity mode is being driven under steady state conditions,
since the impedance of the mode is the Fourier transform of the impulse response.
The loaded Q is therefore also equal to the ratio of the resonant frequency to the
mode bandwidth. Just as the loaded Q is defined in terms of the total power leaving
the mode, separate Q values are defined for each loss mechanism. The loaded Q may

then be written as a function of the separate Q’s,

P = PO+Pext+Pb (2.34)
P _ P0 Pext Pb
wU " wU+ wU +wU (2'35)
1 1 1 1
— = —+—+—. 2.36
QL Q0 Q... Qb ( )

The intrinsic Q (denoted Q0) , external Q (denoted Qext), and the beam Q (denoted
Qb are due to dissipative losses, power leaving through any external ports, and power

taken by an accelerated beam respectively.

27

 

2.3.3 Wall Losses

All hollow, metallic cavities will experience some power loss due to the non-zero
surface resistance of the metal walls. There will be a small tangential electric field
on the metal surface and a correspondingly small energy flux into the metal. These
modes must then in general be represented by a superposition of the Ea’s or Ha’s
since a single member of either set will not satisfy the correct boundary conditions.
However, for all cases of interest the wall losses will be small and therefore the fields
in the interior of the cavity for a given mode will be essentially identical with those of
a single member of the Ea’s or Ha’s. The rest of the superposition consists of a sum
of small, higher-order corrections to satisfy the boundary conditions at the surface.
Since the tangential electric field on the metal surface is non-zero, it is convenient to

use Equation (2.23), setting the E3 x H term equal to zero:

d2h
Tip—3 + wgha = mill-:14; [f (E x Ha) - da]. (2.37)

The intrinsic impedance of a material in which the conduction current is much larger

than the displacement current is

17 = (1+ j) fi—E. (2.38)

The intrinsic impedance may be used to eliminate the electric field from the previous
equation:

d2}; d

If there are no losses, the magnetic field pattern would be that of Ha and the frequency
would be am. In the case of wall losses, the frequency of the mode will be slightly
different due to the higher-order contributions of the superposition. To find the nth-
order correction to the frequency requires use of the (n -— nth-order correction of the
field pattern. For all cases of interest, the wall losses are small enough that a first-

28

 

order correction to the frequency is sufficient and therefore the zero-order correction

to the magnetic field may be used in the equation above,

 

H = haHa, (2'40)
.12}. n d
fi+w3ha+mlhaf<Ha'Haldal = 0’ ‘2‘“)
d2ha '7 2 dha 2
_ _ = , 2.42
dt2 +u(j[Hada) dt +waha 0 ( )

Assuming that ha varies harmonically with time, the equation above may be solved

for the complex angular frequency,

in 2 1 2 77 2 2
w = 2—[2- Hada :l: -2- 4%, + ; Hada . (2.43)

The first term on the right side contains an imaginary portion and is therefore re-

 

sponsible, to first order, for the decay of the fields. We can therefore solve for the Q

after substituting Equation (2.38) using w,- (the real part of w),

 

“£25 = Im [37:1 f Hgda] , (2.44)
“i = /fl 2
Q 2!” fHada, (2.45)
l / 1 6
z) a = Zwra'p ngda = Eng. (246)

The distance at which the fields in the metal have decayed by the factor % is defined
to be 6, the skin depth. The expression, correct to first order in é, for the angular

frequency of a mode with wall losses is then

wa lwa
w=UJa—_+

2Q 2Q'

 

(2.47)

In this case there is a first-order shift in the natural frequency of the mode, in contrast

to the second-order shift in the case of dielectric losses.

29

 

I -1...-

2.3.4 External Ports and External Q

A beam pipe must be added to a cavity in order to introduce charge bunches to be
accelerated, and the beam pipe may also be viewed as a waveguide with a circular
cross section. According to Equation (2.1), a waveguide mode will not have a real axial
wave number, and therefore propagate energy down the guide, unless it is driven above
a cutoff frequency which is determined from the value of the critical wavenumber for
the guide mode. To ensure that the cavity accelerating mode only excites the beam
pipe modes evanescently, the frequency of the accelerating mode is set below the
lowest cutoff frequency of the beam pipe. Various cavity HOMs are excited by the
beam as it passes through and it is desirable to minimize this excitation as much as
possible to prevent instabilities which arise from further interactions with the beam
and to reduce the cryogenic load. Ideally, a cavity would have the frequencias of all
HOMs above the cutoff frequency of the beam pipe: It would then be possible for the
HOM fields to propagate away from the cavity and be damped by a resistive load or
coupling antenna connected to an external load. In addition to high frequency, a mode
must have sufficient coupling to the beam pipe mode for damping to be effective. The
coupling strength is quantified by the value of Qext- A low Qext value means that a
relatively large amount of power leaks out of the cavity through the beam pipe.
The system of interest is a hollow cavity connected to a circular waveguide. We
wish to develop an expression for the input impedance seen looking into the cavity
through the guide. The surface, S’, is far enough away from the cavity that evanescent
fields are negligible. Then the cavity field, Ea, at the surface S’ is proportional to the

transverse field of the waveguide mode

30

 

The magnetic field at S" may then be expressed as

H = z’ZHt. (2.49)

The value of 2' determines the input current which excites the cavity mode, resulting
in a particular value of ea and therefore 22. The ratio of the equivalent voltage to the
equivalent current at S’ is the input impedance of the cavity seen at 3’. To calculate
this impedance it is necessary then to formulate an expression for ea. Equation (2.22)

reduces to

 

d2e 1 d
dt2a + wgea = -23 [f (Ea x H) . da] . (2.50)

Substituting Equations (2.48) and (2.49) into the integral, we obtain

— f (Ea x H) -da = — /S,[(vE,) x (.2110) - (—nda) = vi, (2.51)

so that Equation (2.22) becomes

d2ea 2 ld(vz')
dt2 +waea_2 dt '

 

 

(2.52)

The driving term, 2', will vary sinusoidally, so ea is also assumed to vary sinusoidally

with the same frequency,

 

—w2ea + wgea = (95—!) vi. (2.53)
We can now solve for ea .
(12) m
e — e (2 54)
a (23 — w2
or, equivalently,
.11z_
— _ “a . (2.55)

 

Now that the electric field has been determined, E = eaEa, the impedance may
be calculated by forming the ratio of the amplitudes of the transverse electric and

magnetic fields, Z, = g. Substituting Equations (2.49) and (2.48) produces

68321313 632)

 

Z _ 2.56

’ 52H, 2' ’ ( )
i

z, = “a . (2.57)

 

' .01. _ 43.0.
J (5.. w )
The total impedance generally consists of a sum over all of the cavity modes,

2%.

21:: ewa , (2.58)

. 1(62—2)

 

however for frequencies near a particular mode the contribution from other modes is
small and it is convenient to lump them together into an additional term Za:
v2

2,: 5; +20. (2.59)
1(2-2)

 

The natural frequency of the cavity mode may be found by terminating the guide
with the characteristic impedance, Z0. The impedance dascribed by Equation (2.59)

will then be the negative of this value,

 

Z, = —20, (2.60)
.223.
E a _
.22 2a. +Za+Z0 _ 0, (2.61)
J(... - 5)
2
é; 1— fl = 0 (2 62)
Zo+Za Ma LL) .

. . . 2 .
It IS convenient to define g + Jb = 2.2—f2; and Q17 = $2 where Z 18 the wave

32

 

impedance of the guide,

 

 

(.22 — of, + (“é—‘3) w - (35:0)..1 = 0, (2.63)
w2— [jwag+jb)]w—wg = 0. (2.64)
V

The cavity mode “rings” down as energy is propagated out through the guide, pro-
viding a loss in an otherwise lossless cavity. The cavity frequency will then generally
be complex, with the imaginary part describing the decay rate of the mode. Solving

the last equation for w results in

 

 

. . - 2
Maggywaflt..- (w_a<,gQ;w>]. 5...)

To first order in g+JVb , the frequency is then

lwa (9 + lb)
2QV

 

The impedance Za represents the combined impedance of the other cavity modes
which will be small if the resonant frequencies of the other modes are all sufficiently
far from the mode of interest. However, a surface 5' may always be found where
Za = O and therefore 9 = 225' In this case QV = Qext-

The derivation of the input admittance is similar to that of the input impedance.
The magnetic field on a surface, S, is defined in terms of the transverse magnetic
field of the dominant waveguide mode and the electric field imposed on S is defined

in terms of the transverse electric field of the dominant waveguide mode,

E = 61/13,. (2.67b)

33

f

The driving term in Equation (2.23) is evaluated,

[3 (E x Ha) -da = [S [(vYEt) x (th)] - (—nda) .—. —vz' (2.68)

The value of ha may then be obtained using Equation (2.23),

vi
—7'

ha: a . (2.69)
it“:- )

By using Equations (2.67a) and (2.67b) the expression for the input admittance be-

 

a];

comes

 

.2
z
. u '—7'
W=E=M1=£a1= We , (2.70)
J Jr — w
a

The general expression for the input admittance when a particular mode dominates

is then

 

Yz= W“ , +Ya. (2.71)

To solve for the frequency of free oscillations it is convenient to define r+ j a: = 17%;]?!-

-2
z = 1 - - . . . .
and W Q)” If S is terminated w1th a characteristic admittance, Y0, then

Y1 = —Y0, (2.72)

werewew = .,

j (r :57) wé. (2.74)

=>w = wfl+

 

A location may be found for S such that Ya = 0. In this case, a: = 0, r = 1, and

QI = Qext~
The fields of a lossless cavity with a waveguide port may be found numerically

and two different solutions can be found corresponding to the different boundary

34

 

conditions of S and 5’. Since S and S’ have the same location, the value of Qext
will not generally be equal to QV or Q]. From Equations (2.66) and (2.74) a single
expression for the Qext may be formulated with the assumption that wa and w; are

both approximately equal to tag,

 

 

1 _ i
Qext — QV (2.75)
1 1'
Qext — a}. (276)
=> (9 + 7‘) Qext = QV + 621- (2-77)

The sum g + 1" may then be expanded and simplified:

 

 

 

 

. Z Z Z(Ro-lXa) Roz
9 J Zo+Za Ro+1Xa R3+Xg g Rg+xg( )
, Y Y Y(Go—jBa) GOY
T+ a: = = . = T=—_. 2.79
J Y0+Ya Go+JBa C(2,+Bgz GE+B§ ( )
1 R02 R2X2
_ ___R_o_Z__ _§Z__ +7332“
9+? — 2 2+1 1“ 2 2 ' (2'80)
0 a
If the guide is connected to a matched load then Z = R0 and
R2 X2
g+r=——-—(2)+ (21: _ (2.81)
R0+Xa

Substituting this value into Equation (2.77) gives a relatively simple expression for

calculating the Q of a cavity mode from the rasults of two simulations,

9.... = Qv + 621 = E’s—{”3 + 5%; (2.82)

There are two potential advantages when applying Slater’s method to the problem

of predicting external Qs from simulation results:

0 for the case of a heavily damped mode, wa z w; z wo is no longer valid but

35

 

['9

an expression similar to Equation (2.82) may still be derived with a frequency

dependent coefficient next to Q I and QV;

o the method may, in principle, be extended to describe the case in which the

resonance of neighboring modes overlaps with that of the mode of interest.

2.3.5 Slater’s Theorem

If the field pattern of a cavity mode is slightly perturbed by introducing a small
metallic bead or by slightly deforming the cavity at a small localized region on the
boundary, then the mode frequency will shift by an amount which depends on the
field level at the location of the perturbation. If the perturbation is small, then the
new field pattern will be a superposition of primarily the old field pattern with small
contributions from other modes. The surface currents associated with the discontinu-
ity in the magnetic field on the perturbed boundary will be the source of a driving

term on the right sides of Equations (2.22) and (2.23),
5 (E, x H) -da’, (2.83)

where the integration is over the area of the small perturbed volume. To find the

first-order shift in frequency, the zeroth-order field pattern (H z haHa) is used:

5(Ea x H) -da’ = (15150139. X Ha) - da’, (2.84)

and the divergence theorem is used to convert the surface integral to a volume integral:

113% (Ea x Ha) -da' = ha / V . (E, x Ha) (17’. (2.85)

36

 

Using Equations (3.1) and (8.2), the volume integral may be written as,
ha/ [Ha . (V x Ea) — Ea . (V x Ha)] (17’ = haka/ (Hg —— E3) (17". (2.86)

Substituting this form into Equation (2.23) and assuming harmonic time variation of
the fields results in
—w2 + 52,2, = 51,2, / (H3 — 133) dr’. (2.87)

The first order fractional change in the mode frequency therefore becomes,

 

w—wa~_l_ 2_ 2 I
.52 / (Ha Ea) (17'. (2.88)

wa

2.4 Cavity Figures of Merit

There is no single, universal cavity shape which will always provide the best perfor-
mance for any given accelerator. Each cavity shape is optimized as needed to achieve
the design goals of a particular machine. The optimization generally requires a com-
promise between competing design goals. The purpose of this section is to briefly

describe some figures of merit which are used in the design of the cavity shape.

2.4.1 Accelerating Gradient and Transit Time Factor

A charged particle passing through a cavity will gain energy from the accelerating
mode. The voltage gain is calculated by integrating the electric field of the mode
along the path of the particle,

8f
V=/ E(s,t)-ds=Re
8

i

 

t .
[t f eJ(wt+¢) E(s) - v(t)dt . (2.89)

The cavity fields oscillate sinusoidally with angular frequency w and phase (13. The in-

tegral in Equation (2.89) cannot be evaluated further without knowing how E varies

37

 

along the particle’s path. Additionally, the particle’s path is not known without solv—
ing the full dynamical problem. Once the particle trajectory is known and V is cal-

culated, then an average accelerating gradient may be defined as

(2.90)

Eacc =

s~|<

with d = [3? ds. The actual voltage gain is always lower than would be expected
for the case in which the same electric field pattern was somehow able to remain
fixed at its maximum amplitude and pointing in the direction of particle motion at
all times while the particle travelled through the cavity. The transit time factor, T,
is a measure of how much the voltage gain is reduced because of the variation of the
cavity field with time. The transit time factor is therefore always less than or equal

to 1 and is defined by the following relation,

1

v = T [3 8’ (13(5)) -ds. (2.91)

If the particle enters the cavity with a speed close to the speed of light or if the energy
gain is small, then the particle velocity may be approximated as being constant. This
approximation simplifies the calculation of the voltage gain, however the variation of

the cavity field with time must still be taken into account,
. s .
v = Re [em / f eJ(%)3E . nsds] . (2.92)
3i

Since the portion of the accelerating mode fields that penetrate into the beam pipe
diminishes exponentially with the distance away from the cavity opening, the limits
of the integral in Equation (2.92) may be extended to infinity. The integral is then
equal to the Fourier transform of the electric field component along the particle path,

V = Re [ej ‘1’ F (3)] . (2.93)

’U

38

The Fourier transform is generally a complex function of the real variable “5’ and the
RF phase (I) is an independent parameter. Equation (293) indicates that the voltage
gain varies sinusoidally as a function of RF phase, as one would expect. The maximum
voltage gain is achieved when the RF phase is set equal to the negative of the phase
of the Fourier transform. In this case, the voltage gain is equal to the magnitude of
the Fourier transform of the electric field along the particle path:

V = IF Ci)" (2.94)

‘U

2.4.2 Quality Factor and Geometry Factor

By combining Equations (2.18), (2.45), and (2.30), an expression for the power dissi-

pated in the cavity walls may be derived:

_ wU _l flay ~2
P0 — Q0 — 2 20 {Ha da. (2.95)

An expression for the intrinsic Q may now be written in terms of the magnetic field

integrals:

 

U = 125/171,215, (2.96)
_ /w_&
Rs _ 2;, (2.97)
who [12;sz
Qo = (2.98)

R3 f fifda

R3 is the surface resistance of the metal boundary (either normal conducting, for
which Equation (2.97) is valid, or superconducting). When it is factored out of the

right side of Equation (2.98), what remains is referred to as the geometry factor,
~2
_ WHO I H a (IT
f Hfda
39

G

 

(2.99)

 

The value of the geometry factor is determined by the shape of the cavity, regardless
of the size. It is seen to be proportional to the ratio of cavity volume to surface
area weighted by the value of the square of the magnetic field magnitude. For the
cavity to have a large intrinsic Q, there must be a large amount of stored energy with
very little dissipated power. The total stored energy is equal to twice the average
energy stored in the magnetic field, and the dissipated power depends on the square
magnitude of the magnetic field on the surface of the cavity. Therefore large values
of G are desirable since it indicates that the energy stored in the magnetic field is
concentrated in a region with comparatively little field on the inner surface of the
cavity. So to maximize Q0, the shape should be optimized such that the volume
containing the high magnetic field region is maximized while the surface magnetic

field is minimized.

2.4.3 Geometric Shunt Impedance

Just as the geometry factor is a size-independent parameter which characterizes the
magnetic field of a cavity mode, the geometric shunt impedance is a parameter which
characterizes the electric field. Borrowing terminology from lumped circuit theory,
the shunt impedance R0 of a mode is defined in terms of the dissipated power and
the voltage gain:

V2

P0 _ R0. (2.100)

When this expression for P0 is substituted into Equation (2.30) and V expanded using

Equation (2.91), the ratio of the shunt impedance to Q0 is

s 2
120 v2 _ (Tfs,f |E|(S)°dS)
66 _ Eff- — wU '

 

(2.101)

The ratio 5501 is referred to as the geometric shunt impedance. The stored energy in

the mode is equal to twice the average energr stored in the electric field. Large values

40

of 3% are desirable then since it indicates that the energy stored in the electric field
is concentrated in the field tangential to the particle path. However, Equation (2.101)
shows that transit time effects also affect the value of 5%. For example, the cavity
could be perturbed slightly in a high magnetic field region causing a significant shift
in the resonant frequency of the mode with comparatively little change in the electric
field pattern. The detuning of the mode would cause the voltage to drop and therefore
lower £13. The single parameter which affects the 5% the most is the radius of the
beam tube. A larger beam tube pulls the electric field lines away from the central

axis and therefore lowers the shunt impedance.

2.4.4 Surface Electric Field Ratio

Field emission of electrons will reduce the cavity Q and reduce the maximum achiev-
able accelerating gradient. High pressure rinsing the interior of the cavity with ul-
trapure water reduces the risk of field emission but does not completely eliminate
the possibility. The peak surface electric field, Ep, typically occurs near the iris of
an elliptical cavity. A larger beam tube radius generally produces a larger Ep value

for a given Ea“; value. A low value of is generally desirable since it reduces the

25151

possibility of field emission.

2.4.5 Surface Magnetic Field Ratio

A large peak surface magnetic field, Bp, is undesirable for two reasons. First, a large
magnetic field near a defect in the cavity surface can cause a small region of the
cavity to go normal conducting. The increased resistance causes the temperature
to further increase, which in turn causes more of the surrounding niobium to go
normal conducting. This process continues rapidly, causing the entire inner surface
to go normal conducting with the cavity fields simultaneously collapsing in the new

lossy environment. This thermal breakdown will prevent the cavity from reaching

41

 

higher fields and is therefore undesirable. Second, when the surface magnetic field
reaches a critical value, superconductivity ceases. This is a fundamental feature of
superconductivity and cannot be improved upon with processing techniques. The
theoretical critical field therefore represents a hard limit which can only be raised by
using different materials. A low value of Big: is therefore desirable for achieving the

maximum possible gradient before the transition to the normal state.

2.4.6 Cell-to-cell Coupling

When identical elliptical cells are welded together at the iris, the TM010 mode of one
cell couples to the TM010 mode of the neighboring cells via the electric field. The
cell-to-cell coupling (kcc) causes the frequencies to shift by typically a few percent,
forming a narrow passband with as many resonances as there are cells in the cavity.
Since the width of the passband is determined by the value of 1:00, increasing the
number of cells will increase the density of the modes as more modes occur within
the same frequency band. The cell-to-cell coupling is calculated from Vyr and V0, the

frequencies of the w-mode and 0—mode,

 

V7;- - u

kcc = (V7r + V3). (2.102)

It is preferable to weld many elliptical cells to form the longest possible multicell

cavity. The motivation for doing so is simply to reduce costs. Reducing the total

number of cavities reduces the number of separate cryomodules, input power couplers

and higher-order mode couplers in addition to reducing the overall length of a linear
accelerator.

The number of cells chosen for a given cavity design is generally a compromise

between competing design goals. The fields in the end cells are reduced by a factor

of sin (23V) for the {ff-mode, where N is the number of cells. As N increases, the

fi-modes of the HOM passbands couple out less effectively because of the weaker

42

fields in the end cells. This reduces the damping of the HOMs and can lead to beam
instabilities or excessive heat loads. A larger cavity must supply more beam power;
the maximum power carrying capability of input couplers can limit the number of
cells. It is impossible to fabricate perfectly identical cells, and the small variations
in shape from cell to cell will cause the 7r-mode field pattern to be distorted. A
multicell cavity must be adjusted cell-by-cell to tune the accelerating mode and to
produce a flat 7r-mode. The adjustments are done iteratively until the magnitude
of the accelerating field in each cell is approximately equal; typically variations less
than 3% are sufficient. The distortions are proportional to the square of the number
of cells in the cavity but inversely proportional to kcc, therefore large values of kcc

are desirable. The iris radius is the primary parameter which determines the value of

kcc.

2.4.7 Longitudinal Loss Factor

When a charge passes through a cavity, it will excite any cavity mode which has
an electric field component in the same direction as the velocity of the charge. By
superposition, it is seen that the wake left by a particle is proportional to the charge of
the particle. For very high average current beams, the power dissipated by the HOMs
can be a large fraction of the total heat load. Monopole HOMs are of concern in this
respect since the electric fields on axis will overlap significantly with the ideal particle
trajectory. Equation (2.22) may be used to calculate how much a given beam current
will excite a particular mode. The solution for the case of a single point charge passing
through the cavity is found first. This impulse response may then be convoluted with
the beam current to find the wake of a stream of charge bunches.

