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Summer Student ReportAugust 14, 2015
[email protected]
RF Bead Pull Measurements of the DQW Cavity
Jaume GuillaumeCERN, Geneva, Switzerland
Keywords: HL-LHC, DQW Cavity, BeadPull Measurements, CrabCavity,
RF Measurements, Multipolar Expansion, SlaterPerturbation
Theory
Summary
This report was written within the framework of the CERN Summer
Student Program. It is focused on theRadio Frequency study of the
Double Quarter Wave Crab Cavity [1] considered for the
crab-crossing schemeof the LHC Luminosity upgrade [2]. HFSS
simulation [3] and Bead-Pull Measurements technique were usedfor
the characterization of the higher-order terms of the main
deflecting mode.
1
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Contents1 Introduction 3
2 RF Model of the DQW cavity 32.1 Resonant Frequencies of the
DQW Cavity . . . . . . . . . . . . . . . . . . . . . . . 3
2.1.1 Simulated Resonant Frequencies . . . . . . . . . . . . . .
. . . . . . . . . . 32.1.2 Measured Resonant Frequencies . . . . .
. . . . . . . . . . . . . . . . . . . 4
2.2 Electromagnetic Field and Force in the DQW Cavity . . . . .
. . . . . . . . . . . . 42.3 Transverse Momentum and Higher Order
Modes Identification . . . . . . . . . . . . 6
2.3.1 Transverse Momentum Theory . . . . . . . . . . . . . . . .
. . . . . . . . . 72.3.2 Transverse Momentum Calculation from the
Simulation . . . . . . . . . . . 7
3 Bead-Pull Measurements 93.1 Slater Perturbation Theory . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2
Introduction to the software used to acquire Bead-Pull Measurements
. . . . . . . . 103.3 The Needle as a Perturbing Object . . . . . .
. . . . . . . . . . . . . . . . . . . . . 10
3.3.1 Theory of the Needle as a Perturbing Object . . . . . . .
. . . . . . . . . . . 103.3.2 The example of a 30 mm long, 1.2 mm
width Needle . . . . . . . . . . . . . 12
3.4 The Sphere as a Perturbing Object . . . . . . . . . . . . .
. . . . . . . . . . . . . . 123.5 Post Process Procedure . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.6
Potential Vz obtained with the measurements and identification of
the Higher Order
Components . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 14
4 Conclusion 16
5 Acknowledgements 16
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1 IntroductionThe Double Quarter Wave cavity development is part
of the High-Luminosity LHC (HL-HLC)project aiming to increase the
luminosity of the beam collisions. The DQW cavity is a 400 MHzcrab
cavity designed to impart a transverse kick to the bunch travelling
in the beam-pipe. Due toits compact shape, the cavity does not have
axial symmetry, it gives rise to higher order multi-polarcomponents
of the main deflecting mode being non-zero.Radio Frequency (RF)
properties of the cavity were characterized for the simulation and
then com-pared to the measurements. The bead-pull measurement
technique, based on perturbation theory,was used to acquire all the
data. [4].The obtained electro-magnetic fields were afterwards
treated to derive the evolution of the potentialin the beam-pipe
cross-section and identify the higher order terms.
2 RF Model of the DQW cavityA superconducting prototype of the
Double Quarter Wave cavity was built in Aluminium to measurethe
fields on and off-axis and compare it to the simulation.In the
cavity model shown in Fig. 1, the bunch propagates along the
beam-pipe axis for a distanceof h = 546 mm (along the longitudinal
axis), the elliptical body of the cavity has a semi-majoraxis a =
175 mm, a semi-minor axis b = 147 mm and a height of h = 245 mm.
The beam pipecross-section is r = 42 mm.A 3D model of the cavity
was created using HFSS, a finite element method solver for
electromagneticstructures [3]. All the simulations were realized
using this software in the following parts.
(a) Front View (b) Global View (c) Side View
Figure 1: Technical drawing of the DQW cavity.
2.1 Resonant Frequencies of the DQW Cavity2.1.1 Simulated
Resonant Frequencies
A simulation in the eigenvalue mode, that finds the resonant
frequencies of the structure and theelectro-magnetic the fields was
used [3]. The fundamental frequency was find at ≈ 400 MHz whichis
the frequency of the LHC. The results for the first 5 modes are
presented in Tab. 1.A second simulation is then performed in the
driven modal mode that calculates the modal-basedscattering
parameters in order to visualize the transmission parameter S21.