According to Equations (2.13) and (2.17), the electric field may be written as a

sum over the electric fields of the cavity modes. We begin by finding the integrated

43

 

voltage gain,

fE-ds=;ea\/%/Ea-ds=§c;ea (ngwdz), (2.103)

as a function of time when a uniform line current exists in the cavity. The line current
lies on the z axis which is also chosen to be the central axis of the cavity. The
time dependence is contained completely in the expansion coefficients ea. Multiplying
Equation (2.22) by the combination of constants appearing above and setting the

surface integrals equal to zero gives

2—21E<(;;/EE>J+E Mtg/Em)
= —-<t;:/EE>:E

The value of ja may be expressed in terms of the line current I (t):

35 = f-J EadT—\/_ / J Ead7=\/2—U:/I(t)6 y)()Eazd1(2.105)
= (5%] Eazdz) It

Substituting into Equation (2.106), we obtain

21—;[€a(\/;/Eazdz):l+wa[ea(\/;/Eazdz)] (2.106)
= 770135.12)” :5

By setting I (t) = q6(t), with q equal to the charge of the point particle, the impulse
response of the mode may be found. This expression for the current is acceptable
in modelling a point charge in the sense that the instantaneous pulse of current at
t = 0 carries the total charge q through the cavity. However, the integrated voltage

gain f Eazdz from the ath mode for a spatially uniform line current is greater than

44

 

the voltage gain a point particle would experience because of transit time effects.
By modifying Equation (2.107) with the substitution f Eazdz —+ Ta f lEazldz, the
impulse response due to a point charge may be found. In the case of the voltage V

acting on a point charge, Equation (2.103) becomes

5 / 6
V = Z 83 fiTa / Eade = Z 63 "Z‘I—J—a'Va, = gva, (2.107)
a. a

with the following definitions,

Va = Ta/Eazdz. (2.1083)

va = ea, Eff—av, (2.108b)

Substituting Equations (2.108) into Equation (2.107) gives a differential equation for

the voltage of the ath mode,

 

(1%,, 2 V,2 d]
— = -——--—. 2.10
dt2 + wava 2Ua dt ( 9)
Using Equation (2.101), the above equation takes the form
325,, 2 wa R d1
= -— — —. 2.11
dt2 + ”av“ 2 (Q a dt ( O)

The impulse response due to a point charge q is then zero for t < 0 and

v0) = §va(t) = Ea: £1922 (g)acos(wat) for t 2 0. (2.111)

According to the fundamental theorem of beam loading, the energy a charge deposits

in a mode is equal to one half of the product of its charge and the voltage it induces.

45

The total energy a point charge loses to all of the HOMs is then equal to

 

q all HOMs qgw R
—-AE - t: ——'V = a (—) . (2.112)
pmn 2 a 4 Q a
The longitudinal loss factor k” is defined in terms of AE and q as
k = T—AE (2.113)
ll (,2

The loss factor for a point charge may be expressed as a sum over the individual loss

factors of the cavity modes,

k” =;klla =;% (3)“. (2.114)

The general expression for a mode loss factor suggests there are two ways to reduce
the amount of energy a charge loses. The value of 5 depends only on the shape of the
cavity, so a shape change such as reducing the g by increasing the radius of the beam
pipe and iris reduces Is“. Also, simply scaling the same shape to a larger size would
reduce wa and therefore reduce the k” as well. The voltage for a charge bunch can
be calculated by convoluting the induced voltage of a point charge with the bunch

charge distribution.

46

 

Chapter 3

Six-Cell Elliptical Cavity

Historically, hard x—rays for scientific studies have been produced via synchrotron
radiation by electron storage ring sources. Since the brilliance and quality of the x—
rays produced is tied to the electrons which produce them, the accelerator physics
challenges for storage rings are related to producing large electron currents with low
transverse and longitudinal emittances. Because electrons in a storage ring undergo
thousands of revolutions, the final emittance is the cumulative result of many per-
turbations: magnetic forces in the ring lattice, emission of synchrotron radiation,
electron-electron interactions, and periodic acceleration through RF cavities [15]. In
storage rings, the lattice effectively determines the beam emittance.

The beam emittance from a laser driven photocathode source can be lower than
the equilibrium emittance of a storage ring, but to preserve the low emittance, the
beam must not be recirculated. For a 500 mA beam at 1 GeV, this would result in
an unacceptable 500 MW of beam power being dumped. By recirculating the beam
once and recovering the energy in the cavities of the main linac, high-current, high
emittance beams with a low emittance may be produced. Research and development
for ERL based light sources is underway at Cornell and 4GLS at Daresbury, U.K.
X-ray production is not the only application which can benefit from high electron

beam currents with low emittance. The electron beam may also be used for electron

47

cooling of hadron beams, which lowers the emittance of the hadron beam, and for
collision experiments involving polarized electrons and hadrons [16]. Research is un—
derway at Brookhaven National Laboratory on a superconducting electron cooler and
collision experiments are envisioned as possible upgrades to existing facilities, ELIC
for CEBAF at Thomas Jefferson National Laboratory and eRHIC at Brookhaven
National Laboratory.

The general scheme is outlined in Figure 3.1. A photocathode produces the low
emittance beam. The electrons pass through a series of two-cell injector cavities.
The limitation to low cell numbers is due primarily to limitations on the maximum
power capabilities of RF input couplers: e. g. a 500 mA beam gaining 1 MeV would
require 500 kW of RF power. The electrons would attain high energy by passing
through a series of six-cell cavities and then be used for the intended application:
x-ray production, electron cooling, or collision experiments. The choice of six cells
per cavity is determined by higher order mode damping requirements. The electrons
are then recirculated back to the main linac consisting of six—cell cavities where they
are decelerated, giving most of their energy back to the main linac cavities. The low
energy electrons are then sent to a beam dump. The power delivered to the beam
by the injection cavities is then roughly equal to the power used for the application
and the power dissipated at the beam dump. The remaining beam energy is stored in
the main linac. Superconducting cavities are a natural choice for large current ERLs
because of their large irises and superior performance at large duty cycles. The lower
dissipated power of superconducting cavities is also desirable for the main linac of the
ERL, shown in the block diagram of Figure 3.1.

The two main challenges to achieving average beam currents on the order of
100 mA are in avoiding excessive amounts of beam-induced HOM power dissipation
and avoiding multipass transverse beam breakup. A 6-cell cavity was designed which

addresses both of these concerns and is suitable for use in the main linac of an ERL.

48

 

 

 

 

 

 

 

 

 

 

 

 

 

000000 "
000000

 

 

 

 

if

 

 

 

 

 

 

 

Electron cooling I FEL

 

Figure 3.1: The main components of an ERL consist of a source of electrons, a series
of small injection cavities operating with full beam loading, a main linac of multi-
cell cavities operating with little or no beam loading, the insertion device for the
application, and a beam dump.

3.1 The Approach to High Beam Currents

One approach to reducing HOM power and avoiding beam breakup is to fill every
available RF bucket with a beam bunch, i.e. set the bunch frequency equal to the
RF frequency frf of the accelerating mode. By increasing the bunch frequency, a
given average current corresponds to a smaller amount of charge per bunch. The

beam-induced HOM power, from Equation (2.113) is

with [avg = q frf- By reducing the bunch charge, a given beam current is achieved with
lower HOM power dissipation. A simplified way to see physically why this occurs is
to imagine a single charge q having passed through the cavity and inducing a voltage
2V corresponding to an effective voltage of V (from the fundamental theorem of
beam loading) and an energy loss of qV. The same amount of charge is divided into

two pieces and transferred separately through the cavity. The smaller bunch charge 3
49

induces a smaller effective voltage g- resulting in an energy loss of 92—f- per bunch. The
total loss of both bunches 12‘: is one half of the single bunch energy loss. Reducing
the bunch charge ensures that, at any given moment, less charge is flowing through
the cavity, and therefore perturbing the cavity less, reducing the amplitude of the
wake field.

Increasing the bunch frequency can also simplify HOM damping and increase the
threshold current for beam break up (BBU). The top of the cartoon in Figure 3.2
depicts some common features of the magnitude of the impedance for a multicell
cavity as a function of frequency. The diagram shows a resonance for each mode
of the cavity which has an electric field in the direction of the charge motion. The
lowest frequency modes are in the TM010 passband with the frequency of the 1r—
mode denoted as frf' The next two passbands contain higher order modes. All three
passbands contain very high-Q modes with narrow resonances. The frequencies of
these modes are below the cutoff frequency for the lowest waveguide mode. Since the
cavity cannot couple to a propagating waveguide mode at these frequencies, these
modes lose energy through dissipation in the superconducting walls only. The cavity
modes above the cutoff frequency are able to couple out through the beam pipe and
can be readily damped by either a ferrite load or a HOM coupler and therefore have
much broader resonances. The beam current is assumed to consist of a steady state
series of bunches with a Gaussian charge distribution. The Fourier transform of a
single Gaussian bunch is itself a Gaussian, and forms an envelope for the Fourier
components of the beam. From Equation (2.110), it is seen that the beam acts to
excite the HOMs. The spectra of the cavity and the beam are overlapped at the
bottom of Figure 3.2, showing that the beam can drive the high-Q modes below
cutoff.

The cartoon in Figure 3.3 illustrates some of the benefits of a high bunch frequency.
The bunch charge has been reduced, resulting in less HOM power at the same beam

current, and the spacing between the harmonics of the beam is larger. The shape of

50

J)nn)M_

) 7 frequency
I
l

f”) CUTOFF

. I F -9

frequency

 

Figure 3.2: Top: coupling impedance of a multi—cell cavity as a function of frequency.
Middle: Fourier spectrum of a beam with low bunch frequency. Bottom: superposition
of the coupling impedance and beam spectrum. At low bunch frequencies, the bunch
charge is high and the beam harmonics are close together, so all HOMs must be
heavily damped to avoid driving them to high levels.

the cavity may be designed to take full advantage of decrease in the density of the
beam harmonics to simplify HOM damping. By making the radius of the beam tube
large enough such that the cutoff frequencies of the lowest frequency monopole and
dipole waveguide modes is less than 2 frf, the beam harmonics will only be driving

HOMs in which energy may propagate away through the beam pipe. Both a 2-cell

and 6—cell cavity were designed to meet this condition.

51

 

J‘L JlJ [ fi\ WMA- % frequency
'\‘/ ‘V / ’

 

 

 

* frequency

JlJ) Mm...“

f“. CUTOFF

 

 

Jr frequency

 

 

 

 

Figure 3.3: Top: coupling impedance of a multi-cell cavity as a function of frequency.
Middle: Fourier spectrum of a beam with high bunch frequency. Bottom: superposition
of the coupling impedance and beam spectrum. At high bunch frequencies, the bunch
charge is low and the spacing between the beam harmonics may be large enough so
as only to drive propagating HOMs.

3.2 Six-Cell Design

The cavity shape for the main linac must be designed for electrons travelling at speeds
1) approximately equal to that of light (fl = % as 1). The length l of each cell is then
determined from the condition that it takes half an RF period for an electron to
traverse a cell: this ensures that the electrons will only experience electric forces in

the direction of motion when the cavity is operated in the 7r-mode,

_ v
2frf.

52

z (3.2)

The resonant frequency of the TM010 1r-mode was set equal to 2.45 GHz by adjusting
the value of the equatorial radius. An advantage of choosing a relatively high acceler-
ating mode frequency of 2.45 GHz is that an inexpensive 200W solid state amplifier
could be used for vertical RF testing. Also, a smaller cavity is generally less expensive
to fabricate. However, if the cavity shape were used for an ERL, the size could be
scaled to a lower frequency. As explained in Section 2.4.6, a large number of cells is
desirable but, with many cells, there is an increased risk that some HOMs will not
couple strongly enough to the beam pipe. A prudent choice of six cells was used to
ensure adequate HOM coupling. The angle that the straight section makes with the
vertical is set equal to 6°. Smaller angles benefit the electromagnetic performance of
the accelerating mode, but a lower limit exists because of the need to have fluids drain
out of the cavity during processing. Inadequate draining could result in microparti—
cles remaining in the cavity after high pressure rinsing and ultimately lead to field
emission. The mode of propagation with the lowest cutoff frequency for a circular
waveguide is TE11. It was shown in Section A.2.2 that the cutoff wavenumber for the

TE11 mode is
1.841

 

 

k = , (3.3)
c Riris
which corresponds to a cutoff frequency of
1.8410
fc = . (3.4)
2TrRiris

The speed of light is c and Riris is the radius of the cavity iris which is set equal
to the radius of the beam tube. The beam tube radius must be large enough so
that fc < 2 frf to ensure that only propagating dipole HOMs can be excited by the
beam. Monopole HOMs will not couple to a dipole waveguide mode. The monopole

waveguide mode with the lowest cutoff frequency is TM01. From Section A.2.4, it is

53

seen that the cutoff wavenumber for this mode is

 

 

2.405
k = , (3-5)
C Riris
which corresponds to a cutoff frequency of
2.405c
f = . (3.6)
c 27rRiris

To ensure that fc < 2 frf, the value of the iris radius for the 6—cell design must satisfy

2.405c

' - > — = 2.34 cm. 3.7
ll‘lS 47rfrf ( )

The risk of field emission is reduced in cavities with a low ratio of peak electric
field Ep to accelerating field Ea, while the risk of thermal breakdown is reduced in
cavities with a low value of peak magnetic field Bp to accelerating field Ea. The
aspect ratio of the ellipses primarily determine the values of these peak-field ratios,
and it is generally found that choosing a shape which reduces one peak-field ratio
invariably increases the other. The ratio of Ep to Bp serves as a useful measure of
the compromise made between reducing the chances of field emission versus thermal
breakdown. The TeSLA Test Facility (TTF) [17] cavity shape has been very successful
at achieving high fields reliably, so the TTF ratio of g; was adopted for the 6-cell
cavity. Since the 6—cell cavity will have a larger iris than the TTF cavity, the values
of Eli—c and If}: for the 6—cell cavity will both be higher than those of the TTF
cavity. To reduce the possibility of field emission or thermal breakdown, the increase
in the peak-field ratios of the 6-cell cavity did not exceed the TTF values by more
than 10%.

SUPERFISH [18] was used for the cavity design calculations. Finding a cavity
shape to meet these criteria required hundreds of simulations, with relatively mi-

nor adjustments in the cavity shape between simulations. In addition, the different

54

elements of the cavity shape must be joined together smoothly, 2.6. the elements
must join at locations tangent to one another. A program was therefore developed
in Microsoft Excel to generate the cavity shape for a given set of parameters. The
geometrical parameters for a half—cell are shown in Figure 1.2. The user supplies the
values of B , g, the vertical height h = D — B of the center of the equatorial e1-
lipse, the vertical height 2 = b + Riris of the center of the iris ellipse, L, Ririsv d,
a, and frf: the origin of the coordinate system is located at the lower left of the
diagram in Figure 1.2. The values of the equatorial radius D, the coordinates of the
equatorial-ellipse-to—straight-section union ($1,311), the coordinates of the iris—ellipse-
to-straight-section union ($2,312), the major and minor semi-axes of the equatorial
ellipse (A, B), and the major and minor semi-axes of the iris ellipse (a, b) are then

computed with the following equations:

 

 

 

 

—1
m = tan(0z) (3.8)
h—R- -
x _ [L—d+ m ] (39)
61
1+ m
y1 = m(1'31 - L + d) + Riris (3.10)
B 2 2 2
B = (71-) 1:1 +(y1— h) (3.11)
B .
A = (—B_ (3.12)
X
2
_ 7%?2' +m(L_d)+Z" Riris
£132 — 2 (3.13)
El
m + (:7,
312 = m(:1:2 - L + d) + Riris (3.14)
b = z - Riris (3-15)

(3.16)

55

Table 3.1: Final geometrical parameters for the mid—cell of the 6-cell cavity (left) and
corresponding figures of merit for the 7r-mode of an infinitely long periodic structure
(right).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A 2.216 cm

B 1.884 cm

a 0.813 cm 1

b 1.165 cm 753% 2-15
2 3.555 cm 32 4 63 mT
L 3.062 cm 30° MTn—Y
d 0.7 cm G 275 Q
HMS 2.39 cm kcc 4.18%
Requator 5-58 cm

a 60°

 

 

 

 

A screen shot of the interface is shown in Figure 3.4.

A sketch of the half-cell is produced by the program and a SUPERFISH input
file is created for that cell shape. Short-circuit boundary conditions are placed on
the left plane (the plane of the equator), while open-circuit boundary conditions are
placed on the right plane (the plane of the iris). The resulting field pattern from SU—
PERFISH corresponds to that of the 7r-mode for an infinitely long periodic structure.
The figures of merit are extracted from the SUPERFISH output file and displayed
by the interface. A second input file is created in which the boundary condition
on the right plane is changed to a short circuit. The resulting field pattern corre-
sponds to the O-mode of an infinitely long periodic structure. The frequencies of
the two simulations are then used to compute kcc according to Equation (2.102).
These final values are shown in Table 3.1.

The figures of merit in Table 3.1 are valid for an infinitely long periodic structure,
however the same mid-cell field pattern may be reproduced in a real cavity with beam
tubes attached by varying the shape of the end-cells. Leaving the end-cell geometry
equal to the mid-cell geometry most likely will cause the peak surface fields in the

cavity to be located in the end-cells. Additionally, it is likely that the axial electric

56

Table 3.2: Final geometrical parameters for the end-cells of the 6—cell cavity (left) and
figures of merit for the 7r-mode of the full 6—cell cavity (right).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A 2.312 cm

B 2.820 cm

h 2.757 cm 3 535-2 9
a 0.813 CID J 219
b 1.165 cm

2 3.555 cm E 4-68 [637
I 3.062 cm G 275mg
Rir's 2-39 cm T 0.77

uator 5.58 cm
a 9.93°

 

 

 

 

field will be higher in the end-cells. The values of Ep and Bp determine the maximum
achievable gradient for a cavity, so untuned end-cells ultimately result in lowering the
maximum gradient. The goal of the end-cell tuning is therefore to reduce the peak
field values to a level at or below the peak fields in the mid-cells while simultaneously
providing “flat” axial electric fields. Each cell in a cavity with completely “flat” fields
would have the same magnitude for the peak axial electric field.

Table 3.2 shows the geometrical parameters for the end-cell shape used for the
6—cell cavity. The figures of merit for the full 6-cell cavity are also shown. Figure 3.5
shows a picture of the cavity shape with the 7r-mode electric field lines predicted by
SUPERFISH. The peak electric and magnetic fields are seen in Figure 3.6 to occur in
the midcells and the axial field magnitudes shown in Figure 3.7 are “flat” to within

3%. The flatness was measured by forming the ratio,

20%“ 4133311))

lEz‘gaxl + lEfliinl ’

 

(3.17)

where Ezo is the peak axial electric field within a cell. The transit time curve for the

6—cell cavity is shown in the next chapter (Figure 4.4).

57

3.3 Higher Order Modes of Interest

The six-cell cavity is designed such that, when operating CW in the energy recovery
mode with every RF bucket filled, the beam will only drive HOMs which have frequen-
cies above the cutoff frequency of the beam pipe. It is desirable for the modes which
can be driven by the beam to have strong coupling to the beam pipe in order to ensure
that the energy of these modes may couple out. To help address the question of cou-
pling strength, a HOM analysis following the technique for the BNL “Ampere Class”
cavity was done [19]. The calculations were done using Analyst [20]. For the six-cell
HOMs, simulations were carried out by Analyst: one set of simulations used electric
boundary conditions for the end of the beam pipe while another set used magnetic
boundary conditions. Electric boundary conditions mean that the tangential electric
field and the normal magnetic are both zero on the boundary. Magnetic boundary
conditions mean that the tangential magnetic field and normal electric field are both
zero on the boundary. A strongly coupled HOM will have high field throughout the
beam pipe. In this case, the different boundary conditions will result in different fre-
quencies and this frequency difference may be used as a rough meaure of the HOM
coupling strength. The frequency from the electric boundary condition simulation fe
and the frequency from the magnetic boundary condition fh are combined in a single

expression to measure the coupling strength:

fe+fh [

X = log [m (3.18)

Low x values indicate high coupling.

The 8 was calculated by integrating the voltage gain over a path along the central
axis of the cavity and between the cavity end cell irises. Simulation results for 700
monopole HOMs below approximately 20 GHz are shown in Figures 3.8 and 3.9. In
energy recovery mode with every bucket filled, there are two charge bunches for every

RF period, hence the beam spectrum consists of multiples of 4900 MHz. It is worth
58

noting that the simulations for the HOM spectrum of the six-cell cavity were extended
to a higher frequency (20000 MHz) than in the BNL 5—cell study (1800 MHz with
an accelerating mode frequency of 703 MHz). If the Ampere-class cavity is rescaled
for an accelerating mode frequency of 2450 MHz, then the original 1800 MHz range
HOM study is rescaled to a range of 6273 MHz. The HOM analysis of the six-cell
cavity below 6273 MHz may then be compared to the Ampere-class results.

The six-cell cavity does have some weakly coupled modes below 4900 MHz, but
the X values near 4900 MHz are below 3, which is the maximum x value for the
Ampere-class cavity. It is therefore reasonable to expect the dangerous HOMs of the
six-cell cavity to propagate as effectively as those of the Ampere-class.

Simulation results for 700 dipole HOMs below approximately 20000 MHz are
shown in Figures 3.10 and 3.11. The 5 was calculated using an integration path
between the cavity end cell irises and offset from the central axis by 5 mm. Comparing
these results with those of the Ampere-class, where the same type of calculations
resulted in x between 3 and 4 for the dipole modes, similar performance is again
expected for the six-cell cavity. For the six-cell cavity, X is less than 4 for the dipole
modes at frequencies which are multiples of 4900 MHz. It is seen in Figure 3.11 that
each dipole g is less than 30 c—ffiz’ whereas the Ampere-class contains dipole modes
with gs as high as 55 (373—17. According to Equation (1.5), the threshold current for
beam breakup is inversely proportional to the HOM shunt impedance. It is therefore

not unreasonable to expect slightly higher threshold currents for the six-cell cavity.

59

 

 

 

 

 

 

 

 

 

 

 

 

 

wovvmnv—mm.
omvn

mEo

 

 

 

 

 

 

 

 

 

 

 

 

 

2.2m =8

_, )1.J,,.))_J 4):).1.--)0) ufi)uJ)...):m:)

—ZO.D|— OBI-0.2)?-

    

 

 

 

    

        

. I] 1]!) JV) [1].] (1|!) ll:

.m, .i ,_ a. x

 

 

 

 

 

 

D

 

 

 

ohm-15:61? .H 90

3.3.3

nonflln l:.u~.1i. z. 3;.9. WW! :«zeuo..:.c_vflaa bdifiufin:

 

 

SJ.) - h3£§s§31).fl. Emma

1.x....__:..._>_.) _ _. .x. :: 1...:7.

 

%, h, z, and d uniquely determine the cell shape

displayed at center and the corresponding figures of merit displayed on the right.

B
I)

Figure 3.4: Five free parameters

60

-' —-——_,_—-" ____,_I—

V"— -‘. _.z—' -- -r

— , -_h _—-" “-— —r"—r l ———- .4, —”. - ‘ —w
. \«_.—'{<‘,'. _. ‘~a____—/,.x_-, ..-.\__~—_—r{/, (x ._ —s__r-f, . ,_ .__,—"/, . .\\‘--~_.—-
_ .’ \\>\‘ J . \ . ~ J/ . - \ .

 

‘ \~__. -~"/' 1 - [011‘ ~. . \. —-’;’/.“.: ‘1' fin", .; \an 2.- «C:- _ ,r' '
MLJ’ ‘W/,-'HT‘\.:—_“—r"jux\» -x,’ a.~.\_ Lu) ' \_:_—.._./
\-—5.___r-""" ’- \‘ \M__h'____,___r‘ . \___\\ _’__, '1 W __.,_-" [\K J“ l “
_ ,.r' “ .r" _ —r’ .._- .—.—/ _ __ Pn" \.

 

Figure 3.5: The shape of the 6—cell cavity with electric field lines calculated by SU-
PERFISH.

 

35 j I I 7 I I

 

 

    

 

 

I lE' I
30 _ c B ...... _
25 - "
surface — . ........ . . ......... . ,,,,,,, .. _
field 20 -
(Mir!) 15 "z ‘
10 7" ~
5 r a
0 I i :1 1 1" 1
0 5 10 15 2 25 30 35 40 45

distance along surface (cm)

Figure 3.6: Magnitudes of the electric and magnetic fields for the 6-cell cavity as a
function of distance along the surface. The origin is located at the middle of the cavity
on the iris. The path moves along the surface and toward the beam pipe. The fields
are normalized to an accelerating gradient of 15.3 M31.

61

 

35

I

30

25 *-

IEzl 20
(M)

m 15

10-

I

5.—

 

l

 

 

 

 

 

 

 

 

l

l l

 

 

0 1
—40 —30

—20

1
—10

0 10

distance along axis (cm)

20 30

40

Figure 3.7: Variation of the magnitude of the axial field for the 7r-mode for the 6-cell
cavity as a function of the axial distance. The fields are normalized to an accelerating
gradient of 15.3 Mfiya The origin is at the middle iris of the cavity.

fe+f
103(1—1‘72 fe—fh )

OHNOOtD-Cflaflm

 

 

 

 

 

 

. . ;

. 4' o .l '5 .0 .. .

. . . :' ' ~ '. 0. o I. .‘:

. :. 0". . . u . I"... . | 4

a , ‘ 5 3° 4' - $4., -:..5o;::-.f 4'3" . ) .I‘

.- ..\. :j l 05’}. o '0’! Y. '5‘, '

.1. - : a: .u . eigfiflt. 8'.- 34 ruffle-u 01. .1. . |
IOL I I .1... '0. .0... o. :1 q ..

.’- o a

 

 

 

 

 

 

 

 

 

4900

9800

frequency (MHz)

14700

19600

Figure 3.8: Two different simulations were done for each monopole mode: one with
electric boundary conditions on the beam pipe and the other with magnetic bound-
ary conditions on the beam pipe. Large differences in the frequencies for these two
simulations indicates large field in the beam pipe and corresponding large coupling
of the HOM. It should be noted that the zero values seen correspond to numerically

divergent results where the frequencies of the two simulations are equal.

62

 

100

 

 

 

 

 

 

 

 

 

 

 

80
60
R/Q (9)
40
20
0 : -1‘-..-'.._-".'.:-;--.~'.;:-‘ £-_,,-_--.',~,",' 'c' 11C”..- ,; 33.11.31;
0 4900 9800 14700 19600

frequency (MHz)

Figure 3.9: The R/Qs for the monopole modes as a function of frequency.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

8
7
6
5
fe+fh
10g(2 f€_fh) 4 I .: . . '-
3 {if ' ':.:- 1;“ ,. . .
2 a s L...» {’“L‘ ‘E‘fllnfiiffih #33,}flh 1:!
1
0 __
0 4900 9800 14700 19600

frequency (MHz)

Figure 3.10: Two different simulations were done for each dipole mode: one with elec-
tric boundary conditions on the beam pipe and the other with magnetic boundary
conditions on the beam pipe. Large differences in the frequencies for these two sim-
ulations indicates large field in the beam pipe and corresponding large coupling of
the HOM. It should be noted that the zero values seen correspond to numerically
divergent results where the frequencies of the two simulations are equal.

63

 

300000

 

250000

 

200000

 

R/Q (n/m2150000

 

100000

50000

 

I": '

 

 

 

 

 

 

 

I b“.
o I ‘g '
I “an. -i‘ -Au‘-!-.p'

0 '. . ' ' ' .. .. '
0 4900 9800 14700 19600
frequency (MHz)

 

Figure 3.11: The R/Qs for the dipole modes as a function of frequency.

64

Chapter 4

Two-Cell Elliptical Cavity

4.1 Two-Cell Cavity Design

4. 1 . 1 SUPERFISH Simulations

The 2—cell cavity will share the same design criteria as the 6—cell cavity:
0 2.45 GHz 7r—mode frequency
0 TMOl cutoff < 1.95frf
o TE11 cutoff < 1.5frf
0 6° wall angle
. g2 equal to TTF value.
P

In addition, the 2—cell cavity was designed to accelerate electrons with low initial
energy. Electrons from a gun with an energy of approximately 50 keV could be accel—
erated by two 2-cell cavities to an injection energy between 1 and 2 MeV for injection
into the ERL main linac.

A final cell geometry was determined using SUPERFISH simulations and the half-

cell parameters are listed in Table 4.1 along with the figures of merit. In the 2-cell

65

case, a single half-cell geometry was used; the half-cells which are welded to the beam
pipe have the same shape as the half-cells welded together at the middle iris. The
cavity shape is shown in Figure 4.1. The magnitudes of the electric and magnetic
fields on the surface of the right half of the cavity are shown in Figure 4.2. It is seen

that the peak electric field occurs on the middle iris.

Table 4.1: Final geometrical parameters for the 2-cell cavity (left) and figuras of merit
(right) for 6 = 1. The value of kcc was calculated using the same cell geometry but
assuming an infinitely long periodic structure.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A 1.643 cm
B 1.479 cm
h 4.120 cm 5 164.6 (2
a 0.732 cm
b 1.049 cm FE}; 2.16
2 3.439 cm _§r 4 59 mT
I 2.462 cm ace ' WIT?
d 0.630 cm G 228 Q
Tiris 2.390 cm kcc 5.62%
1297,11,,“ 5.599 cm
a 60"
I.- .‘I
g E:
V \\-/9

 

Figure 4.1: The shape of the 2-cell cavity with the electric field lines for the 1r-mode.

Figure 4.3 shows a plot of the magnitude of the axial electric field Ez along the
axis of the right half of the cavity. The field goes through zero at the middle of
the cavity since it is the 7r-mode, and trails off exponentially with the evanescent
fields in the beam pipe. For an infinitely long structure with the same cell shape, the
field would go through zero at every iris. In this case, maximum energy gain would
occur for particles with a constant speed of 6 = 4—fgLL, the so called geometric 6.

66

 

35 I I I I 7 I ElI
c B ......

d

30

25

surface 20

field
(1g) 15

10

 

 

 

 

0 2 4 6 8 10 12 14 16
distance along surface (cm)

Figure 4.2: Magnitudes of the electric and magnetic fields as a function of the distance
along the surface. The origin is located at the middle of the cavity on the iris. The
path moves along the surface and toward the beam pipe. The fields are normalized

to an accelerating gradient of 10%.

Substituting frf -- 2.45GHz and L = 2.462 cm results in a geometric B of 0.805 for
the 2—cell cavity. Adding a large-diameter beam pipe cause the field to fringe out into
the beam pipe. The cell length L = 2.462 cm was chosen so because most of the high
field region is then contained within a distance of 9,3- = 6.12 cm from the center. This
improves the transit time factor for velocity of light particles. For a particle to travel
the iris-to-iris distance in exactly half a RF period would require a constant speed of
,6 = % = 0.805. It is the extension of the axial electric field into the beam pipe which
causes the energy gain for 6 = 1 to be larger than 6 = 0.805.

Figure 4.4 shows a plot of the transit time factor as a function of particle 6 for
both the 2-cell and 6—cell cavities. The transit time factors were calculated using a
constant velocity approximation. By squeezing the 2-cell cavity to a geometric B of
0.805, the transit time factor is comparable to the 6-cell cavity at particle speeds
equal to the speed of light. Electrons will quickly approach the speed of light and
efficient acceleration at these speeds is therefore desirable. Also, if a more elaborate

electron source such as a superconducting photocathode is used, then the particles
67

 

 

 

l l 1 l l J

0 2 4 6 8 10 12 14
distance along axis (cm)

 

Figure 4.3: Magnitude of the axial electric field along the central axis of the 2-cell
MV
0?.

cavity, normalized to an accelerating gradient of 1
would enter the injection cavity with speeds comparable to that of light, and efficient
acceleration at B = 1 is again desirable.

Because only two cells are used to form the injection cavity, the transit time
values at low particle speeds are much better than for the 6-cell cavity. This allows
low energy electrons with speeds of approximately 0.5c from a normal conducting
gun to be accelerated by the 2-cell cavity. Limiting the injection cavity to two cells is
also necessary because of the limitations in the maximum power [capabilities of input
couplers. The 2-cell cavity would be operated with full beam loading, in contrast to
the 6—cell cavity which would operate in energy recovery mode. Providing 500 kV of
voltage gain to 100 mA of beam would require 50 kW of input power; this is close to
the power handling limits of input couplers. This translates to a design accelerating
gradient of approximately 5 7MfiY. Operation at 2 K with a Q0 = 5 - 109 is feasible.
Operation at or below 2 K would be necessary if a relatively high frequency such as
2.45 GHz is used, since the BCS resistance increases with frequency.

Table 4.2 compares the figures of merit and aperture dimensions of the 2-cell cavity

to those of the Cornell ERL injector cavity [21]. Both cavities are designed for 100
68

 

 

2-celll
0.7 __ 60811 ......

   
 

0.6 -

.0
01
I

   

 

 

 

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
particle velocity 6

Figure 4.4: Transit time factor T as a function of particle speed calculated with SU-
PERFISH for the 2—cell and 6—cell cavities. The particle velocity is assumed constant
for the evaluation of T.

mA of beam current and both consist of 2 cells. The Cornell cavity has a better 8,
Egg? and 331% due primarlly to Its smaller ll‘lS radius, the 2-cell cav1ty ll‘lS radius 18
28% larger when both cavities are scaled to have the same frequency. An advantage
of the larger iris radius of the 2-cell cavity is a lower overall loss factor, and it is worth

noting that 15% and 6%: are only 11% and 7% larger respectively.

4.2 Higher Order Modes of Interest

The two-cell cavity is designed such that, when operating CW with every RF bucket
filled, the beam will only drive HOMs which have frequencies above the cutoff fre-
quency of the beam pipe. It is desirable for the modes which can be driven by the
beam to have strong coupling to the beam pipe in order to ensure that the energy of
these modes may couple out. The technique for analyzing the 2—cell HOMs is identi-
cal to that in Section 3.3 for the six-cell cavity. For the 2-cell cavity, beam loading is
present and Fourier components of the beam are present at multiples of 2450 MHz.

Simulation results for 425 monopole HOMs below approximately 30 GHz are shown
69

Table 4.2: Comparison of the 2-cell cavity figures of merit to those of the Cornell ERL
injector cavity(also a 2-cell cavity).

 

 

 

 

 

 

 

 

 

cavity 2-cell Cornell Injector
R 164.6 12 218 Q
Fifi 2.16 1.94

B ” mT mT
Ea}: 4.59 MY 4.28 m

m m

kcc 5.62% 0.7 %
frequency 2450 MHz 1300 MHz
left beam pipe radius 2.39 cm 3.9 cm
iris radius 2.39 cm 3.5 cm
right beam pipe radius 2.39 cm 5.3 cm

 

 

 

 

 

in Figures 4.5 and 4.6. The 8 was calculated by integrating the voltage gain over a
path along the central axis of the cavity and between the end cell irises. The relatively
weak coupling seen in Figure 4.5 for modes near 17,150 MHz, 22,050 MHz, and 26,950
MHz may lead to high resonant HOM fields if adequate damping is not present.
Simulation results for 400 dipole HOMs below approximately 30 GHz are shown
in Figures 4.7 and 4.8. The 8 was calculated using an integration path between the
ends of the cavity and offset from the central axis by 5 mm. For the two-cell cavity, x
is highest between 6 to 7 for the monopole modes and between 4 to 5 for the dipole

modes at frequencies which are multiples of 2450 MHz.

70

fe+f
loE(W—Afl fe-fh )

Figure 4.5: Two different simulations were done for each monopole mode: one with
electric boundary conditions on the beam pipe and the other with magnetic bound-
ary conditions on the beam pipe. Large differences in the frequencies for these two
simulations indicates large field in the beam pipe and corresponding large coupling
of the HOM. It should be noted that the zero values seen correspond to numerically

OHMWQU‘IGNOO

 

 

 

 

I

 

I".

' H

‘ |
' . .0. I .J. \ I. .
a 3 . . II ’ I.
'G:,-. . . o I I I.

 

O
I
.~'d I.
I0

- 1‘}: .-I c

I.

I. .
.5 .-.l.‘.:AQ .'
‘
P.:": .f‘ .-
-.00 0‘
I .

 

 

 

 

 

A

 

 

 

 

 

 

 

4900

9800

14700
frequency (MHz)

19600 24500 29400

divergent results where the frequencies of the two simulations are equal.

140
120
100
30
R/ Q (9)
60
40

20

Figure 4.6: The R/Qs for the monopole modes are plotted vs. frequency.

 

 

 

 

 

 

 

 

 

) "

 

0,-0..II

. I
I I
AAA...-- L-

 

' I.

I. ' 0
II I
lama-An n A.

 

..I...I
1f..2£3

 

 

 

.
..’. If... E” l..I-. i.

 

0
0

4900

9800

1470
frequency (MHz)

71

19600

24500

29400

 

fe+f
log(2|fe-fhl)

 

OHMWIRU‘QKIW

0 4900 9800 14700 19600 24500 29400
frequency (MHz)

Figure 4.7: Two different simulations were done for each dipole mode: one with elec-
tric boundary conditions on the beam pipe and the other with magnetic boundary
conditions on the beam pipe. Large differences in the frequencies for these two simu-
lations indicates large field in the beam pipe and corresponding large coupling of the

HOM.

 

160000
140000
120000
100000
R/Q (Q/m2)80000
60000
40000 . . _. _ .
20000 3* I: ,. _.:'f ?' f_'fil..-“-:[ r

\-
.l .0...
I

 

 

 

 

 

 

. .'-., ..' . 11...... 2”; ‘,-:....__.:;;j)+.. .

0 “H“ .'- '-"°-'1"-..-. -'.'n".:'-'.u_.'.' 11.123333? Lat-aw;

0 4900 9800 14700 19600 24500 29400
frequency (MHz)

 

 

 

 

 

 

 

 

 

 

 

Figure 4.8: The R/Qs for the dipole modes of the two-cell cavity as a function of
frequency.

72

Chapter 5

Prototyping

5. 1 Fabrication

Two 2-cell cavity prototypes were fabricated from 250 RRR niobium. They were deep—
drawn from 4 mm thick sheets with a manually operated 25 ton hydraulic press. A
separate coining die was used after deep drawing to compensate for spring back of
the niobium half cells. Interlocking equatorial weld preps (total thickness 2 mm) were
machined with an end mill. The flanges were made of N bTi. For vacuum sealing, 3—3 / 4
inch knife edges were machined into the flanges. The niobium tubes were formed by
spinning, using a mandrel made of 7075 T6 aluminum. The flanges were tungsten-
inert-gas (TIG) welded to the beam tubes and all remaining cavity welds were done
with an electron beam at Sciaky, Inc. in Chicago, Illinois. A photograph of the beam
tubes, flanges, and half-cells is shown in Figure 5.1. Small titanium dishes attached
to the flanges are present for the purpose of welding a helium vessel around the cavity
in the future. An additional 0.015 inches of material was left at the weld preps to

allow for electron beam weld shrinkage.

73

 

Figure 5.1: Niobium half cells after stamping and machining for weld preps.

5.1.1 Electron Beam Welding

This discussion expands upon notes on electron beam welding from Robert Wag-
ner [22]. The weld preps were cleaned just prior to welding by soaking the parts in a
1:122 acid BCP for 5—7 minutes. This removed a layer of material approximately 10 pm
thick. After a rinse with distilled water and ultrapure water, the parts were assembled
for welding. The welding of a half-cell to a beam tube is shown in Figure 5.2. The
parts are held in a rotating fixture. The electron gun (shown on the left in Figure 5.2)
can be moved and rotated. For the 2-cell welds, the gun remains stationary and the
cavity parts rotate over 360°. Electron beam welding of high RRR niobium requires
a better vacuum than is typical for electron beam welds. The pressure used in the

Sciaky welder was 50 utorr. The aluminum fixture never touched the cavity parts

74

directly to avoid diffusion of foreign material into the hot niobium: small niobium
tabs are placed between the part and the fixture (see Figure 5.2).

The iris welds were done first: The beam tubes were welded to the end half-cells
and the two middle half-cells were welded together at the iris forming a “dumbbell”.
As illustrated in Figure 5.2, the iris welds were inside welds (with the electron beam
incident on the inside surface of the cavity). This means that the weld did not need
to penetrate through the entire thickness of the niobium to produce a good RF joint.
It takes approximately 1 hour for the 68” x68” x84” vacuum chamber to pump down.
The rotating table has chucks for 4 independent assemblies, allowing several iris welds
to be done in one pump-down. The welding itself was not time consuming, taking
approximately 20 seconds. But, to avoid contamination from air diffusing into the hot
niobium, the welded parts were allowed to cool under vacuum for about 40 minutes.
A small amount (100-200 torr) of nitrogen was then introducedto aid in cooling. The
total cool-down time was about one hour. The gun voltage was 50 kV and, after an
initial tack weld to keep parts aligned, a partial weld around each iris was done with
30-35 mA of beam.

The final round of welding was for the equators of the cells. The cavity was assem-
bled vertically on the rotating chuck in the same orientation as shown in Figure 5.2.
The beam came from the side and was incident on the outside surface of the cavity.
Therefore the weld had to penetrate through the entire thickness of the cavity, which
was 2 mm at the weld prep, to ensure a smooth inner surface. The beam current
was carefully chosen to ensure that it was large enough for full penetration but low
enough to avoid a “blow through”. The beam was rastered to slightly defocus it, and
avoid weld spatter. The fixture used springs to help maintain contact while shrinkage
occurred during the welding. A spot weld of approximately 2.5 cm was done at 4
locations on both the top equator and the bottom equator to keep the parts aligned.
After spot welding, a shallow (40 mA) seal weld was done at a rate of 18 inches per

minute along the entire circumference of the top equator. The full penetration weld

75

 

Figure 5.2: Setting up for an iris weld inside the vacuum chamber of the electron
beam welder. The electron gun is visible on the upper left. Parts for a single cell
6 = 1 cavity are shown; the welding of the 2—cell cavity was done the same way.

was started with a beam current of 52 mA at 12 inches per minute. The current
was reduced to 50 mA for the remainder (approximately 90%) of the weld. The top
equator was welded first to help avoid misalignment due to weld shrinkage. A seal
weld and a full penetration weld were then done in the same manner for the bottom
equator.

After cool—down it was noticed that the there was a sector of the top equator weld
which did not have full penetration, while the bottom equator had full penetration
throughout. This difference was attributed to the heating from the top weld decreasing
the beam current necessary for full penetration of the bottom weld. The top weld was
redone with 12% more beam current along a. 90° sector centered on the gap. The I
second cavity was welded with 52 mA for the top weld and 50 mA for the bottom

76

weld. The welds in both cells looked fine after cooldown and inspection. A photograph

of one of the complete cavities is shown in Figure 5.3.

 

Figure 5.3: A completed 2-cell cavity.

5.1.2 Waveguide Coupling Measurements and Choice of Beam

Tube Length

For vertical testing of a cavity, the niobium beam tube is made long enough to ensure
that the field levels at the stainless steel connecting flange are negligible. One of the
beam tubes of the 2-cell cavity was deliberately made shorter so that the fabricated

cavities could be used for additional experiments. One such experiment would be
77

to test the feasibility of a circular normal-conducting copper waveguide as an input
coupler for a superconducting cavity. As illustrated in Figure 5.4, the input coupler
would be placed outside of the helium vessel and the beam would pass through the
waveguide.

Figure 5.5 is an top-view cartoon. The waveguide is shorted at the top and the
electric field of the TE11 waveguide mode couples to the TM010 mode of the cavity.
One would expect the coupling strength to increase as the distance separating the
cavity and the waveguide decreases. However, there are mechanical constraints on
how close the cavity and waveguide may be placed to each other with the waveguide
outside the helium vessel. One basic question was whether sufficient coupling could
be achieved in such a configuration.

One of the 2-cell cavity beam tubes was to be made short enough to be used to
test such a coupler. Bench top RF tests were done with a single-cell copper model
to measure the coupling between the cavity and the waveguide. The copper model
has the same cell shape as the 2-cell cavity. These measurements were then used to
determine the beam tube lengths for the 2-cell cavity. Figure 5.6 shows a diagram of
the test setup.

First, with the cavity removed from the waveguide, the RF power from the network
analyzer was applied to feedthrough B. Reflections from the feedthrough were reduced
to approximately zero by adjusting the length of a small wire antenna attached to the
center conductor of the feedthrough. An antenna length was found for which the real
part of the input impedance at B is 50 D. To match the entire input impedance at B to
50 Q, the imaginary portion must be eliminated. This was accomplished by sliding the
position of sliding short D. The same procedure was used to provide a matched input
at feedthrough A by adjusting the position of sliding short C. The position of this
short remained fixed throughout the testing to provide a matched connection between
the waveguide and the network analyzer. The single-cell copper cavity F and collar

G were next attached to the waveguide. RF power from the network analyzer was

78

n“
11.7,)
1“

A

”I ‘,
'4': 1] 11'" I!)
_;;___ "‘

 

' I
| s "1‘ \ en “in u
. I -
‘ .-
295.1312“? g "E'
——-

 

 

 

Figure 5.4: The feasibility of a proposed normal-conducting waveguide input coupler
may be tested with the 2-cell cavity. The TE11 wave in the copper waveguide would
couple to the TM010 mode of the cavity and the beam would pass through the
waveguide. The waveguide feed is below the beam line, with a waveguide short above
it.