Two ports are assignedas shown in Fig. 2(b), the calculation was
made in a frequency range of 200 MHz. The fundamentalmode is
therefore highlighted by plotting the magnitude of the S21
parameter as a function of thefrequency (See Fig. 2(a)).
3
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(a) Simulation of the S21 parameter versus frequency. (b) 3D
model Cavity with the assigned port.
Figure 2: Driven modal mode simulation.
2.1.2 Measured Resonant Frequencies
To determine experimentally the resonant frequencies of the DQW
cavity, a Virtual Network Anal-yser (VNA) was connected to 2 ports
as shown above in Fig. 2(b). From it, the 5 first resonantmodes
were extracted and are displayed in Fig. 3. For more precision, the
VNA is itself connected
Figure 3: Experimental first 5 resonant modes for the DQW cavity
at room temperature.
to a software capable of centering the analysis close to each
resonant modes and therefore extract anaccurate value (cf Sec.
3.2). The results are also presented in Tab. 1. The difference
between thesimulation and the experiments is partially explained
because of fabrication errors.
2.2 Electromagnetic Field and Force in the DQW CavityThe Double
Quarter Wave cavity (DQW) was designed to impart a transverse
momentum to thebunch travelling along the longitudinal axis. The
bunch will be therefore kicked (in the x-axis)proportionality to
the strength of the fields ~E and ~H [5].
4
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Mode Simulated Frequencies [MHz] Measured Frequencies [MHz]
1 398.6 398.32
2 578.2 577.74
3 672.6 672.04
4 701.6 701.23
5 752.1 752.18
Table 1: Resonant frequencies for the first 5 modes of the DQW
cavity obtained with the HFSSsimulation and with the measurements
(VNA).
The total force for a charged particle q in an electromagnetic
field is given by the Lorentz lawexpressed by Eq. 1.
~F = q( ~E + ~v × ~B) (1)
where q is the charge of the particle, ~E and ~B are
respectively the electric and magnetic field and ~vis the velocity
of the particle.The transverse force created is directly derived
form Eq. 1 and is given by Eq. 2.
F⊥ = e(Ekick + µ0.~c×Hkick) (2)
where Ekick = E⊥cos(ω0c z), Hkick = H⊥sin(ω0cz), E⊥ and H⊥ are
respectively the transverse
electric and magnetic fields, c is the speed of light, e is the
elementary charge and µ0 is the perme-ability of vacuum.The desired
force is obtained with the contribution of E⊥ and H⊥.The main
contribution to the forceis from the electric field such that the
resulting force is in the x̂ direction.This is shown by Fig. 4(b),
where a strong transverse electric field is visible in the XZ
plane. Theinfluence of the magnetic field is highlighted in the YZ
plane (see Fig. 4(a)).
The magnitude of the fields is normalized by the stored energy
in the cavity calculated using Eq. 3.
(a) Magnetic field on the XZ plane (b) Electric field on the YZ
plane
Figure 4: Electromagnetic field in the cavity.
U =1
2�0
∫V
|E|2dv = 12µ0
∫V
|H|2dv. (3)
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where E and H are respectively the electric and magnetic field,
�0 and µ0 are the permittivity and thepermeability of vacuum and V
denotes the volume of the cavity.HFSS was used to obtain for the
total stored energy U = 4.912.10−15 J. This result can be
doublechecked by first integrating ~E on the cavity and in the
other hand ~H (see Eq. 3). The differencebetween for both results
is less than 10−17 J.U is then used to normalize ~E as follow :
√E2
U, the same procedure is applied for ~H .
The profile for ~H and ~E was extracted from the simulation as
shown in Fig. 5. It shows the normal-ized electric and magnetic
field magnitude along the longitudinal axis on and off-axis. The
shaperemains the same along the x-axis for both ‖ ~E ‖ and ‖ ~H
‖.
(a) Normalized electric field magnitude along the lon-gitudinal
axis.
(b) Normalized magnetic field magnitude along thelongitudinal
axis.
Figure 5: Results obtained by using an HFSS model (eigenvalue
mode) of the DQW cavity. Themesh was created such that the
tetrahedra density is higher close to the longitudinal axis and
ensureaccurate results in this zone.