79

Figure 5.5: Top view of the electric field in the waveguide (shown as a cartoon)
coupling to the electric field of the TM010 1r-mode (from SUPERFISH).

supplied to the cavity at the N-type feedthrough E. The input at E is also matched
to 50 Q to maximize the cavity field level. The RF detection port of the network
analyzer was attached to feedthrough A. By measuring the transmission from point
E to point A, the coupling strength, or Qexta between the cavity and the waveguide
at the collar G was inferred. There are two parameters which affect this coupling
strength: the distance d seperating sliding short D from the beam line and the length
of the collar at G. Collars of various lengths were used to vary the separation between
the cavity and the waveguide, and thereby vary the Qext- For each collar length, Qext
was measured at different d. Since a two-cell cavity will have twice as much stored
energy as a single cell, the measured Qext values were multiplied by two. The results
are plotted in Figure 5.7.

Envisioning the transverse electric field of the TE11 waveguide mode coupling
to the axial field of the TM010 cavity mode as in Figure 5.5, it is reasonable to
anticipate a maximum in the coupling when the short is located approximately one
quarter RF wavelength away from the beam line and a minimum in the coupling when
the short is located approximately one half RF wavelength away from the beam line.

This was in fact measured, as seen in Figure 5.7. The strongest coupling occurs when

80

 

 

 

 

Figure 5.6: A copper waveguide with moveable shorts at either end is used with a
network analyzer to measure the coupling strength of the TE11 mode of the guide to
the TM01 mode of the cavity as a function of the short to beam line distance d: A)
N-type coaxial feedthrough B) N-type coaxial feedthrough C) sliding short D) sliding
short E) N-type coaxial feedthrough F) copper single cell G) removable collar.

81

 

Qext

 

 

 

 

 

 

6 .
100.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
separation between short and beamline

(RF wavelengths)
Figure 5.7: The Qext of the waveguide coupler measured as a function of the posi-

tion of the sliding short closest to the cavity. The other short remained fixed. The
measurement was repeated for several different distances between the cavity and the
waveguide:1.935 in, 2.435 in, and 2.685 in. The Qext values shown are for a 2-cell
cavity, scaled from measurements on a single—cell cavity.

the Qext is lowest. Another consideration which affects the beam tube length is the
power dissipated in the copper waveguide. The beam tube must be short enough to
establish adequate coupling, however the cavity will be loaded by the copper more
if it is closer to the waveguide. The decrease in Q0 due to dissipation in the copper
walls is approximately found from SUPERFISH simulations for a two-cell cavity with
long beam tubes by considering a long beam tube which is niobium near the cavity
and copper past a chosen distance L away. The simulation uses the surface resistance
of niobium and the calculated values of the surface magnetic field to determine the
power dissipated in each segment of the cavity geometry. If the copper waveguide is to
be located a distance L from the cavity, then the additional heating in the copper may
be estimated by first finding the dissipated power in one of the beam tubes, starting at
the distance L from the cavity and extending to the end of the tube, then multiplying
this power by the ratio of the surface resistance of copper to the surface resistance of
niobium. The total dissipated power was then found by adding the estimate of the

power dissipated in copper to the power dissipated in the remaining segments, which

82

 

 

 

 

 

1 . 9
10 g Cu lbading . I ' 10
' ext 2
100 1_
Cu losses _
tota losses

10_1 g

10—2 1 n 1 1 1 106

0 1 2 3 4 5 6
beam tube length (cm)

Figure 5.8: The percentage of power that is dissipated in the copper waveguide is
compared with the coupling strength of the waveguide. Both quantities have a strong
dependence on the length of the gap separating the cavity and the waveguide. The
stars indicate the chosen final beam tube length of 3.5 cm.

are taken to still be niobium. The Q0 was then calculated as

Q0 = L_wU( PNb )=QNb(__f_)N_b___), (5,1)

 

PNb + Pcu — PNb PNb + Pcu PNb + Pcu
Q0 PCu
_ = 1 _ ———————. (5.2)
QNb PNb + PCu

The lowest measured Qext for a given cavity-coupler separation and the fraction
of power dissipated in copper are plotted in Figure 5.8. For the calculation of the
dissipated power, a niobium surface resistance of 15 nil and a copper surface resistance
of 12.861 m9 were used. To be conservative, the value for the surface resistance of
copper is based on the resistivity at room temperature; the power dissipation in the
copper can be expected to decrease in the case of copper at lower temperature. At large
tube lengths, the Q loading due to copper is small, but the Qext is large. At short tube
lengths, the Q loading due to copper is large, but the Qext is small. The exponential

dependence of the Qext on tube length is expected since the evanescent fields of the

2.405 _.

TM010 mode diminish exponentially in the beam tube. The rate of decay 23931—1 —

1.01 cm’1 is roughly consistent with the 6 dB/ cm seen in Figure 5.8. The beam tube

83

blzTr-ansllsslon Phase 2.I'l Raf -75.II'
P20”

zeal—.— \

 

fleaslzflkrl 2~63.25l HHz '
-BS.162‘

 

 

 

 

 

 

 

 

 

 

 

 

I

‘l

‘1
##fl

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5.9: A small metal bead was pulled along the axis of the 2-cell cavity with fish-
ing line. The bead causes the resonant frequency to shift by an amount proportional
to IEZI2 and this in turn causes the measured CW phase to decrease.

length used for the 2-cell cavity was chosen to be 3.5 cm long, corresponding to Qext =
107 and Q6135 = 40%. The beam tube which would not be connected to a coupler
was chosen to be 6.0 cm long, corresponding to copper losses less than 3% of the
total. The measured waveguide coupling is not strong enough for heavily beam-loaded
cavities, such as an injection cavity, where Qext z 105 is needed. However, cavities
operating in energy recovery mode often have Qext z 107 to increase the stability of
the accelerating mode in the presence of microphonic fluctuations. The 2-cell cavity
with a tube length of 3.5 cm may therefore be useful for future investigations of the

feasibility of the waveguide coupler.

5.1.3 Bead Pulls

After completing the welding of the 2-cell cavities, a bead pull was done to measure
the field flatness of the 7r-mode. The method is based on Slater’s theorem [14]. The
cavity is excited with the network analyzer and a transmission measurement is used

to identify the resonant frequency of the 1r-mode. The phase of the transmitted wave

84

is then detected and displayed. The phase decreases by 71 when the RF frequency
is increased through the cavity resonance. This change is more rapid for higher Q
cavities and is approximately linear for frequencies close to the resonant frequency.
Phase measurements therefore have the advantage of increased sensitivity: a small
change in the resonant frequency will cause a large change in the detected phase.
Instead of continuously sweeping through a frequency range, the network analyzer
was then set to measure transmission at a fixed frequency, the resonant frequency
of the unperturbed cavity. A metallic bead travelling along the beam line will shift
the resonant frequency down, resulting in a decrease in the phase measured at the
original, unperturbed resonant frequency. Since, for small perturbations, changes in
phase are linearly related to changes in frequency, the measured relative changes in
phase are equal to the relative changes in the resonant frequency.

A 1 mm long section of a hypodermic needle was threaded onto a fishing line
and fed through the cavity along the axis. A small bead is desirable in this case
since the perturbation must be small and the cavity itself is rather small. The bead
removes a small region of the electric field energy density as it moves along the
axis. This perturbation will reduce the frequency of the cavity mode according to
Equation (2.88). The measured drop in frequency is therefore proportional to the
square of Ez at the location of the bead. Figure 5.9 shows the network analyzer trace
of the phase as a function of time. The entire time for the trace was approximately one
minute. By taking the square root of the measurement and calculating the deviation
of the peak axial fields from the average of the peaks, a field flatness within 3% was
calculated for both cavities. No tuning was done to either two—cell cavity since field

flatness within 5% is adequate for multicell cavities.

85

5.2 Testing

Figure 5.10 shows a diagram of the setup for vertical testing of the 2-cell cavity. The
bottom flange is connected to the short beam tube and has openings for a vacuum
line and the input coupler. The top flange is connected to the long beam tube and
has a single opening for the pickup coupler. The top and bottom flanges are both
made of stainless steel, and the power dissipation in these components would be high
during testing if nothing were done to shield the flanges from the cavity fields. A thin
niobium disk was therefore made to function as both a gasket for the knife edges of
the vacuum seals and to shield the inner surface of the stainless steel flange from the
cavity fields. The niobium disk was used on the short beam tube since this endcap
has higher fields than the long beam tube end cap. The Nb disk therefore had two
holes cut into it for the vacuum line and input antenna. A thin copper disk with a
center hole for the pickup antenna was used for the long beam tube flange at top.
Copper was used in an attempt to simulate the conditions of possible future tests
with a copper waveguide coupler. SUPERFISH simulations predicted a 20% drop in
Q0 due to the presence of the copper.

The RF input and pickup couplers used antennae with a fixed length for the
vertical testing. The lengths of the antennae needed for the desired external Q’s
were determined by benchtop measurements after the expected Q0 at high fields was
estimated. Recent tests at MSU of a single-cell TTF shape cavity with a TM010
frequency of 2.45 GHz resulted in a measured Q0 of 1.8 - 1010 at 1.5 K. Using the

TTF value for G, the corresponding surface resistance was calculated to be
Rs = — = —— = 15 mi). (5.3)

Since the 2—cell cavities were made from the same stock of niobium and operate at the

same frequency as the TTF single cell, 15 110 was used as the estimate of the 2—cell

86

 

Figure 5.10: The 2—cell cavity is positioned vertically for testing with two fixed-length
antennae. The input coupler and vacuum connection are located on the bottom and
the pickup probe is located on the top flange. NOTE: The antenna lengths in the
picture are not the ones actually used for testing.

cavity surface resistance. The estimated low-field Q0 for the 2-cell cavity is then,
Q0 = — = — =1.5-1010, (5.4)
including an additional 20% drop due to the copper disk on the top flange, we predict
Q0 = 1.2 . 1010 (5.5)
for the 2-cell at 1.5 K.

5.2.1 Antenna Measurements

The Q0 is expected to drop at high field, however it is not possible to accurately

predict by how much. A rough estimate for the high-field Q0 of one third of the

low-field Q0 was used to determine the coupling strength of the input and pickup

antennae. The goal was to pick an input Qext that is close to Q0 at high field. When
87

these Q’s are equal, the cavity impedance is matched to the characteristic impedance
of the input line and all of the power available from the amplifier goes to the cavity
without reflection. Since the the input antenna length is fixed, a perfect match will
not generally exist, but a Qext close to Q0 will still minimize reflections and wasted
power.

The Q0 of the 2-cell cavity at room temperature was measured with a network
analyzer to be 6620. The Qext of the input coupler was then measured by using it as a
pickup antenna to measure the transmission while the network analyzer was exciting
the cavity with a matched input. The desired Qext for the input coupler was 4 - 109,

corresponding to a coupling coefficient of

Q0 6620 —6
= = —— = 1.655 - 10 5.6
fl Qext 4 ' 109 ( )

 

at room temperature. The coupling coefficient, written in terms of the transmission

coefficient [321] = Biff“, is

P t |S21|2
s = 12$- : —-———— : (5.7)
P0 1- |S21|2

Pext is the power transmitted through the coupler; Pf is the forward power feeding
into the cavity from the network analyzer; and P0 is the power dissipated in the

copper cavity walls. The desired value of [521] is therefore,

2 _ 5 ~ = , —6
19211 _ —"1+5 [3 1.65510 (5.8)

= —57.81 dB. (59)

Figure 5.11 shows a plot of the transmission measurements for several antenna
lengths. As one would expect, the transmission coefficient and Qext increase exponen-
tially as the antenna length increases. An antenna length of 1.61 inches corresponding

to [821] = —57.81 dB was extrapolated from the measurements.
88

 

‘30 . I I I I
Input
« 57 .81dB *

 

 

I
.1;
O

1

A
D.-
w
v
| I
U1 A
O 0"
I I

I
C11
U1

I

-60 1 1 1 a
1.5 1.6 1.7 1.8 1.9 2 2.1

antenna length (inches)

 

 

)—

 

Figure 5.11: The length corresponding to Qext = 4 - 109 (star) is extrapolated from
the measured Qext’s for several different input antenna lengths (circles).

For the vertical test, the pickup antenna’s purpose is to sample the field level
in the cavity without significantly loading the Q. Having the pickup Qext >>> Q0
makes the calculation of Q0 from the measured values simpler. The coupling of the
pickup antenna was therefore chosen to be a factor of 100 weaker than the input
coupler, resulting in a desired Qext of 4 - 1011. The procedure for determining the
antenna length needed for this coupling strength is the same as for the input coupler,
where now |821| = —77.81 dB is desired. Figure 5.12 shows the measured values
used to interpolate the necessary antenna length. As with the input antenna, the
transmission coefficient increases exponentially with antenna length. Note that the
[5211 values for the input and pickup antenna do not lie on the same line because the
beam tube length and radial offsets are different. The Qext values for the fabricated
antennae were verified to be in good agreement with the predictions: for example,

Qext = 3.3 - 1011 for the pickup antenna and Qext = 2.7 - 109 for the input coupler.

89

—70

 

 

picku I
- 77.81d *
_75 -

...........................................

—80

S21

—90

I

I

I

 

—95

 

 

l l

l l l
l 1.1 1.2 1.3 1.4 1.5 1.6 1.7
antenna length (inches)

—100 '

 

Figure 5.12: The length corresponding to Qext = 4 - 1011(star) is interpolated from
the measured Qext’s for several different pickup antenna lengths (circles).

5.2.2 Cavity Preparation
Chemical Etching

The first of the two 2—cell cavities was prepared for testing by first undergoing a chem-
ical polish with buffered chemicals (BCP) which consists of a mixture of hydrofluoric,
nitric, and phosphoric acids with a 1:1:2 volume ratio. When this mixture of chemi-
cals is chilled below 18°C, the amount of hydrogen released during the etch is limited.
Limiting the diffusion of hydrogen into niobium inhibits the formation of hydride
precipitates on the niobium surface at temperatures between approximately 60 K to
150 K. If large amounts of hydride form on the surface, the residual resistance will
increase and the Q0 will decrease, a phenomenon known as Q-disease. Hydrofluoric
acid is very unforgiving, and potentially deadly, if handled improperly. Additionally,
the chemical etching produces corrosive fumes which must be adequately vented. For
these reasons, an isolated acid room was constructed, the interior of which is shown
in Figure 5.13. It has a sloped floor which would drain any chemicals into a catch

basin in the event of a large spill.

90

u

4
b
3

71f}: ‘

‘- A.

v
,.

i
E
ii
.1"

a;
5 i

 

Figure 5.13: Preparing to etch a cavity in the acid room.

Teflon flanges are attached to the cavity end flangas, see Figure 5.14, allowing the
cavity to be connected to a closed-loop sytem which continuously circulates the acid
between the cavity and a chilling tank. The temperature of the acid is measured at the
chilling tank to ensure it does not exceed 20°C. The closed loop is contained within a
large plexiglass glovebox which provides a protective barrier, contains any fumes, and
allows for safe handling of manually operated valves in the loop. Any fumes produced
are directed away from the acid room through a “scrubber”, which is a tall column
filled with neutralizing beads. The neutralized fumes at the top of the scrubber are
simply vented outside. The cavity was etched for one hour and 45 minutes with the
acid at 13°C, removing approximately 175-195 pm of material from the surface.

The acid loop was then valved off and valves connecting the cavity to a supply of
ultrapure water and a drainage tank in the acid room were opened. The cavity was

rinsed with ultrapure water for ten minutes using 55 gallons of water. This dilutes

91

 

Figure 5.14: The 2-cell cavity is ready for BCP.

92

 

Figure 5.15: The 2—cell cavity underwent high—pressure rinsing with ultrapure water
in a class 100 clean room for a minimum of one hour. The high pressure water jets
remove foreign particulates from the inner surface to prevent field emission at high
RF field.

any residual acid and removes acid residue in the cavity.

High-Pressure Water Rinsing

The rinsed cavity was then taken to the class 10000 clean room, where the teflon
flanges were removed in a stainless steel basin and the cavity was given a final, brief
rinse with ultrapure water. The cavity was then moved to the class 100 clean room
for high-pressure (1000 to 1100 psi) rinsing (HPR) with ultrapure water. Figure 5.15
shows the arrangement for HPR: the nozzle remains stationary and continuously
sprays water in various directions while the cavity moves to enable water to contact
every portion of the cavity interior. The cavity is held by an aluminum fixture which
moves up and down. A motor attached to the fixture rotates the stand which is bolted
to the cavity. This provides simultaneous relative rotation and vertical translation
between the cavity and nozzle. The cavity was rinsed in this way for 70 minutes and

allowed to dry overnight on a table in the class 100 area.

93

Assembly and Leak Checking

The cavity was assembled to the insert inside the class 100 clean room, see Figure 5.16,
and pumped down to approximately 1.4 . 10‘7torr. After placing the insert in the
small Dewar, pumping, and backfilling the Dewar with He gas, a leak was detected
with the residual gas analyzer (RGA). The insert was removed from the Dewar and a
leak check revealed a leak at the connection between the vacuum pipe and the elbow.
The insert was placed back in the class 10,000 clean room and the copper gasket at
this joint was replaced. A leak was then detected at the bottom cavity flange; the
vacuum-pipeto—elbow flange was observed to still have a leak also. Tightening the
flanges eliminated the cavity flange leak and reduced the vacuum-pipeto—elbow leak.
The insert was then placed back in the Dewar for testing.

The cavity vacuum was stable enough for testing at temperatures at or above 2.2
K, however testing below the A point of liquid helium was not possible. The absence
of viscosity in superfluid helium allowed helium to leak into the cavity, ruining the
vacuum. After the initial round of testing, the first cavity was removed from the
Dewar. Leak checking revealed that, in addition to the leak at the vacuum-pipeto—
elbow flanges, thermal cycling to cryogenic temperatures and back had opened new
leaks in both the top and bottom flanges of the cavity. Inspection of the knife edges
on the cavity flanges showed that the surface finish could be improved by polishing.
Inspection of the knife edge on the vacuum tube flange revealed a large nick. The
vacuum tube flange was cut off and a new one welded on. No leak was detected
at room temperature or at 90 K after polishing the cavity flanges. The cavity was
reetched, removing approximately 60 pm of material, followed by 75 minutes of HPR.
During the second vertical test, the cavity vacuum was stable down to and below the
A point. A second 2-cell cavity had approximately 190 pm of material removed with
BCP, followed by 75 minutes of HPR.

94

 

Figure 5.16: Assembly of the 2-cell cavity on the insert in the class 10,000 clean room.
The vacuum pumping line and RF feedthroughs are connected to the cavity flanges.

95

5.2.3 Vertical Tests

The two—cell cavities were tested in a vertical helium Dewar which is surrounded by
a liquid nitrogen jacket. Multiple baffles located below the top of the insert act as
thermal radiation shields. Germanium temperature sensors were placed on the top

flange and bottom flange during testing.

RF System

A superconducting cavity has a Q0 value on the order of 109 and the frequency
is on the order of 109 Hz; the resonance bandwidth is therefore on the order of
(109 Hz) / 109 = 1Hz. The fluctuation of the resonant frequency due to small cav-
ity shape changes from external sources of noise is larger than the mode bandwidth.
This means that the resonance curve will shift in frequency over time in response to
external vibrations. In order to maximize the cavity field, the RF generator must con-
tinually adjust its frequency to match that at the peak of the resonance curve. This is
accomplished by phase-locking the generator to the cavity. Figure 5.17 shows a block
diagram of the components used for the RF measurements during vertical testing.
The low-level RF is provided by a signal generator which provides two parameters
which the user may adjust during testing: the output power may change in sub-dB in-
crements and the gain of the feedback loop may be adjusted by changing the volts / Hz
ratio at the modulation input. Fine tuning of the RF power is useful for measuring
the cavity field level at which observed phenomena occur and increasing the loop gain
can enable the generator to remained locked to the cavity during processing events,
116. multipacting or field emission. The step attenuator is used to change the forward
power level by 1 dB or more. The phase shifter allows fine adjustment of the generator
frequency within the bandwidth of the cavity resonace for the purpose of maximizing
the cavity field. The function generator produces square pulses in which the frequency
(typically on the order of 1 Hz) and time between pulses are both adjustable. The

pulses switch the forward power on or off, allowing for the direct observation of the

96

cavity’s transient behavior; this information is ultimately used for determining both
the cavity Q0 and Qext- The transmitted power level may be viewed on the spectrum
analyzer while the forward and reverse power levels are displayed on an oscilloscope.
The numeric values of the forward, reverse, and transmitted powers are displayed by
three separate power meters. Separate calibration measurements are done to account
for the attenuation in the cables and the effect of the directional coupler and the
RF splitters; the calibration coefficients are used to convert from the measured power
levels to the actual forward power, reverse power, and transmitted power coupled into
and out of the cavity. The generator frequency is measured by a frequency counter.
Any electron current intercepted by the pickup antenna is detected with a bias tee
and measured with a picoammeter. A bias tee is a device which separates an incoming
signal into two branches: one branch contains the AC (or RF) component while the
other branch contains the DC component if it is present in the original signal. The
low-level RF output of the generator is amplified by approximately 52 dB with a 200
W solid state amplifier. The RF output of the amplifier is sent to the input coupler

of the cavity.

RF Measurement Technique

The cavity signal is mixed with the generator signal and the mixer output is fed into
the frequency modulation input of the generator. The mixer output is proportional
to the cosine of the phase angle between the two inputs. If the inputs have different
frequencies, then the mixer output varies sinusoidally with a frequency equal to the
frequency difference of the two inputs. The voltage level at the modulation input of
the generator determines the frequency of the generator, which will therefore also vary
sinusoidally. As the generator frequency approaches that of the cavity, the frequency
of the mixer output reduces until a steady DC voltage is eventually reached: at this

point, the generator and cavity have the same frequency and the generator is said to

97

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

mm WM ”‘9'
m
[RFOUT Men m \ m
are m union m R 3:: 3w /‘ m
W m m w W will! m
l and! ma
3% I” ]
.... m... 1 1
L0 500
”E w
as
m 319 am his)
m m RF no

 

 

an
1 oc
W W
m
m

 

 

 

 

 

 

Figure 5.17: The signal generator is phase-locked to the cavity by mixing the trans-
mitted signal from the cavity with the generator signal. The generator (L0) is locked
to the cavity (RF) when the frequencies of the two signals are equal. In this case, the
mixer output (IF) is a DC voltage, the value of which depends on the phase difference
between the cavity and generator.

98

be locked to the cavity. Once a lock is established, the cavity field may be maximized
by a final fine adjustment of the cavity frequency via the phase shifter. The cavity
phase changes by approximately 17 through a frequency range on the order of the
relatively small resonance bandwidth. Coarse changes in phase, by adjusting the phase
shifter, therefore result in fine changes in the generator frequency. The phase shifter
is adjusted until the generator frequency is set at the resonace peak, which occurs
when the cavity field is maximized. The maximum in the cavity field is observed as a
maximum in the transmitted power since the latter is proportional to the square of
the cavity field.

At this point with CW RF power applied, the forward power, reverse power, and
transmitted power are recorded as well as the cavity frequency, helium bath pressure,

and temperature. The coupling coefficient 61 for the input coupler can be calculated:

_\/-Ffi\/T’;
—\/Ff3=s/'PT

In Equation (5.10), the top sign is used when the cavity is overcoupled while the

61 (5.10)

bottom sign is used when the cavity is undercoupled. To determine if the cavity is
overcoupled or undercoupled, the forward power is pulsed using the function generator
and the traces of the forward power and reverse power as a function of time are
displayed with the oscilloscope. At the beginning of the pulse, the reverse power will
increase suddenly to a value proportional to the square of the magnitude of the wave
principally reflected from the cavity. The energy stored in the cavity increases to
a steady state value after which point the pulse ends. Immediately after the pulse
ends, the forward power is zero and the undriven cavity emits power through the
input coupler, causing the reverse power to rise suddenly. If the emitted power signal
immediately following the end of the pulse is greater than the signal of the principally
reflected wave, then the cavity is overcoupled. If the emitted signal is less than the

principally reflected signal, then the cavity is undercoupled. When the two signals

99

are equal, the cavity has unity coupling. The power dissipated in the cavity walls is
given by
Pd = Pf — Pr — Pt. (5.11)

The value of Pd calculated from Equation (5.11) can then be used to calculate the
coupling coeflicient 62 of the pickup antenna:

51;

fi2 = Pd. (5.12)

To extract the values of Q0 and Ease from these CW power measurements, the Qext
values for the input coupler and pickup antenna must be measured. This is done by
measuring QL, the loaded Q, while the function generator is modulating the forward
power. The forward power is zero after the pulse ends and the undriven cavity loses
its stored energy at an exponential rate due to three loss mechanisms: dissipating
power in the walls, emitting power through the input coupler which is measured as
reverse power, and emitting power through the pickup antenna which is measured as
transmitted power. The trace obtained with the spectrum analyzer of the exponential
decrease in transmitted power is used to measure the rate of decay of the cavity field,
from which the value of QL is calculated. The time t and signal power level a at two

points are recorded and used to calculate QL:

21rfAt

QL = - A1n(a) (5.13)

 

where f is the cavity frequency. The results of Equations (5.10), (5.12), and (5.13)

can then be used to determine Q0:

Q0 = QL(1+ 61+ 32) (5.14)

The stored energy may be calculated directly in terms of Pd and the measured fre-

100

quency:
Pd
= —, 5.15
27, f ( )
SUPERFISH simulation results are used to predict what electric and magnetic fields
exist in the cavity for a given stored energy. Finally, the external Q’s may be calcu-

lated:

Qextl = 0— (5.16)
1

Qext2 = %9 (5-17)
2

In principle, these values are dependent only on the cavity and coupler geometry,
which remains constant throughout the test. In practice, there are slight variations in
the calculated values of the external Q’s due to a sytematic error in the measurement
of the forward and reverse power: the dual directional coupler is not ideal in the sense
that the fraction of the forward and reverse signals sent to the power meters has a
small dependence on the magnitude and phase of the reverse travelling wave. If there
are large reflections because of a large mismatch with the cavity, then the difference
between the measured values of Pf and Pr can be on the order of the systematic error,
resulting in erroneous values of Pd calculated by Equation (5.11). To reduce the effect
of the systematic error on the calculated value of the Qext’s, the modulated measure-
ments are made at low cavity field. Additionally, the measurements are repeated as
the helium bath is cooled by lowering the bath pressure with a roots blower. The
calculated values of the Qext’s remain nearly constant at lower temperatures where
the input coupling coefficient is closer to unity. At higher temperatures, there is more
variation in the values of the Qext’s because the cavity Q0 is lower, producing a larger
mismatch.

After the modulated measurements are done, a series of CW measurements may

be made at various forward power levels and cavity temperatures. Two expressions

101

may then be used to calculate Q0 using the measured CW power levels and the Qext

values obtained from the modulated measurements:

 

_ Pt
Q0 — Qext2 (Pf _ Pr _ Pt) (5-18)
Q0 = Qextl (519)

Q P '
22.73257)”

The field level is determined from the stored energy which is given by

 

U 2 19t (929%?) . (5.20)

Equation (5.18) is more prone to systematic errors since it relies on calculating the
difference between Pf and Pr, while Equation (5.19) does not. Additional discussion of

this point and derivations of Equation (5.19) and (5.20) can be found in Appendix C.

First Cavity Test Results

Figure 5.18 shows the CW measurements of Q0 as a function of the accelerating
gradient, Eacc at 2.3 K. At 2 —1\%’-, the CW measurements were not as stable and a
DC current of —50 nA was detected by a bias-tee connected to the pickup antenna
with no accompanying x-rays. The DC current increased with higher cavity fields.
Past 4 1%V— the cavity began self-modulating with the DC current reaching —180nA
during the transient. At 6 7% it was possible for the transmitted power to be stable
for brief periods of time with no DC current. The DC current and the RF amplitude
fluctuations might have been due to soft multipacting barriers. Since the fields were
stable for brief periods, it was possible to reach higher fields by pulsing the forward
power. After a few minutes of pulsed power, a new stable CW measurement at higher
fields was possible. This technique was used to reach the final limiting field of 8.5 1%};
Above this limit, the cavity lost phase lock spontaneously.

A vacuum leak prevented testing of the first 2—cell cavity below 2.2 K. The next
102

 

1010

 

 

 

2.3K .
109 E. 0 II. 0 .0 0 Q 0"... I . .. .
: "as...
Q0 : ”'0‘-
108 .-
107 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9
acc m

Figure 5.18: Quality factor as a function of gradient for the first test of the first
2-cell cavity at 2.3 K. The maximum Eacc was limited: increasing Pf resulted in a
spontaneous loss of phase—lock to the cavity.

round of measurements were done after repairing the vacuum leak, as described in
the previous section. Figure 5.19 shows the CW measurements of Q0 at various tem-
peratures. At 1.83 K, the cavity fields were stable up to 6 kiln-X, where the transmitted
power would drop suddenly and spontaneously recover. The forward power was then
reduced and gradually increased, reproducing the same Q0 values up to 6 Ag. At
6.5 ¥nV—, the cavity pressure rose dramatically, so it was decided to try to reach higher
fields in the 0-mode. A limiting field was reached in the O-mode in which any at-
tempt to increase the forward power would cause the transmitted power to drop by
approximately 17 dB. The cavity vacuum was stable again, and upon returning to the
1r-mode it was possible to reach a limiting field of 7.4 %. Any attempt to increase
the forward power beyond this limit would cause the cavity to transition to a new,
stable state at 1 £13,— where the Q0 was reduced by about a factor of 20. A compari-
son of the peak magnetic surface fields reached in the 7r-mode and 0-mode before the
transition to the low-Q state is shown in Figure 5.20. As can be seen, a higher peak
surface magnetic field is reached in the vr-mode. The field and Q0 values for the two

modes are similar after the transition.

103

 

 

 

 

 

E * * ’5 r 7* .* 1k 4 1M
. ‘ ‘k
109 g . ' ' " °- .. °
: spontaneo’11§,....
Q0 : * fluctuations 0-mode
; conditioning
108 - I, + + + +
: "N 4K +
1.83K >5
2.3K -
107 1 1 1 1 114K 1*
0 1 2 3 4 M5V 6 7 8 9
12.5%?)

Figure 5.19: During the second round of testing, the first 2-cell cavity experienced a
transition at high field to a low-Q state. Reducing Pf in the low-Q state eventually
caused a sudden transition back to the high-Q state.

The next set of measurements at 2.3K (see Figure 5.19) had spontaneous fluc-
tuations in the transmitted power occurring at 5 @7131. At this point, the forward
power was switched off and, after it was restored, the transmitted power overshot
its previous CW value by approximately 1 dB. After about 1 second the transmitted
power returned to its previous, lower value, but oscillating with a variation of about
0.8 dB. The transition to the low-Q state occurred at 6.9 h—[Inym Increasing the forward
power enabled the low-Q state to reach a limiting field of 1.7 %V—, while reducing the
forward power resulted in improving the Q0 at the same field level. Reducing the
forward power further resulted in a sudden transition back to the high-Q state. A
final set of measurements was done at 1.4 K. No instabilities were observed during

the measurements at 1.4 K until the low-Q state transition above 7 MmV—.

Second Cavity Test Results

The second 2-cell cavity was tested in hope that the results would facilitate deter-
mining the source of the low-Q state. The results are shown in Figure 5.21. During

the vertical test at 2 K, self-modulation occurred at 6.6 Mm—V with a small amount of

104

 

 

 

 

1010 =
109 g
Q0
108 r
E .
. Pl-Mode -
Zero-Mode **
107 1 1 1 1 1 1
0 5 10 15 20 25 30 35 40

Bpk (mT)

Figure 5.20: The first 2-cell cavity experienced a transition to a low-Q state in both
the O-mode and 7r-mode, however the peak magnetic fields reached in these modes is
different.

x-rays (0.5-1.5 mR/ hour was measured inside the radiation shield). The cavity field

was reduced to 1.4 —Mm—V- and the forward power was then gradually increased, showing

DAV.
m

an improved Q0 value at 4 %. This variation in Q0 values at 4 was also seen in
the first cavity test (Figure 5.18), and suggests a possible soft multipacting barrier.
The transition to the low-Q state occurred at 8.7 LQIX. A limiting field of 2 Mfiv— was
reached in the low-Q state. Reducing the forward power caused the Q0 to improve
while the field level remained constant. As the forward power was further reduced,
a sudden transition back to the high-Q state occurred, as we observed for the first
cavity.

The field at which the transition occurred was not always reproducible. The fol-
lowing day, an attempt was made to measure more accurately the field level at which
the low-Q transition occurred. However, in this case the transition occurred at a lower
limit of 8.5 May. In a subsequant measurement at 1.4 K, the low-Q transition occurred
a15171%§1.

Measurements on the 0-mode were done at 1.5K. The comparison with the 77-

mode is shown in Figure 5.25. A transition to a low-Q state occurred for both the

105

 

1010 .

 

 

 

: 3 1° _ . *° .. ~ ° '11.
109 5’ * 11
C se -
Q0 I , modulating
108 :7
' 21< «
. 2K Limit 24hrs Later *
107 m 1 1 1 1 1 1'5K .
0 1 2 3 4 5 6 7 8 9

E... (Mn-‘4)
Figure 5.21: The second 2-cell cavity also experienced the sudden transition to a
low-Q state at high field.

0—mode and Tr-mode. The peak magnetic field reached in the 0-mode was smaller than

than for the 1r-mode, as was the case for the first cavity.

Surface Resistance

The low-field Q measurements which were done while pumping the Dewar down to
lower pressures and temperatures may be used to calculate the surface resistance of
the 2-cell cavities. The surface resistance Rs in terms of Q0 and the geometry factor
G is given by

G

.Hg===2§6. (5121)

A plot of Rs versus 1 / T, where T is the temperature, for each of the three cavity
tests is shown in Figure 5.22. At high temperatures, the surface resistance is seen
to follow the BCS form of Equation (1.4). The residual resistance after the low-Q
transition appears to increase for both cavities, with the second cavity showing the
most dramatic change of the two. The residual resistance for the second cavity is

estimated to be in the range of 50 Q to 60 Q.

106

 

 

 

 

 

10000 _ I , I
. first cav1ty, first test x

first cavity, second test 0 1

second cav1t I 4

)

1000 :7 j
g
I
Rs(n9) .
L.

100 1 4,

J~ -1

I )

I 1
I

low-Q transition points
I -1
10 M l l l
0 0.2 0.4 0. 0.8 1
1 /T (K—1

Figure 5.22: The low-field Q0 values measured from the three cavity tests are used
to calculate the surface resistance as a function of temperature. Fits to the data
are shown where the surface resistance is the sum of a BCS term and a residual
termst = A(27rf)2exp [—7cA—é%2] +1213 where A = 1.13, A(0) = 17.67, and f =
2467 MHz. The BCS contribution is the same for each fit but the residual resistance
has values of 50 HQ, 60 nil, and 110 mi]. The residual resistance of both cavities
appears to have increased after experiencing a low-Q transition.

107

Frequency Shifts

The helium bath temperature is lowered by reducing the pressure with a roots blower.
The pressure differential between the helium bath and the cavity vacuum squeezes
the cavity. This squeezing results in small displacements of the cavity surface near the
iris. Since the iris is a region of high electric field, the squeezing reduces the resonant
frequency of the cavity, by Slater’s theorem. The 7r-mode frequency was recorded for
each Q measurement while pumping down and the change in resonant frequency with
pressure may be seen in Figure 5.23. This data, combined with similar results for the
second cavity, gives the shift in resonant frequency with pressure to be approximately
—213 Hz / torr. This value would be useful when fabricating the 2—cell cavity for use
in an accelerator, as it is important to ensure that that design frequency can be
reached at the design temperature (and the corresponding bath pressure). A small
frequency shift with pressure is preferable, since fluctuations in the bath pressure can
produce fluctuations in the resonant frequency. The measured shift in frequency with
pressure is larger than the value of 10 Hz/mbar (13 Hz/torr) reported for a stiffened
1.3 GHz ,8 = 1 multi-cell [17]. It is consistent with the values measured for 805 MHz
6 = 0.47 single-cell and multi-cell cavities, which ranged from +0.46 kHz/torr to
-1.0 kHz / torr, depending on the number of cells, the presence or absence of stiffening
rings, and external constaints from a support frame or helium vessel [23].

Lorentz force detuning occurs when the radiation pressure of the cavity field is
sufficiently high to deflect the cavity walls enough to change the resonant frequency
by a measureable amount. The deflections are small, but the resonance bandwidth of
a high—Q superconducting cavity mode can also be very small. When the amount of
detuning is greater than the bandwidth, a measureable shift in frequency occurs which
is proportional to the square of the field level. Figure 5.24 shows the shift in frequency
that occured during testing of the second cavity at a constant bath pressure of 25.8
torr and a bath temperature of 1.99 K. From this data a Lorentz detuning coefficient

of —4.22 Hz / MV2 / m2 is obtained. The measured Lorentz detuning coefficient is larger
108

 

 

2467.38 553* I I I I first cefvity l><
2467.36 - 'x, linear fit ...... -

2467.34 ' - x _ —
2467.32 '

2467.3 .
2467.28 1 x
2467.26 r —
2467.24 ' 'X - . _ -
2467.22 A ' '

2467.2 ’ ' - -x

2467.18 1 l l l l l l l
0 100 200 300 400 500 600 700 800 900

pressure (torr)

1 I
X
1

I
X

frequency
(MHz)

I

T
L

 

 

 

Figure 5.23: The accelerating field of the first cavity is maintained between 0.70

% and 1.12 Mfilv— while cooling down. The measured frequencies of the 7r-mode

show the resonant frequency shift by —212 Hz / torr. The same measurements on the

second cavity, maintaining Eacc between 0.56 3% and 1.48 Lily—1 give a value of —214
Hz / torr.

than the value of (-)1 Hz/(MV/m)2 reported for a 1.3 GHz 6 = 1 multi-cell cavity [17].
It is close to the measured value of -3.5 Hz / (MV/m)2 measured for a B = 0.81 multi-
cell cavity [24]. Note that the latter cavity’s )6 value is similar to the 2-cell, but the
frequency is lower (805 MHz), stiffening rings were used, and a Ti support frame was

attached.

Discussion

The transition from a high-Q state to a low-Q state at high power levels and the tran-
sition from the low-Q state back to the high-Q state when the power is reduced might
be explained by a superconducting region of the cavity going normal-conducting at
high field. One plausible location for this to occur is the Nb disk on the input flange.
The thin outer edge is the only part of the disk in direct contact with liquid He. It is
possible that this is not sufficient cooling for the disk to remain superconducting at

high field. Prior to testing, simulations were done to predict the Q0 for the case in

109

 

 

 

 

2465.11265 1 I 1 , 1
second caVIty 0
Q“ « . , linear fit ------
2465.11260 — <> ~0_ -
2465.11255 _ O 7
frequency f _
(MHZ) 246511250) 0
2465.11245 — 0* _ -
0.
2465.11240— ' .. —
‘0
2465.11235 1 l l l l
0 10 20 30 40 50 60
2
E... (a)

Figure 5.24: The frequency of the 7r-mode for the second cavity is measured at different
field levels while maintaining a constant bath pressure of 25.8 torr. The shift in the

resonant frecguency due to radiation pressure of the cavity field is seen to be —4.22
Hz / (MV2 / m ).

which both the input and pickup flanges used Cu disks and for the case in which only
the pickup flange used a Cu disk while the input flange used a Nb disk. Table 5.1
compares the measured ratios of Qlow/thgh to the ratio of the Qo’s from the simula—
tions. The measured Q0 ratios are similar to that predicted when the cavity contains
two normal—conducting disks for the low-Q state and one normal-conducting disk for
the high-Q state. It is also worth noting that the magnetic field at the mechanical
joint of the Nb disk and the NbTi flange is 0.9 mT with a peak magnetic field in the
7r-mode of 40 mT. This is comparable to the 0.7 mT reached at the Nb—NbTi joint for
a quarter-wave cavity tested at MSU [25] and the 1 mT reached in subsequent testing

of the same quarter-wave cavity.

110

Table 5.1: The measured Q0 drop is comparable to that predicted from simulations
for the cases of 2 Qu disks and 1 fig disk

 

 

 

 

 

 

 

 

 

 

 

Cavity Temp- (K) Qlow/thgh (‘70)
13t cavity 7r-mode 1.83 6.59
1St cavity 1r-mode 2.3 9.52
lst cavity 7r-mode 1.4 6.47
2nd cavity 1r-mode 2.0 4.38
2nd cavity O-mode 1.5 6.30
Simulation 1.5 6.04
1010 E
E
P O 0 O . $ . o ‘ o o o . c
109 r
C 0
Q0 .
108 :—
E
: Pi—Mode é
Zero-Mode 1*
107 1 I l 1 1 1
0 5 10 15 20 25 30 35 40
Bpk (mT)

Figure 5.25: The second 2-cell cavity experienced a transition to a low-Q state in both
the O-mode and 7r-mode, however the peak magnetic fields reached in these modes is
different.

111

Chapter 6

Single-Cell HOM Cavity

6.1 Concept

It was seen in Section 1.2.1 that the variation in the field magnitude of individ—
ual cells in a multicell elliptical cavity could be understood by modelling each cell
as a resonator coupled only to its nearest neighbors. The field magnitude varies si-
nusoidally from one cell to the next, with different modes having a different phase
advance from cell to cell (see Figure 1.5). In this model, the lowest frequency mode
has a phase advance of f7 where N is the total number of cells in the cavity. This
mode has the disadvantage of having the lowest field magnitudes in the end cells.
The low fields reduce the coupling of the mode to the beam pipe and to any HOM
pickup antennae. This problem worsens for higher cell numbers, further reducing the
end cell field strength. Another challenge which develops at higher cell numbers is
more tuning being required, since the sensitivity of the 7r-mode field pattern to shape
imperfections will increase with the square of the number of cells. An attempt to
remedy this situation by approximately reproducing the accelerating field pattern of
a muticell 7r-mode using the TM012 mode of a single-cell cavity is illustrated with the
cartoon in Figure 6.1. The simplicity of the single-cell design also has the advantage

of reducing the number of electron beam welds or possibly eliminating the welds by

112

 

i- w m V x.” I - ' “x . , "“4. fl ’-

. fl

‘ Q—_ .“ Iva/”Hg ,r’~"‘~ ‘,H-_‘\
\ 1‘ I I j W
1 . 1

Figure 6.1: A single-cell cavity operating in a higher order TM012 mode produces an
accelerating field pattern on axis that superficially resembles that of a 3—cell multicell
7r-mode.

hydroforming. The simple shape may also be better suited for depositing thin films
of niobium on copper.

The shape of the cavity is essentially that of a large cylindrical waveguide which
is shorted at both ends. Many of the characteristics of the HOM cavity may be
understood by examining the field patterns and figures of merit for a “pillbox” cavity,
which may be solved analytically. However, an important aspect of adding beam pipes
to the pillbox deserves mentioning. For an ideal pillbox with no beam pipes using
TM01p as the accelerating mode, the length of the pillbox should be close to that of
p cells of a multicell cavity that has a 1r-mode at the same frequency. The sinusoidal
variation of the electric field along the axis would then be approximately equal for the

two cavities. Adding beam pipes to the HOM cavity causes two significant changes:
1. The electric field penetrates evanescently into the beam pipe.
2. The peak surface electric field will be located along the edge of the opening.

The iris must be rounded to minimize the peak surface electric field, with a larger
radius of curvature required for a larger beam pipe radius. The length of the cavity
therefore increases by an amount that is twice this radius of curvature. The final
length and field pattern for an HOM cavity with beam pipes operating in the TM01p
mode is then closer to that of p + 1 cells of a multicell cavity that has a 7r-mode at

the same frequency.

113

6.1. 1 Pillbox Trends

The procedure for designing an HOM cavity shape begins with an initial pillbox shape.
The pillbox shape is completely specified in terms of g, the ratio of the length to the
radius. The corners are then rounded to reduce the peak magnetic field. A beam pipe
is added and the end shape is optimized by adjusting the parameters of two ellipses
connected by a straight section as in elliptical cavity design. Two questions must be

initially addressed:
0 Which HOM to use, z'.e. what is the value of p in TM01p?

o What should the initial shape of the pillbox be once the value of p is determined,

i.e. what should the initial value of El,- be?

Fields

To answer these questions, the field patterns and figures of merit for the TM01p mode
of a pillbox will be solved for as a function of the axial mode number p and the aspect
ratio g. In Section A.2.4, the case of a TM01 waveguide mode travelling in a guide
of circular cross-section was considered. The fields for a TM01 wave travelling in the

positive 2 direction with axial wave number 6 was found to be

E+ = EngoUcc/Jk—jflznz - €9ng (kcp)e_jflznp (6.1a)
c
H+ = —l2;—;Jk—:0J1(kcp)e—jfizn¢. (6.1b)

Cylindrical coordinates ¢ (azimuthal), p (radial), and z (radial) are used. E0 is the
magnitude of the electric field; J0 and J1 are Bessel functions; 17 is the characteristic
impedance of free space; kc is the critical wavenumber; and k0 is the wavenumber for
a plane wave in free space. A standing wave may be established by superimposing

a wave with the same amplitude but travelling in the opposite direction with axial

114

wave number —fi,

E_ = E22J0(kcp)€j’6znz + %’0'£EJ1(kcp)€jfiznp (6.2a)
C
— _. £21182 '32

The sum of the two travelling waves is

 

 

E+ + E‘ = E0J0(kcp)cos(fiz)nz — EZ)6J1(ka)sin(fiz)np, (6.2c)
c
H+ + H‘ —] E0 E)-J1(kcp) cos(fiz)n¢. (6.2d)
7? kc

This standing wave pattern will remain unaltered if a pillbox of length l is formed by
placing shorting plates at locations where the transverse electric field is zero. Every-
thing exterior to the pillbox may now be removed and the field pattern remaining in

the pillbox is that of the TM01p mode with p = £762.

Voltage Gain

The voltage gain of a charge travelling at the speed of light along the axis of the

pillbox is found by integrating the axial electric field along the axis:

 

 

 

 

V = fol E0 cos (ID—TE) eJ sz dz , (6.3)
V _ .1522 j: [ J(ko+2;-T-)z+ J(ko—2F)z] dz , (6.4)
V — 701 sinc [5 (k0 + 1%)] + (_1)p sinc [— (k0 — 213)] l (6 5)

 

 

k3 = (2.405)2 + (p771)? (6.6)

115

k0 can be eliminated to obtain an expression for the voltage gain in terms of p, l, and

a:

 

E01
V:—
2

((3%)2(é)2+(”-§)2+% +
44—64... «$315)? + ($2 _ I;

where sinc(:c) = m.

I

 

 

 

Transit Time Factor

The transit time factor T may be calculated by dividing Equation (6.7) by f6 IE 2 (z) |dz:

 

T:Z

¢(%%°§)2(2)2+s>2+5 +
+(—1>v.... Kai)? (2)2 + (122:)? _ a;

It is a function of the two independent variables p and 215' The transit time factor for

 

 

(6.8)

 

a multicell cavity is maximized by setting the midcell length equal to the distance a
charge travels in half an RF period. This ensures that the charge experiences forces
applied only in the direction of motion. For a charge travelling at the speed of light,
this condition may be written as

g = ll or k0 = B, (6.9)

cell

Equation (6.9) can never be satisified by the TM01p mode of a single-cell cavity

because of the cutoff frequency associated with establishing a standing wave in the

116

 

0.9 1 1 T 1 1 I
0.8
0.7
0.6
0.5
0.4
0.3 F -
0.2
0.1

O l l l l
0 0.5 1 1.5 2 2.5 3 3.5

fl

Figure 6.2: The transit time factor T for the TM012 mode of an ideal pillbox with a
length to radius ratio of 5 is plotted as a function of particle speed. The maximum T
occurs at the physically unattainable % = 1.92.

l
l

l
l

l
l

T
l

l
l

I
l

I
J

 

 

L.

 

radial direction,

 

(%)2=(2'i”5)2+(%’5)2 . .1.

Since lcell should be set equal to %, Equation (6.9) is incompatible with Equa-
tion (6.10). Agreement with Equation (6.9) improves as the pillbox radius (1 increases.
In the limit a —> 00, the fields in the pillbox become those of free space and Equa-
tion (6.9) is exactly satisfied. Some of the challenges in designing an HOM cavity may
be understood from Figure 6.2, which shows T for the TM012 mode as a function of
the particle velocity %.

As anticipated from Equation (6.10), the maximum energy gain occurs for a parti-
cle velocity greater than the speed of light. Additionally, for % z 1, T is approximately
0.14 ,which is significantly lower than what may be achieved (T m 0.7) for multicell
elliptical cavities. A low T implies a low voltage gain and figures of merit such as
5, 125%, andEEg; will therefore degrade as a result. For this reason, it is desirable

to choose a pillbox shape which has a local maximum in T near the design particle
117

 

 

 

 

 

0 1 2 3 4 5 6 7 8 9 10
length / radius

Figure 6.3: The transit time factor T for the TM011 and TM012 modes of an ideal
pillbox is plotted as a function of the ratio of length to radius. T is calculated with a
constant particle speed of % = 1.

speed. Figure 6.3 shows that the shape which maximizes T generally depends on the
chosen value of p.
. . . . B R .
It 13 therefore useful to have explic1t expressrons for both 35:: and Q 1n terms of

p and g: the goal is then to determine which values of p and g produce a sufficiently

. R B
high Q and low Bi)?
Figures of Merit

The peak magnetic field may be found from Equation (6.2d):

 

__ E01...

Cc

Bp J1(1.84). (6.11)

E13?— as a function of g and p is then

8C0

Bp _ E011...

E... chc J1(1.84). (6.12)

118

The energy U stored in the mode may be calculated by evaluating the average energy

Uh stored in the magnetic field,

U=2%=%/%m

2 27rl
= %(—EnOk—l?) [0 /O‘/0aJi?(kcp)cos2(fiz)pd¢dzdp. (6.13)

This integral may be evaluated using the following property of Bessel functions:

 

a 2 a2 2
/0 PJn(kP)dP = ‘2" [Jn(ka) _ Jn—1(ka)Jn+1(ka)] a (6-14)
resulting in a final expression
71' 2 Eoko 2 2
c

Equations (6.7) and (6.15) may be used to form an expression for g as a function of

pand g,

R 13 (5)2

Q wU mk0,€0(%rcz)2.112(2.405)
.(—.)p..... flags)? (92 + (a)? — %

Shunt Impedance

 

ll

 

2 2
, 2.405 I prr 2 p7r
5‘“ ((7) (a) +(7) +2 +

2

 

 

(6.16)

 

The 5 of the TM01p mode is generally lower than that of p + 1 cells of an elliptical
multicell cavity, despite having similar electric field patterns. The reduction of the
TM01p 5 is due primarily to transit time effects. It may be expected then that
TM01p modes with low p values and correspondingly shorter lengths would possess
larger gs. This trend can be seen in Figure 6.4. Low values of p are clearly preferable

since a large g is desirable.

119

220 1 1 1 I
200 r 0 0.475 a
180 " ‘
160 r' "

R 140 f . 0.350 1
Q 120 L ~

(9) 100 _ p.300 _

0.250
80 ' ° 0. 225

60 — .0 225 0 200
40 -

20 l l l l
0 2 4 6 8 10

TM01 axial mode index, p

 

0 200 0.175“
. _

 

 

 

Figure 6. 4: TheR Q is seen to decrease with the mode index p. The values of“ T which

maximizes Q are used and printed next to the data point.

Geometry Factor

Also, Equation (6.15) may be combined with an explicit expression for f |H|2da to

determine the geometry factor G:

 

 

 

2f lHl da
Using Equation (6.2d),
2 27f l EOJ1(2.405)k0 pvrz 2
/|H| da = [0 A; [ cos(—l—)] ad¢dz
27f E 77k
+2 / j [ 0 —-°-:]( J12 (kcp1pd4dp (6.18)
2
: ”la [E0J1(2.405)k0:|2 +27m2 [EOJ1(2.405)k0] . (6.19)
"kc ”kc

120

 

1200 I I I I I I

I
O
1

1000

800

I
O
l

G’ (52) 600

I
O
l

l

O -
400 O

200 00 _

 

 

0 l l l l l l
0 2 4 6 8 10 12 14

TM01 axial mode index, p

 

Figure 6.5: The values of G for an ideal pillbox with a length to radius ratio of 5
increase as the axial mode number increases.

Substituting Equations (6.19), (6.15),and (6.10) into Equation (6.17) results in

51¢ [2.405 (.3)

2+

 

2 + (m)2

E

G:

 

(6.20)

SDI“-

Thus the geometry factor depends on the mode index p and the length to radius

ratio. Values of G as a function of p are shown in Figure 6.5 for a fixed l/a ratio.

Peak Magnetic Field

The ultimate theoretical limit to the gradient of a superconducting cavity is deter-
mined by the value of Bane. For this reason, it is useful to examine how this figure
of merit depends on the choice of the mode index and initial pillbox shape. Equar
tion (6.12) determines the value of £5; for a given mode in a pillbox as a function of
the aspect ratio g. The equation is evaluated at discrete values of é separated by a
numerical difference of 0.025. The cell shape which minimizes E1355 is then recorded
for each value of the mode index p. The results for the first nine HOMs are shown

in Figure 6.6. Low values of p are clearly necessary if the HOM cavity is to have a
121

 

 

 

 

13 I I I I
12 - . 0.2011
11 _ . 0.225 _
0.225
B 10 ~ “0250 -
E... 9 r 0.275 -
T _ ‘0.300 -
(iv): .
m 7 — 0.350 _
6 ~ '0425 -
5 — . _.
4 0.575. . . .
0 2 4 6 8 10

TM01 axial mode index, p

Figure 6.6: The lowest value of 12% for a TM01p mode is seen to increase with the

mode index p. The value of -‘f which minimizes BEEC— is printed next to the data point.

reasonable value of E15312? When comparing the single-cell HOM cavity to a multicell
elliptical cavity, the value of p corresponds roughly to the number of cells in a compa-
rable multicell cavity. It should be noted that while £2ch generally increases with p
for the HOM cavity, a similar trend of increasing E25; with larger cell numbers does
not occur for multicell cavities. The reason is that frequency f of the accelerating
mode of a multicell cavity is determined by adjusting the cell radius and the axial
wave number ,8 is determined by the cell length. Independent control of these two
parameters enables multicell cavities to be designed such that an accelerated charge

only experiences forces in the direction of its motion:

= l__ (6.21)

1.) 7r
v cell
where v is the particle velocity. As described earlier, independent control over the

values of the HOM cavity frequency and fl is not possible since Equation (6.10) must

be satisfied. Equation (6.21) can never be satisfied for the HOM cavity, so the particle

122

will experience forces in the direction opposite to its motion; the voltage gain, and

therefore Ea“, will be lower; and both EBi-L and Egg- will increase. This effect is worse
ECG BCC

for higher p since the particle will become less synchronised with the accelerating field

as it travels further through the HOM cavity.

Parasitic Modes

The cartoon in Figure 3.2 illustrates the general situation for multicell cavities in
which entire HOM passbands may be below the cutoff frequency for the beam pipe.
These parasitic modes would have high Q’s if their stored energies were not properly
removed with a coupler. This situation would also occur when forming a single-cell
HOM cavity with the exception that there are no longer passbands. However, using
TM01p (p > 0) as the accelerating mode guarantees the existence of lower order modes
(LOM) which will also be parasitic. The number of LOMs below TM01p in frequency
increases for larger values of p. Figure 6.7 shows that the number of parasitic modes

increases linearly with the value of the mode index. For a given value of p, the cavity

 

 

 

 

T j l l
o0.200
200 ” . 0.225 ‘
. 0.225 %
““13fo 150 f . 0.250
parasitic . 0.275
modes 100 F 0 0.300 _
. 0.350
50 ‘ . 0.425 ‘
. 0.575
0 J l l l
0 2 4 6 8 10

TM01 axial mode index, p

Figure 6.7: The number of parasitic modes is seen to increase with the mode index p.

The values of % which minimize 7gp— are used and printed next to the data point.

80C

123

shape which minimizes 72% was used and the size of the pillbox was determined by
setting the frequency of the TM01p mode to 703 MHz. The cutoff frequencies for the
waveguide modes were determined by choosing the radius of a hypothetical beam pipe
to be 12 cm. A pillbox Tanp (TEmnp) mode is considered parasitic if its frequency
is below the cutoff frequency of the TMml (TEml) beam pipe mode. Since there are
an infinite number of field patterns to count, only pillbox modes with m S 2 and
n S 2 were considered.

The parasitic modes must be sufficiently damped to avoid beam instabilities and
excessive cryogenic heat loads. The challenge of damping a large number of parasitic
modes increases considerably if a large value of p is chosen for the accelerating mode

of the cavity. Low values of p are therefore desirable.

6.2 Cavity Design

The analytical results for an ideal pillbox indicate that a single-cell HOM cavity
will have higher peak magnetic fields, a lower 5, and more parasitic modes than a
multicell elliptical cavity. A multicell cavity designed for a high-current linac cannot
therefore be simply replaced with a single-cell HOM cavity. The high peak magnetic
field of the single cell would reduce Eacc too much to compete.

However, the single-cell HOM cavity may be suitable for use where high gradients
are not necessary, such as in a high-current storage ring. For such an application,
the cavity would only need to provide a voltage gain on the order of 1 MV. Since
the length of the HOM cavity would be longer than a TM010 single-cell cavity, the
required accelerating gradient would be reduced. So, while the ratios Egg: and Egg;
are high, the peak fields in the HOM cavity can be small. This can be facilitated
by choosing a relatively low value of p for the accelerating mode, which has the
additional advantage of limiting the number of parasitic modes to a reasonable level.

The 5 is low, however this does not necessarily imply that the power dissipated by

124

 

Figure 6.8: The relative magnitudes of the magnetic field for the TM010 and TM013
are shown, as predicted by Analyst, with red corresponding to the largest values and
blue corresponding to the lowest.

the accelerating mode will be excessive. This is due to the fact that the geometry
factor for a TM01p mode is typically higher than for a multicell cavity. The dynamic

power dissipation for the accelerating mode is

 

 

It is the product (8) G which therefore determines the dynamic losses. The higher
G values for a TM01p mode may be understood by comparing the magnetic field
pattern to that of the TM010 mode in a multicell cavity, as shown in Figure 6.8.
The high magnetic field region of a multicell cavity is distributed along a significant
fraction of the surface, whereas the high magnetic field regions of the TM013 mode
are more in the volume of the cavity, with less contact with the surface. Therefore
the ratio of stored energy to dissipated power is higher for the TM013 mode. This
property of the TM01p modes also explains why G for an ideal pillbox increases with

p, as can be seen in Figure 6.5.

125

Table 6.1: The best values of 752% and g for TM012 occur at different pillbox cavity
shapes, measured with the ratio of radius over length %.

1 7.53% (5%) 8(9)

 

 

 

m
B
Min. FEE 0.425 6.21 114.0
Max. , 0.350 6.68 131.8

 

 

 

 

 

 

 

The TM012 mode was chosen as the accelerating mode of the HOM cavity. The
iris radius is set equal to that of BNL’s “Ampere Class” cavity which is designed for
2100 mA of electron beam current. The “Ampere Class” iris radius is 8.5 cm with

an accelerating mode frequency of 703 MHz.

6.2.1 Rounding the Corners

The next step is to choose an initial pillbox shape for the TM012 cavity, but it is not
entirely apparent which value of ff to start with. The pillbox shape which mlm' 'mizes
E-P—icc for the TM012 mode has % = 0.425 (Figure 6.6), while the pillbox shape which
maximizes g has % = 0.350 (Figure 6.4). The values of these figures of merit are listed
in Table 6.1. A further complication is that both figures of merit will change as the
. . . a _ . . . B
corners of the pillbox are rounded. The pillbox shape w1th T — 0.425 nnmmizes Efi'
The corners of this pillbox were rounded by adding circular arcs, and the radius of
curvature was varied. The values of 51,5, and G for each case are listed in Table 6.2.
B 800

The reduction in Elam which occurs as a function of the radius of curvature may
be seen in Figure 6.9. It is seen that the pillbox shape with 5f = 0.425 and a radius
of curvature at the corners of 10.5 cm has a E-P—B which is lower and a g which is

80C

higher than any case listed in Table 6.1. Figure 6.10 shows the increase in (g) G as
a function of radius of curvature.

The pillbox shape with 5f = 0.350 maximizes g. The corners of this pillbox were

rounded by adding circular arcs, and the radius of curvature was varied. The values

126

Table 6.2: Rounding the corners of a pillbox(% = 0.425) decreases the minimum

, increases the maximum R, and increases G for the TM012 mode. The values

artifcfrom SUPERFISH simulations. It is worth noting that the results for zero radius
of curvature are consistent with the analytically-calculated values in Table 6.1.

6.05

Figure 6.9: The minimum value of Eli“: for the TM012 mode of a pillbox (% = 0.425)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

radius

of B T R
curvature(cm) Egg: (E12) Q62) 0(9)
0 6.22In 113.6 365.8
1.0 6.21 113.9 372.1
2.0 6.18 114.8 378.9
3.0 6.17 116.2 385.9
4.0 6.13 118.1 392.8
5.0 6.12 120.3 399.7
6.0 6.10 122.8 406.2
7.0 6.08 125.5 412.6
8.0 6.07 128.3 418.7
9.0 6.06 130.9 424.8
10.0 6.051 133.4 430.7
10.5 6.049 134.5 433.6
11.0 6.05 135.5 436.5
12.0 6.06 137.1 442.2
14.0 6.14 138.0 453.7
16.0 6.31 135.2 465.2

 

 

 

 

 

l l

 

 

l 1

 

6 8
radius of curvature (cm)

I
10

is seen to occur at a radius of curvature of 10.5 cm.

127

12 14 16

 

65000 I I I I I I I

60000
R 55000
Q G

(92).)...

45000

 

C

40000 I l l l l l l
0 2 4 6 8 10 12 14 16

radius of curvature (cm)

 

 

 

Figure 6.10: (g) G for the TM012 mode increases with larger radii of curvature at
the pillbox (% = 0.425) corners.

B R . .
of HEE’Q’ and G for each case are llsted 1n Table 6.3.
Trends similar to the % = 0.425 case are seen in Figures 6.11 and 6.12, however
12% does not improve substantially and is still higher than the 6.049% achieved
75"

with % = 0.425. For this reason, the TM012 cavity was chosen to have an initial

pillbox shape with % = 0.425.

6.2.2 Adding the Beam Tubes

The ends of the TM012 cavity were next modelled as two ellipses connected by a
straight section. The geometrical parameters were varied and the cavity figures of
merit were optimized using the same SUPERFISH interface program described in
Section 3.2. A plot of the cavity shape with the electric field lines of the TM012 mode
is shown in Figure 6.13. Adding beam pipes reduces the accelerating gradient and
results in a £5: which is larger than is shown in Table 6.2 for a radius of curvature
of 10.5 cm. It was possible to limit the increase in 61% by increasing the length of
the vertical semi-axis to approximately 19 cm while keeping the horizontal semi-axis

128

 

Table 6. 3: Rounding the corners of the pillbox (% = 0.350) decreases the minimum

E—L, increases the maximum Q’ and increases G for the TM012 mode. The values

are from SUPERFISH simulations. It IS worth noting that the results for zero radius
of curvature are consistent with the analytically calculated values 1n Table 6.1.

 

 

 

 

 

 

 

 

radius
of B T R

curvature (cm) Ear-E: (151117) 0(9) G (9)
0 6.68 131.7 361.0

1.0 6.68 131.8 367.1

2.0 6.67 132.3 373.7

4.0 6.66 133.9 387.1

8.0 6.68 135.4 413.4

9.0 6.69 134.5 420.0

10.0 6.74 132.7 426.8

 

 

 

 

 

 

 

 

 

 

 

6.65 l l l 1
0

radius of curvature (cm)

Figure 6.11: The minimum value of E856: for a pillbox with a = 0.350 is still much
higher than the 6.049 {61% achieved with 4 = 0.425.

H1

129

 

 

57000
56000
55000
54000

Rasamo

(92)52000
51000
50000
49000
48000c

47000 1 J ‘ '
0 2 4 6 8 10

radius of curvature (cm)

 

 

 

 

Figure 6.12: (g) G for the TM012 mode increases with larger radii of curvature at
the pillbox (% = 0.350) corners.

length at approximately 10 cm, as shown in Table 6.4. The large Egg: and E158;
values seen in Table 6.4 prevent the TM012 cavity from achieving high accelerating
gradients. It therefore would not be suitable for linacs. However, in applications where
electron bunches are repeatedly fed back to the same small number of cavities, such as
in an electron storage ring, a modest accelerating gradient may be acceptable. Also,
the large iris and correspondingly low k“ would enable the use of large beam currents.

The magnitude of the axial component of the electric field Ez is plotted in Fig-
ure 6.14. There are 2 zeroes of E2 along the axis, therefore the mode is TM012.
However, adding beam pipes allows the electric field to leak out and of the cavity,
where it decreases exponentially. As a result, the variation of Ez more closely resem-
bles that of 3 cells in a 7r-mode structure. The total length of the TM012 cavity from
iris to iris is 61.6 cm, which is close to the length of 3 cells in a 7r-mode structure
with 376“ = 64cm. Multicell elliptical cavities are normally tuned to have a flat 7r—

mode: the field magnitude on axis is the same for each cell to within a few percent.

In comparison, the TM012 cavity has very unflat fields. This field unflatness is a

130

 

Table 6.4: The geometrical parameters and figures of merit for the TM012 cavity are
listed, corresponding to a frequency of 703 MHz. The parameters describe the end
cell shape as illustrated in Figure 1.2. Note that the definition of l in the table is
the axial distance from the beginning of the equatorial ellipse to the end of the iris
ellipse. The distance from the middle of the cavity to the beginning of the equatorial

ellipse is [straight'

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A cm 9.956

B cm 18.916

h cm 14.098 1 Q 55_()

(1 cm 2.947

b cm 2.947 hi"L 3-06

2 cm 11.464 _§cc mT 8.40
acc m

[straight CID 13.392 [[1

1 cm 17.415 G 2 476

d cm 2.654 k” SC 0.49

0 degrees 6.0 T 0.63

Tiris cm 8.517

Requator cm 33.014

 

 

   

 

Figure 6.13: The electric field lines for the TM012 mode as predicted from SUPER-
FISH.

131

 

manifestation of the fact that the frequency and cell length of a HOM cavity cannot
be varied independently as is possible with multicell ellitical cavities. The distance
separating the two zeroes of Ez is greater than £3- = 21.3 cm and a particle would
experience a force opposing its motion, reducing the net voltage gain. For larger p,
this would be true between all zeroes in Ez in the interior of the cavity. To minimize
the reduction of the overall voltage gain by these detuned peaks in Ez, the magnitude
of the peaks would be smaller in a final optimised shape of a general TM01p mode.
Adding beam pipes to the HOM cavity causes Ez to fringe out, resulting in peaks
whose length is greater than those in the interior. The ratios 7%: and 533% depend
strongly on the shape of the ends of the TM012 cavity since, as was noted before,
the peak fields occur there. It is understandable then that a shape which producas
low 13%,; and high 8 is one in which the field intensity is highest in the ends: this is
where the proper shape can simultaneously reduce the peak fields and vary the length
of the Ez peaks. Figure 6.15 shows the magnitude of the electric and magnetic field
that are present on the inner surface of the cavity. The initial point is located at the
middle of the cavity on the equator. The path then moves along the inner surface of

the cavity and toward the beam pipe.

6.2.3 Loss Factor

The total longitudinal loss factor was numerically calculated with the program ABCI [26].
In the ABCI simulation, a charge bunch with a Gaussian profile is sent through the
cavity on axis, maintaining a constant velocity c. The value of a in the Gaussian
profile is 1 cm and the total length of the bunch is 5 a. Figure 6.16 shows a picture
of the cavity, the charge bunch, and the electric field lines of the wake at six different
times. Maxwell’s equations are numerically solved and the sum of the forces acting

back on the charge bunch gives the loss factor.

132

 

 

25 T l l l I l l

l
l

20

T
l

IEzl 15

(Mmllm

I
l

 

 

 

O l L l l l
—80 ~60 —40 —20 O 20 40 60 80

distance along axis (cm)

Figure 6.14: A SUPERFISH calculation of the axial electric field magnitude along the

beam axis, normalized for an accelerating gradient of 10%.

 

 

 

 

l L

0 10 20 30 40 5O 60 70 80 90 100
distance along surface (cm)

 

Figure 6.15: The magnitudes of the electric and magnetic fields are plotted as a
function of the distance travelled along the surface. The origin is located at the
middle of the cavity on the equator, and the path moves along the surface and toward
the beam pipe. The values are calculated from SUPERFISH and normalized to an

accelerating gradient of 10%.

133

 

Figure 6.16: A charge bunch, which has a Gaussian distribution with a = 1 cm,
enters the TM012 cavity from the left in the upper left frame and proceeds through
the cavity, exiting in the lower right frame. ABCI was used to numerically calculate
the wake and the associated longitudinal loss factor.

6.2.4 Comparison with SOLEIL Cavity

An example to illustrate the conditions under which the TM012 cavity may be useful
is provided by the 353 MHz single-cell niobium-coated copper cavity at SOLEIL [27]
which was designed for an average beam current of 500 mA. The TM012 cavity was
rescaled to operate at 353 MHz and ABCI was then used to calculate the loss factor
for the case of a 0 =5 mm bunch. The figures of merit for both cavities, as well as
peak field levels, assuming both cavities provide the same voltage gain of 2 MV, are
shown in Table 6.5. A drawing comparing the two cavity shapes with the correct
relative scale is shown in Figure 6.17. The TM012 cavity is huge in comparison. The
larger size of the TM012 cavity does not however indicate larger dynamic losses. The
geometry factor of the SOLEIL cavity must be 291 Q for the product (g) G to be
equal to that of the TM012 cavity. Since elliptical cells typically have values for G
between 225 Q and 280 (2, less power is expected to be dissipated in the TM012 cavity
accelerating mode. Using the TM012 mode as the accelerating mode allows the size
of the cavity to increase without increasing, or actually reducing, the dynamic losses

of the accelerating mode. Also, the peak field levels in the TM012 cavity are lower

134

 

 

 

Figure 6.17: The scale of the TM012 cavity drawing on the left is such that the TM012
mode would have the same frequency (353 MHz) as the TM010 mode of the SOLEIL
cavity shown on the right.

despite have large values of Egg— and Egg" Because of the additional length of the

C C

cavity, the required accelerating gradient is lower and results in lower peak fields.

Table 6.5: The figures of merit for the TM012 cavity and the SOLEIL cavity are
compared for the case that the accelerating modes of both cavities have the same
frequency (353 MHz) and both are normalized to provide the same voltage gain.

 

 

 

 

 

SOLEIL TM012
1%: 2 3.06
T T
ace 4.5 fi—V 8.40 {Viv
‘9 V
k” 0.38 5C 0.37 pm
5 90 Q 55 a
G 476 Q
Cavity length 42.5 cm 122.7 cm
Beam pipe radius 13.0 cm 17.0 cm
Voltage 2.0 MV 2.0 MV
Ep 942 l‘fil—V 4.99 fig
BL, 21.2 mT 13.69 311T

 

135

 

Chapter 7

Conclusion

7 .1 Summary

Smaller, higher-frequency cavities for accelerating 100’s of mA of beam current for an
ERL are envisioned with every RF bucket filled. This has the advantages of reducing
the bunch charge and simplifying HOM damping by ensuring that all dangerous
HOMs are above cutoff. Two elliptical cavity designs for this purpose are presented
which provide low peak fields and high cell-to-cell coupling. The beam pipe radius is
large enough for any HOM with a frequency at or above twice the accelerating mode
frequency to propagate out of the cavity. A single-cell cavity operating in the TM012
mode can provide useful voltage gain and a low loss factor, but the issue of adequate
lower—order parasitic mode damping must still be addressed.

Two copies of a 2—cell cavity design for high average beam currents have been
fabricated and tested, achieving a maximum accelerating gradient of 8.74 91131. This
limiting field might be explained by a superconducting region of the cavity go-
ing normal-conducting at high fields. One possible area for the superconducting to
normal-conducting transition to have occurred is a Nb disk which was used to shield
the stainless steel input flange during vertical testing. It seems as if some soft mul-

tipacting barriers were encountered at lower fields such as 4 %, but it was possible

136

 

to work through them and achieve gradients suitable for accelerating 100’s of mA of

beam current.

7 .2 Implications

To increase the maximum electron beam current in a storage ring, the cavity iris
must be opened to prevent beam breakup and to limit the increase in induced HOM
power. Increasing the iris of a single-cell elliptical cavity results in a smaller value of
g and ultimately leads to larger dynamic losses. By using a single-cell HOM cavity, a
mechanism exist for upgrading a storage ring to higher beam currents without paying
more for cryogenics. The iris of the HOM cavity can be much larger than that of an
elliptical cell operating in the TM010 mode without increasing the dynamic losses.
Additionally, the HOM cavity can also supply the same voltage gain with lower peak
fields by operating at a lower accelerating gradient.

The small wall angle of the 2-cell cavity was adequate for draining the chemicals
from etching and the ultrapure water from the high pressure rinse. The accelerating
gradient achieved in both 2-cell cavities met the design criteria, providing in excess

of 5.45 MEY, without being limited by field emission or quenching. Any multipacting
MV

barriers encountered below 8.74 F were soft and processed through.

Benchtop measurements on the 2-cell cavity showed that the Qext’s possible when
coupling to a normal-conducting waveguide which intersects the beam line are m
107 — 108. Since this is the same range of values currently used for cavities operating
with energy recovery, the large diameter iris of the 6-cell cavity may be of interest to

researchers wishing to study the feasibility of using such a coupling scheme.

137

 

7 .3 Possible Future Work

The number of projects which could benefit from the results of this dissertation is only
limited by the imagination of the researcher. Some possibilities worthy of exploring

include:

6-cell Cavity

1. F abricate and test a prototype cavity.
2. Tuner design and microphonics experiments with a prototype cavity.

3. Experimentally determine the frequencies of the vibrational modes of the cavity-

tuner combination and study methods of suppressing microphonics.

4. High-field tests of the cavity with a normal conducting waveguide coupler which

intercepts the beam line.
5. HOM coupler or ferrite loaded beam pipe design and construction.

6. Cryomodule design with tuners, power couplers, and HOM couplers/ loads.

2-Cell Cavity

1. Vertical tests with normal conducting beam pipe extensions.
2. Cold cavity tests with high beam currents.

3. Input coupler design for 50 kW or greater.

HOM Cavity

1. Design of a coaxial beam tube with ferrite loads or HOM couplers for extracting

lower order parasitic modes.

2. Calculate the longitudinal loss factor taking into account the coaxial beam tube.

138

3. Fabricate and test a prototype cavity.
4. Design an input coupler and tuner for the HOM cavity.

5. Fabricate and test the input coupler, HOM dampers, and tuner.

139

 

Appendix A

Waveguide Modes

In this appendix, the differential equations will be derived for the modes of propaga-
tion in a waveguide of uniform cross-section, starting with Maxwell’s equations. The
waveguide is assumed to have perfectly conducting walls with the inside under vac-
uum. The particular case of a waveguide of circular cross-section will be considered.
The cutoff frequencies and field distributions for the lowest circular waveguide mode
of each type (the TE11 mode and the TM01 mode) will be derived.

The derivations presented in the following section synthesize key elements of the

texts by Collin [12], Pozar [13], and Slater [14].

A. 1 Maxwell’s Equations

The electric field E and magnetic field B satisfy Maxwell’s equations:

6B

(9D
VXH — W+J (Alb)
V-D = p (A.1c)
v-B = 0. (A.1d)

 

For all cases of interest, the fields will vary harmonically with time, so phasor notation
will be used for the electric and magnetic field components, i.e. the amplitude and
phase of oscillation of a field component is represented by a single complex number.
To obtain the value of any component as a function of time it will be necessary to take
the real part of the product of the phasor and ej‘“. For any two components, A and
B, the relation Re [4251] = (AB) holds, where ( ) denotes averaging over a period.
Differentiation (integration) is performed by multiplying (dividing) the phasor by jw.

In this notation Maxwell’s equations become

V XE = —ij—M (A.2a)
V x H = ij + J (A.2b)
V - D = p (A.2c)
V - B = 0. (A.2d)

The region of interest is the vacuum inside of a waveguide so J, M, and p are all
zero. Additionally, D = (OE and B = poH, where 60 = 8.854 x 10"12 F/m and

p0 = 47r x 10"7 H / m. In this case, Maxwell’s equations reduce to

V X E = — jwpoH (A.3a)
V x H = jwcoE (A.3b)
V - E = 0 (A.3c)
V ' B = 0. (A.3d)

141

 

Ii

Equations (A.3a) and (A.3b) can be decoupled by taking the curl of both sides of
Equation (A.3a) and substituting Equations (A.3b) and (A.3c):

Vx(VxE) = -jwp0VxH (A.4)
Vx(VxE) = V(V-E)—V2E (A.5)
—v2E = — 1°qu (jweoE) (A6)
V2E+w2p0€0E = 0 (A.7)

Taking the curl of Equation (A.3b) gives a differential equation for H of the same

form:

V2H + w2poeoH = 0. (A.8)

Substituting k0 E (.9 p060 into Equations (A.7) and (A8) gives

v2E+k3E = 0 (A.9a)

v2H+k§H = o. (A.9b)

If the shape of the metal boundary is not a function of z, a solution for any component
of E or H can assume the form 8‘ jfiz f (x, y). For this case it is convenient to separate

E and H into transverse and axial components.

= 8(I,y)e_mz+ez(x,y)e—jflz

Ht'i'Hz

Emma
11

= h($,y)e_mz+hz(z,y)e—jfiz

142

 

 

Maxwell’s equations can be similarly simplified by writing the del operator in terms

of transverse and axial components,
V = Vt + V; = Vt - jfinz (A.10)

since n25); = — jfinz. Using this notation, each of Maxwell’s equations can now be

reduced to a form more amenable to solving and classifying waveguide modes

Faraday’s Law

The left-hand side of Equation (A.3a) can be written as
VxE=(Vt—jflnz) x (e+ez)e—j’6‘z (A.11)
and the right-hand side may be written as
— jwp0H= — jwpo (h+hz)ejfiz. (A.12)
Hence, Equation (A.3a) becomes

Vtxe—jfinzxe+VtXez—jfinzxez = ‘lwfloh-lwflohz (A-13)

Vt x e— jfinz >< e+Vt xez = —jwp0h— jwpohz. (A.14)

Equating the axial and transverse components of Equation (A.14) gives the two vector

equations,

Vt x e = — jwpohz (A.15a)

Vt x ez — jflnz x e = - jwpoh. (A.15b)

143

 

 

If

The first term on the left of Equation (A.15b) may be rewritten as

66 86
Vt X 82 = Elana; ‘— ing = —nz X Vtez. (A.16)

Substituting the above relation into Equation (A.15b) gives
112 X Vtez + jflnz X e =jUJ/10h. (A.17)

Equations (A.15a) and (A.17) will be used in the following sections to solve for the

field patterns of waveguide modes.

Ampere / Maxwell Relation

In a similar way, Equation (A.3b) may be separated into two vector equations for the

axial and transverse components:

Vt X h = jweoez (A.18)

nz X Vthz + jfinz X h = — jLUfoe. (A.19)

Divergence of B

The left-hand side of Equation (A.3d) can be expressed in terms of axial and trans—

verse components of H:

V ° B = V ' (I-loH) (A.20)

v - (90H) (v, — jfinz) - (h + hz) ,10 e- W (A21)

144

Hence, Equation (A.3d) can be written as

(Vt.h—jphz)poe—Jflz = 0 (A.22)

oras Vt-h = jfihz. (A.23)

Coulomb’s Law

Expressing Equation (A.3c) in terms of components, we obtain a similar expression

in terms of electric field components:

Vt . e = j pez (A.24)

A.2 Circular Waveguide Modes

A.2.1 Transverse Electric Modes

Transverse electric (TE) modes are by definition modes in which there is never any
component of the electric field in the axial direction, i.e. ez = 0. Under this condition,

the above component forms of Maxwell’s equations reduce to

Vt x e = — jwpohz (A.25a)

fin; x e = wpoh (A.25b)

Vt x h = 0 (A.25c)

nz x Vthz + jfinz x h = — jweoe (A.25d)
Vt - h = jflhz (A.25e)

Vt - e = 0 (A.25f)

145

Equation (A.9b) may be written explicitly in terms of axial and transverse compo-

nents. We write

v2 = v? -62 (A26)
k3 = 1.3—p2 (A27)
so that Equation (A.9b) gives
v26 +k2h — 0 (A 28)
t Z c 2 - -
v§h+k§h = 0. (A29)

The Pythagorean relation between the critical wavenumber kc, the free space wavenum-
ber kg, and the axial wavenumber B in Equation (A.27) provides a simple physical
interpretation of a waveguide mode. Below a critical frequency, we 2 cite, a mode
may only be excited evanescently. The mode can only store energy during one half
of an RF cycle before delivering it back to the RF source, similar to the behavior
of a lumped capacitor or inductor in low frequency circuits. At we, a standing wave
is established in the radial direction. The wave is a superposition of a wave travel-
ling away from the axis and a reflected wave travelling toward the axis. The type of
mode and the shape of the waveguide cross section will determine the value of kc
when Equation (A28) is solved. For frequencies above we, a wave travelling in the
axial direction is established with the wavenumber ,8. The field pattern can there-
fore be viewed as the superposition of two travelling waves which propagate in equal
but opposite directions slightly off axis and continually reflect off of the walls of the
waveguide. The two waves travel with the free-space wavenumber k0 which has he as
a transverse component and ,6 as an axial component.
The first step in solving for the resulting field pattern is to solve Equation (A28)

for hz. Because E and H must not only be solutions of the Helmholz equation but

146

also must satisfy Maxwell’s equations, it can be shown that all of the remaining field
components can be determined from h 2. Since there is no electric current and no axial
displacement current (62 = 0) in the interior of the waveguide, Vt x h must then be
zero by Equation (A.18). If the transverse curl of a transverse field, h, is zero it is
possible to write h as the gradient of a scalar. Additionally, this scalar must involve

hz since V - (poH) must always be zero.
v, x (v, x h) = v, (Vt - h) — V211 = 0 (A.30)

Substituting Equation (A.25e) and replacing Vgh with —kgh shows that h is pro-

portional to the gradient of h z:

The transverse electric field, e, may now be found using Equation (A.25b).

n; x (flnz x e) = nz x (wpoh) (A.32)
—fle = wponz (A.33)
e = —%nz x h (A.34)

The combination of constants appearing in Equation (A.34) can be conveniently

rewritten,
01/10 _ koflo _ 1‘70 #0 k0

 

 

[3 -A./—u.e.—3‘ 23:77" ““3"”
k.

z _ F (A.36)

e = —anxh (A37)

where 17 = ( / [:30 = 376 Q is the intrinsic impedance of free space. It is the ratio of the

147

‘ 1 ' i-.ol.2‘ ‘_ .f'.‘. '1ul‘15]

 

I‘H .

magnitudes of the electric and magnetic fields which are perpendicular to each other
and therefore responsible, by Poynting’s theorem, for the energy flux carried by a
plane wave. Since 6 3 190 from Equation (A.27), it follows that the wave impedance,
Z, of a waveguide mode is greater than 17. Additionally, the phase velocity, “3’, of the
waveguide mode is always greater than the speed of light, 7%.

The procedure for solving for a particular TE mode of a waveguide is generally:

1. Find IL; and kc as a solution of Vghz + 19th = 0 using appropriate boundary

conditions for the given waveguide cross section.

2. The free-space wavenumber, mode wavenumber, and wave impedance are cal-

culated using the following relations:

k0 = 2
C
= (pg—kg
k0 190 #0
Z = _ =_ _
6" 6 so

where w is the angular frequency of the mode and c is the speed of light.
3. The transverse magnetic field, h, is then found using the relation h = —‘]c-c'26-Vt h z.
4. The transverse electric field, e, is then found from the relation, 9 = —an x h.
5. The mode field pattern can then be expressed as,

H = ihe¥332+hze¥mz

E — eeqzjfiz

where a plus sign is used in the exponents if the wave is travelling in the direction

of —z.

148

A.2.2 TE11 Mode

The TE11 mode for a waveguide with a circular cross section will be used to demon-
strate the procedure described above; The final results will be useful in later sections.
First, the eigenvalue equation Vghz + 16th = 0 must be solved. Cylindrical coordi-

nates will be used since the waveguide has a circular cross section:

82h; 16112 1 62}... 2

———+——+————+kh =0 A.38

0.02 pap p26¢2 CZ ( )
This equation may be solved using the technique of separation of variables assuming a

solution of the form h z = R (p) F (¢)- Substituting this solution into Equation (A38)

and dividing by hz results in

1d2R 1dR 1d2F 2
———- ——+——— k =0 A.39
de2 dep p2Fd¢2 + C ( )
This equation may be separated into a part which only depends on p and a part which
only depends on 99:

Biff—R £9.13 2 2 1 W

k =——— .
de2 de p C Fd¢2 (A40)

The only way this equation can be true for any arbitrary values of p and d) is if both
sides are equal to the same constant 112. The two ordinary diflerential equations which

result are

(123 ldR 2 V2
—+-—-+ k —— R = O A.41
— + u2F = 0. (A.42)
The general solution of Equation (A.42) is periodic and has the form

F ((6) = A cos (1415) + Bsin (mp). (A.43)

149

 

To insure that F remains single-valued for values of 63 larger than 21r, u must be an
integer, 11. Equation (A.41) is a form of Bessel’s equation. A solution which remains
finite for all values of p is Bessel’s function of the first kind Jn (kcp) of order n. The

general solution then has the form
I1; = [A cos (11¢) + B sin (n¢)]Jn (kcp) . (A.44)

The condition that must be satisfied at p = a, the metal boundary, is that the
tangential component of the electric field is zero. According to Equations (A.31) and
(A37), this implies that 95%; = 0 at p = a. The only allowed values of kca are therefore
the roots of (13% = 0. The general solution for a mode is specified completely by two
integers, n and m. Modes for which n = 0 are called monopole; n = 1 are dipole;
n = 2 are quadrupole, etc. The value of m determines which zero of 513% is used
and therefore also determines the value of kc. The lowest frequency TE mode is a
dipole (n = 1) and corresponds to the first zero (m = l) of %—I;". The zero occurs
approximately at kca = 1.841, therefore kc = 1%. The solution for h; in the TE11
mode is then,

hz = [A COS(¢) + Bsin(¢)] J1 (1.8415) . (A.45)

Next, h is determined by evaluating h = -—-];thhz with Equation (A.45) substituted
C
in,
jfi th 161.2 )
h = —— ——n + ——n A46
1:? ( 5p p p 59¢ ¢ ( )

Jfl .
h = —;€—g-; [B cos(q§) — A sm(¢)] J1 (kcp) n¢

—M [A cos<¢> + 3.193).]; (kcp) np (AA?)

150

 

 

 

where J[(p) = (U pp . Finally, e is determined by evaluating Equation (A.37):
' k . ' k .
= JZCO [A cos(¢) + Bs1n(d>)] J] (kcp) n¢ — Jkn2p0 [B cos(<p) - Asm(¢)] J1 (kcp) up
0

(A.48)
where Z = %n has been used.

A.2.3 Transverse Magnetic Modes

Transverse magnetic (TM) modes are by definition modes in which there is never any
component of the magnetic field in the axial direction, i.e. hz = 0. The derivation
of TM mode field patterns is essentially identical to the previous case for TE modes

with E and H interchanged. The final procedure for solving for TM modes is

1. Find 62 and kc as a solution of Vgez + 1636;, = 0 using appropriate boundary

conditions for the given waveguide cross section.

2. The free-space wavenumber, mode wavenumber, and mode impedance are cal-

culated using the following relations:

kg:

6)
c
= (/ 1:3 — k?
B .3 #0
z = _ = _ _
(€017 190 60
where w is the angular frequency of the mode and c is the speed of light.

3. The transverse electric field, e, is then found using the relation e = —]§Vtez.
C

4. The transverse magnetic field, h, is then found from the relation, h = %nz x e.

151

 

5. The mode field pattern can then be expressed as

E = ee;jfiz+ieze¥jfiz

H = :thezFJ-fiz

where a plus sign is used in the exponents if the wave is travelling in the direction

of —z.

A.2.4 TM01 Mode

The TM01 mode for a waveguide with a circular cross section will be used to demon-
strate the procedure described above. The final results will be useful in later sections
for describing cavity modes. First, the eigenvalue equation Vgez + kgez = 0 must be

solved. The general solution has the form
62 = [A cos (7105) + B sin (n¢)]Jn (kcp) . (A.49)

The condition that must be satisfied at p = a, the metal boundary, is that 62 = 0.
The only allowed values of kca are therefore the zeros of Jn. The general solution for
a mode is specified completely by two integers, n and m. The value of m determines
which zero of Jn is used and therefore also determines the value of kc. The lowest
frequency TM mode is a monopole (n = 0) and corresponds to the first zero (m = 1)
of Jo. The zero occurs approximately at kca = 2.405, therefore kc = 1211—0; The
solution for 6; in the TM01 mode is then

e; = J0 (2.4055) . (A50)

152

Next, 9 is determined by evaluating e = —a]€—g—Vtez with Equation (A50) substituted
C

in,

jfidez I5

15
———n = —— —J k n A.51
kg dp p kg 1( GP) p ( )

kch (kc/J) np = — kc

where the relation 143-382 = J1 (p) has been used. Finally, h is determined by evalu-

ating h = %nz X e with Equation (A.51) substituted in,
j k0
_ _ __ A. 2
h h¢n¢ flkc J1 (kcp) n¢ ( 5 )

where Z = €020 has been used.

153

 

Appendix B

General Expansion of

Electromagnetic Fields in a Cavity

In this appendix, the electromagnetic field inside a cavity will be expressed in terms of
an orthonormal set of modes. The time evolution of the field in the cavity will then be
expressed in terms of scalar differential equations describing the time evolution of the
field amplitude of each mode. The differential equations will allow for the possibility
of electromagnetic energy being transferred into the mode or out of the mode due to
a beam travelling through the cavity, power being supplied or lost through coupling
ports, or power dissipation in the walls of the cavity. The derivations in this appendix

follow closely the analysis originally done by Slater [14].

B.1 The Complete Orthogonal Set

In order to characterize the electromagnetic performance of a cavity and to describe
a cavity’s interaction with a beam of charged particles passing through it, it is conve-
nient to write any electric or magnetic field which exists in the cavity as a sum over
a complete set of modes. This complete set of modes will satisfy boundary conditions

appropriate for a cavity with no dissipative losses. While this physical situation never

154

 

 

occurs in reality, it is a good approximation for a cavity with low losses, and loss—free

boundary conditions are easier for numerical calculation of the modes.

B.1.1 The Solenoidal Fields

For the closed cavities of interest, any vector field is determined uniquely once the
values of its divergence and curl are known at all points. The vector field may therefore
be written as the sum of two parts: a solenoidal function with zero divergence and
an irrotational function with zero curl. In principle then, a single complete set of
solenoidal functions and another complete set of irrotational functions is all that is
needed to completely describe either the electric or the magnetic field in a cavity.
In practice however, it is convenient to establish two sets of solenoidal functions,
Ea and Ha. The reason for this is due to the electric field and magnetic field in
a cavity satisfying different boundary conditions; On a perfect conducting surface,
there is no tangential component of the electric field, while for the magnetic field
there is no component normal to the surface. A complete set of functions which
have no tangential component along the cavity surface could, for example, be used
to express the magnetic field at all points inside of the cavity. While none of the
vector functions in the set satisfy the correct boundary conditions for the magnetic
field pattern of interest, an infinite series of these functions may still duplicate the
magnetic field pattern arbitrarily close to the surface: The resulting infinite series
would duplicate the magnetic field pattern inside of the cavity and would differ only
in that the value of its tangential component would discontinuously drop to zero
at the surface. To avoid this complexity, two sets of solenoidal functions satisfying
two different boundary conditions are used. In this case, the electric and magnetic
field pattern of an actual cavity mode may be represented primarily by one mode
of each set with small corrections to account for small energy losses. The Ea’s and
the Ha’s each form a set which may be used to describe any vector field, however

the convergence will be much better if the Ea’s are used to describe the electric field

155

and the Ha’s are used to describe a magnetic field. Since, according to Maxwell’s
equations, the curl of the electric field is proportional to the value of the magnetic
field and vise versa, the Ea’s and Ha’s will be defined such that the curl of one field
is proportional to the value of the other. We will assume that the solenoidal fields
satisfy the relations

V X Ea = kaHa (B.l)
V X Ha = kaEa (B.2)

The boundary conditions satisfied by the solenoidal functions will depend on the sys-
tem of interest but the bounding surface will generally satisfy two types of conditions
which are easily implemented in numerical models. Part of the surface, denoted by
S, will have n x Ea = 0 at all points on S while the remaining portion of the surface,
denoted by S’, will have n x Ha 2' 0 at all points on S’. The unit vector normal
to the surface is n. The significant feature for both boundary conditions is that all
components of Ea x Ha must be tangential to the surface since either Ha or Ea is
normal to the surface. When the Ea and H3, correspond to the electric and magnetic
fields of a cavity mode, this means that no energy is transferred through the bounding
surface. In addition, it may be shown that Ea may have normal components only on
S and H3 may have normal components only on S". For example, Ha must be normal

to the surface 5" so integrating over a small region of 5’ gives

[9,(VxHa)-da=fis’Ha-ds=0. (B.3)

But since the area of integration can be made arbitrarily small, Ea must then be

tangential to S" from Equation (8.2):

0 =/ (V X Ha) ' da = ka/ Ea ° da. (B.4)
S" S"

156

 

Using the same argument for S shows that Ha must be tangential to S since Ea is
normal to S.
The differential equation that the solenoidal functions satisfy may be found by

taking the curl of Equations (8.1) and (B2). Starting with Equation (B.1) we obtain

V x (V x E.) = V (V - Ea) — V2Ea (3.5)
V x (V x Ea) = V x (kaHa) = kgEa. (B.6)

This gives
V2Ea + 1.36.. = 0. (3.7)

The same operations on Equation (B.2) results in
V2Ha + kgHa = 0. (B8)

Equations (B7) and (B8) have the same form as Equations (A.9a) and (A.9b). The

orthogonality of the solenoidal functions may be proven by first forming
V - [Eb x (V x Ea)] = (V x Ea) - (V x Eb) — Eb - [V x (V x Ea)] (8.9)

V . [Ea x (V x Eb)] = (V x Eb) - (V x Ea) — Ea - [V x (V x Eb)]. (3.10)

Equation (B10) is then subtracted from Equation (8.9); The right-hand side may be

simplified:
Ea- [V x (V x Eb)] — Eb . [V x (V x Ea)] = (Ea - Eb) (kg — k3,). (13.11)
and the Divergence Theorem is used to evaluate the left-hand side,

/V - [Eb x (V x Ea)]dT = leaf (Eb >< Ha) ~da = 0 (8.12)

157

 

and

[V - [Ea x (V x Eb)]dr = kb f (Ea x Hb) - da = 0, (B.13)

so that

(k2 — kg) / (Ea . Eb) d7 = 0. (B.14)

If a = b then k3 = lab and f (Ea - Eb) (17' may have a nonzero value. If a 74 b
then ka 74 lob and f (Ea - Eb) (17' must be zero and the modes are orthogonal. This
argument only works if the functions are nondegenerate, however it can be shown that
the orthogonality still holds in the case of degeneracies once the degenerate functions
are combined in appropriate linear combinations. Since the case of degeneracies is not
of particular interest here, the derivation will not be done and it will be assumed for
the remainder of the text that the proper linear combinations have been made such
that all solenoidal functions are mutually orthogonal.

For complicated geometries, the Ea’s and Ha’s are found numerically using pro-
grams such as SUPERFISH [18] or Analyst [20]. The numerical solution is generally
given in terms of properly dimensioned electric and magnetic vector fields for the
cavity modes. The electric field E; and the magnetic field H; can be integrated over

the volume of the cavity to obtain the total energy Ua stored in the cavity mode:

3 [13ng = g [1‘1ng = Ua. (3.15)

Since we wish to normalize the Ea’s so that f Egd'r = 1 then the relation between

the fi’s, which are found from simulations and have units of %, and the Ea’s must

be
6 ~
E = — , .
a war), (B 16)

3
The units of Ea must then be m‘?. Additionally, f 13ng = 1 implies that f 11ng = 1

158

since

V-[Eax(V><Ea)] Ea-[Vx(Van)]—(Van)-(Van)

= k3 (E3 — HE) (B.17)

1.3/(Eg— H3) (17 = /V . [Ea x (V x Ea)ld7'
= ha f (Ea x Ha) - da = 0 (B.18)
=> / H267 = / Egdr. (8.19)

The relation between the Ha’s and the Hé’s, which have units of 72-1, must then be

it ~
_ _ B20

_3
and the units of Ha are m 2.

B.1.2 The Irrotational Fields

Since V x Fa = 0, Fa is equal to the gradient of a scalar function,
HaFa = Vt/Ja. (B.21)
Also, assume that $3. satisfies the differential equation

V2d26+n§¢n = 0. (B22)

=>O

V (V2023) + nsz/Ja = vgpa + Age... (B23)

Both F3 and 1123 satisfy the same differential equation, where ma aé ka in general.
The boundary conditions satisfied are 1,03 = 0 and V x Fa = 0 at all points on the

bounding surface. The orthogonality of the (bah and Fa’s may be proven. First, note

159

 

 

that

\7 - (weWb) = M241, + (We) - (W...) = «gem. + (We) - (We) (13.24)

and

V ' (Awe) = wa21/Ja + (VI/1b) ' (Vibe) = -I€g¢b¢a + (VI/Jb) ° (Vwaf (13-25)

Subtracting Equation (8.25) from (B24) and integrating over the entire volume re-

sults in

/[V ' Wavwb) — V - (waz/Jafld‘r = [$311213 (rig — mg) (17'. (B26)

The left side may be converted to an integration over the surface,

f (AW). — waz/Je) -da = (8?, — Ag) / (bat/21,67. (B27)

The surface integral is zero since the values of both ’l/)a and (pb are zero for all points

on the surface. Hence

(n3 — Ag) / 68.1,)de = 0. (B28)

If n3 75 Kb, then f i/Jawbd'r = 0 and the fimctions are orthogonal. Otherwise, Isa = Rb
and f I/Jalbbd‘r is finite. The (pa’s are normalized by setting f #2ng = 1.
The orthonormality of the (03’s implies that the Fa’s are also orthonormal. To see

this, first note that

f V - (weWb) dr = f «A. (We) -da = 0. (13.29)

160

But the volume integral on the left may also be written as
f (We) - (<74me + / ¢3V2111bd11 (13.30)
Substituting Equations (B21) and (B22) results in
nan), / Fa - der — It}; / (4.121)de = 0. (B31)

This relation shows that the Fa’s must form an orthonormal set if the ¢a’S are
orthonormal. Lastly, it can be shown that the Fa’s are orthogonal to the solenoidal

functions. The proof below uses one of the Ea’s as an example.

fV'(1/JaEb)dT = fr/jaEb-da=0 (B32)
fV-(waEbmr = /(V1/)a)'Ede+/1/Ja(V-Eb)d'r

= na/Fa-Ebdr (B33)
=> [Fa-Ebd—r = 0 for all values ofaand b. (B34)

In summary, any vector field inside of the bounded volume may now be represented

by an appropriate linear combination of either the Ea’s and Fa’s or the Ha’s and

Fa,S.

161

 

B.2 Time Evolution of the Expansion Coefficients

The electric field, magnetic field, current density, and charge density in a cavity may

now be written as linear combinations of the solenoidal and irrotational functions:

E = Z<enEa+feFa (3.35)
H = ihaHa (B36)
J = 2030.6.me (13.37)
p = grada- (13-38)

If the only systems of interest are cavities in which no energy ever enters or leaves,
then the time evolution of the expansion coefficients could be determined by sim-
ply substituting Equations (B.35) through (B.38) directly into Maxwell’s equations.
However, we would also like to describe systems in which energy does cross over
the cavity boundary. For example, all metallic cavities have small conduction losses,
therefore there must be a small tangential electric field on the surface where current
is flowing. Power must be supplied to the cavity to compensate for these losses, so
an input port must be provided. In both of these examples, there is now a region of
the cavity where a component of E x H is normal to the boundary. The solenoidal
and irrotational functions still form a complete set and can represent any field within
the original boundary even for the case that power flows across the boundary. For

h cavity mode

example, if the wall losses are small then the electric field for the at
would consist primarily of Ea plus relatively minor contributions from the other Ea’s.
The other Ea’s allow the value of E to be finite right up to the bounding surface with
arbitrarily small precision. However, all of the Ea’s have zero tangential component
on the bounding surface S. So while the value of E is correct at all points within the
cavity, the tangential component must drop instantaneously to zero at the boundary

S, therefore derivatives of the limiting function for E will not be correct at the cavity

162

 

surface. Stated more mathematically, the convergence of E is nonuniform for the case
of wall losses. The nonuniform convergence implies that while E may be represented
by a linear combination of the Ea’s, the derivative of E may not generally be repre-
sented by the same linear combination of the derivatives of the Ea’s. To handle the
more general situation where power is able to flow into and out of the cavity, the curls
and divergences which appear in Maxwell’s equations must be expanded directly.
The coefficients in the expansion of V X E are found by calculating the values of

f (V X E) - HadT for every a. To calculate these integrals first note that

V-[EX(VXEa)] = (VXEa)~(VXE)—E-[VX(VXEa)]

= kaHa - (V x E) — 236.13,, (B39)
so that
/[V - [E X (V X Ea)ld7' = [kaHa - (V X E) (17' — leg/E - Ead'r. (B40)
The volume integral on the left may be converted to a surface integral:

/[V - [E X (V X Ea)]dr = f[E X (V X Ea)] - da = kafm X Ha) ~da. (B41)

Substituting the right side of this relation and ea = f E - EadT into Equation (B40)

results in

f (V x E) - Hadr = kaea + f (E x Ha) -da, (B42)

The left-hand side of the above equation is the ath coefficient of V X E, so
V X E = ZHalkaea + f (E X Ha) -da]. (B43)
a

The second term on the right side of Equation (B43) is zero when E has no component

tangential to the surface S. The expansion for V X E is then what is obtained if the
163

 

 

 

curl of both sides of Equation (B35) is taken. A similar derivation for the expansion

coefficients of V X H results in
V X H = ZEaUCaha — f(Ea X H) ' da]. (B.44)
a

The coeflicients in the expansion of V - E are found by calculating the values of

f (V « E) (bad'r for every (1. We can write

V ' (waE) = 1(1a(V ‘ E) + E ' (Vi/1a) = Via (V ' E) + “(1E ‘ Fa (345)
so that
/ V - (212313) (17 = f d’aE . da = 0 (B46)
and hence
[(V ' E) ¢adT = *Ka/E ’ FadT = “Rafa. (8.47)

Gauss’s Law

Substituting Equations (B47) and (B38) into V - E = '3 gives

2 ‘Kafal/Ja = ‘2 Z 7'aipa- (348)
a

a

Orthogonality implies that

7.
—I€afa = f. (8.49)

The equation of continuity of charge may be used to eliminate ra from Equation (B49).

We can write

6p_
V-J+-3?—0. (B50)

164

 

rm-.
,.
_ ‘

 

Expanding this equation gives

d a
Z "Kagalpa + 23: $aé‘ = 0- (13-51)

8.

Again using orthogonality, we infer that

dra
—_ —_ B52
Raga d t ( )

Taking the derivative of Equation (B49) and substituting the relation above gives

(1
ya = —€7dea-. (B53)

Ampere / Maxwell Relation

Substituting Equations (B44), (B37), and (B35) into V x H = J + 6%?- gives

;E8[kaha — f (Ea x H) . da] = 283$, (ja + 5%) + Fa (ya + 6%) . (3.54)

Equating coefficients of Fa simply reproduces Equation (B53) while equating coeffi-

cients of Ea results in

dea

kaha = ja + f (Ea X H) ° (:13 + 6.217. (13.55)

The first two terms on the right side of Equation (B55) show that the magnetic

field of the ath mode will be excited whenever the volume current or surface current

h

overlap with the electric field of the at mode.

165

 

Faraday’s Law

Substituting Equations (B43) and (B36) into V x E = —p%:lti gives

2H3 [kaea + f (E x Ha) da]= Z— —Ha,1dh—‘1. (3.56)

Hence
dha

dt. (3.57)

kaea=_f(EXHa)' da— [1—

Decoupled Equations for Time-Dependent Coefficients

A single differential equation describing the ha’s may now be found by differentiating

Equation (B57) and substituting the result into Equation (B55),

dea 1 d )1 dzha
— = ——— H -d ——— B58
dt ka dt (E X a) 3 k3 dt2 ’ ( )
d
kaha = ja+j£(EaxH) da—i— f(ExHa)-da
k3 dt
2
_epd ha
, B.59
—ka —d_t2 ( )
d2ha w2 .
3" .372.ng = 7.3 l’aflimb‘ X m'dal
——— [f (E x Ha)- )-da] (B60)
2 k2
where can = #6' Total time derivatives have been used since the expansion coeffi-

cients and the surface integrals do not depend on the spatial coordinates. The same

procedure results in a similar expression for the ea’s:

a!2 1 d 2
(it; +wgea = —;E [121+ f(Ea x H) da] — %— [f (E x Ha)- da]. (8.61)

8.

 

166

 

Appendix C

Analysis of CW RF Measurements

This section describes how measured values from the cavity tests are used to calcu-

 

If”! :|

late the Q0 of the cavity and is reproduced, almost verbatim, from notes by Walter
Hartung [28]. Other derivations and discussions of formulae for analysis of RF mea-
surements can be found in the literature [29, 30].

In a CW measurement, we are able to measure the following quantities:
Pf = forward power,
Pr = reverse power,
Pt = transmitted power,
f = frequency.

Additional information is needed to calculate the stored energy, field, and Q0 of the
cavity. We can obtain additional information by doing a “field level calibration” at low
power. This is done with modulated measurements to supplement the CW measure-

ments. With modulated measurements, the following quantities may be calculated:
Qext,1 = strength of the input coupler,

Qext,2 = strength of the pickup coupler.

167

A numerical calculation of the field pattern (using SUPERFISH, Analyst, etc.) for
the mode of interest is also needed to find out the relationship between the stored
energy and the field. The field is proportional to the square root of the stored energy
U:

amu¢fi (on

The peak surface fields, Bp and Ep, are proportional to the square root of U also.
With the output of the numerical calculation, any field of interest may be found from
U.

If Qext,1 and Qext,2 are obtained from a low-power modulated field calibration,
the CW measurements are overdetermined. So there is more than one way to calculate
the stored energy, field, and Q0. In principle, different methods should give the same

result. In practice, the answers may be differth due to systematic errors.

C. 1 Direct Method

We calculate the dissipated power Pd by subtraction,
Pd = Pf — P1- — Pt. (C2)

The stored energy is related to the transmitted power via the coupling strength of
the pickup,

wU
Qext,2 E 13; (03)

So, knowing Qext,2 and w = 27r f , U may be calculated,

 

Qext,2) .

U = 10t ( w ((3.4)

168

 

Having Pd and U, Q0 may be calculated directly via

 

wU
= —. C.5
Q0 Pd ( )
In terms of measured quantities, the intrinsic Q is
Pt
= . C6

The field may be calculated from Equation (C.1), with the appropriate conversion

factor. Advantages of the direct method include:
1. direct, easy to understand
2. knowledge of Qext,1 is unnecessary.
Disadvantages of the direct method include:

1. makes use of Pr, which may have systematic error associated with it

2. relies on the difference between Pf and Pr, which may be a small number when

not close to unity coupling.

C.2 Indirect Method Using %

An expression for Q0 which does not involve Pr is desired. We begin with Equa-

tion (C.2), re-writing it in terms of scattering parameters,

Pr
|311| = — (C.7)
Pf
Pt
|S21| = —. (0.8)
Pf
Equation (C.2) becomes
P
—d =1-|521|2 - |511|2 (C9)

Pf
169

The input coupling coefficient 31 depends on the reflection coefficient,

«mm

= C.10

1i |511|
—. C.11
fl 11F |511l ( )

In the above equations, the upper sign applies when overcoupled and the lower sign

applies when undercoupled. Solving for lS'lll2 results in

2_ 1-51 2
|311| _(—1+fl1) . (0.12)

Substituting this expression into Equation (C9) and simplifying gives

& ._ ___431 _ |S21|2 (0.13)

Pf (1+31)2

It is not possible to measure Pd directly, so it must be eliminated. It may be expressed

in terms of 52 and I521l,

 

 

B2 = 1% (0.14)
g: = B}; (0.15)
515ch = Eggi (C.16)
Substituting this into Equation (C.13) results in
162;)? = (11%)? " '521'2' ‘0'”)
Define
|R21|=|521|(/%=IS21| 2:: ((118)

170

 

so that Equation (C.17) becomes,

 

IR l2-————4fi¥ warm? (019)
21 (1+fi1)2 ’
sothat
25
lR21l\/1+52= 1 - ((3.20)
1+[31

If [32 << 1 (i.e. weakly-coupled pickup antenna), then

 

2B1
R z . C21
| ml 1 + 31 ( )
Solving for ,61, we obtain
1
lR2ll
But 61 = age—1, so Equation (C.22) reads
ex ,
Q0 :2: #. (C.23)

— 1
|R21l

Substituting Equations ((3.18) and (CS) into the last expression results in an explicit

expression for Q0,
Qext,P 1

QON N 2(Q_LQext1 _
ext,2

Advantages of the indirect method include

 

(0.24)

1. does not use Pr, so systematic errors in Pr measurement are avoided

2. does not rely on calculation of Pf — Pr, which may be problematic to use when

not close to unity coupling.
Disadvantages of the indirect method include

1. derivation of formula is more complicated and its application may be less intu-
171

 

 

itive
2. must use both Qext,1 and Qext,2
3. assumes a weak pickup antenna

4. the formula may not work well when the cavity is strongly overcoupled.

C.3 Graphical Assessment of Systematic Errors

The direct method gives

 

 

Qgirect= Pthxt,2 = |321|2Qext,2 _ (025)
PP Pr - Pt 1- |311|2 - I321|2

The indirect method gives,

- - Q t1
andirect z 29X 1 1. ((3.26)
mm _

Both methods should, in principle, give the same answer, therefore we should have,

 

 

lerect = andirect, (C.27)
which leads to,
Qext,2lS21l2 ~ Qext,1 IR21I2 ~ 1
1—52—5 2N2—1 1—32—3 2~2_'
|21| |11| W |21| |11| W1
(C.28)

Weak coupling was assumed for the pickup, so 1 — |S11|2 >> IS'21I2 and to a good

approximation,
|R21|2 ~ 1

—— ~ —, (0.29)
1- |511|2 1—13221 -1

172

 

 

    
    

|R21| = 1+ |311|

|R21|

|R21| = 1 - |511|
undercoupled

 

 

 

I311| = {1;

Figure 0.1: Measured values of Qext,1 and Qext,2 may be checked for consistency by
inspecting how closely the calculated value of |R21| lies on the “Duality Triangle”.

therefore it follows,

1—|311|2+IR21I2-2|R21| z o, (0.30)
(1-IR21I)2 % |311|21 (C-31)

1-IR21I % ilSnl, (Q32)

|R21| % 1¥|511|- (C33)

Since 0 S lslll S 1, Equation (C.33) tells us that the values of ISnI and |R21|
should be on one of the two equal sides of an isosceles triangle. One might call this
the “duality triangle” (see Figure C.1). When doing CW measurements, the values .
of lslll and |R21| can be calculated. The proximity of these values to the duality
triangle then indicates whether the values of Qext,1a Qext,21 Pf, Pr, and Pt are all

consistent.

173

 

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