2.3 Transverse Momentum and Higher Order Modes IdentificationDue
to its compact shape and lack of axial symmetry, higher order terms
are non-zero in the cavity.It means that the transverse kick for
the particles off-axis will not be perfectly oriented in the
x̂direction. To compensate this lack of axial symmetry, those
higher coefficients need to be identified.It is assumed that the
cavity is symmetric with respect to the horizontal plane XZ. It
means thatall skew components are zero and azimuthal dependence is
simplified to a cos(φ) (where φ is theazimuth angle).A precise
value for the evolution of the potential Vz in the beam-pipe
(x-axis) was determined asexplained in the following part. In the
case where the cavity was perfectly symmetric (no higherorder
terms), the potential evolution would have been linear. Therefore a
fit of the obtained curvesgives directly the contribution of the
higher orders coefficients. It corresponds to decompose
thepotential in a sum of contribution as shown in Eq. 4.
Vz(r, φ) =4∑0
V (n)z rncos(φ) (4)
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2.3.1 Transverse Momentum Theory
Direct Integral :
The transverse momentum change of the particle when crossing the
cavity is given by Eq. 5 [6].∆p⊥ can also be seen as the
longitudinal contribution of the potential Vz.
∆p⊥(r, φ) =1
c
L∫0
F⊥dz (5)
where c is the speed of light and F⊥ is as shown previously in
Eq. 2.
Panofsky-Wenzel :
The potential Vx of the particles is given by Eq. 6.
Vx =
L∫0
E⊥(r, φ, z)sin(ω0cz)dz (6)
From Eq. 6, the transverse momentum can also be derived using
the Panofsky-Wenzel theorem [6]as follow in Eq. 7.
∆p⊥(r, φ) =je
w0
L∫0
∇⊥E⊥(r, φ, z)sin(ω0cz)dz (7)
∆p⊥(r, φ) =je
w0r
L∫0
E⊥(r, φ, z)sin(ω0cz)dz (8)
where φ is the azimuth angle, ∇⊥ is the azimuthal and radial
component of the gradient in a cylin-drical coordinate system , e
is the elementary charge and r is the radius (or offset).Eq. 5 and
8 can be combined to obtain Eq. 9.
L∫0
B⊥dz =1
ec
L∫0
F⊥dz =1
ω0r
L∫0
Eaccdz (9)
By comparing the results from Eq. 9, an estimate of numerical
accuracy is therefore highlighted.
2.3.2 Transverse Momentum Calculation from the Simulation
Values for Ez, Ex and By were first extracted from the
simulation along the longitudinal axis (az-imuth angle Φ = 0) for x
∈ [−35,+35] mm with 5 mm step.By applying Eq. 5 and 8,the
transverse momentum change (or potential) of the particles was
derivedand is shown in Fig. 6.
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Figure 6: Vx, Vz and Vz with Panofsky-Wenzel theorem profile
calculated on the x-axis (φ = 0).
The two curves obtained for the potential Vz fit well specially
for small offsets. The differenceobserved close to the limits of
the beam-pipe can come from the fact that the tetrahedra density
isless important in this region than close to the longitudinal
axis.To improve the accuracy of the potential Vz, data were
extracted in the entire beam-pipe with somedifferent azimuth angles
from [0,+360] with 30 degrees step (see Fig 7). The relationship
betweenthe angle and the potential is given by Eq. 10.
Vz(rn, φ = 0) =Vz(rn, φ1)
cos(φ1)(10)
Values obtained for the same radius rn were afterwards
renormalized using Eq. 10 and averaged.
Figure 7: Extracted points in the beam-pipe cross section (r =
42 mm radius circle).
8
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A 4th order fit of the curves was realized from Fig. 8 to
determine the higher order terms (nonlinearity coefficients). The
coefficients are presented in Tab. 2.
Figure 8: Vz calculated with Φ = [0, 360] with 30 degrees
step.
Order Coefficient
Offset 9.1× 10−7
1st 3.8× 10−2
2nd 4.0× 10−1
3rd 2.0× 10−2
4th 3.0× 103
Table 2: 4th order fitting coefficients of the potential Vz in
the beam-pipe.
3 Bead-Pull Measurements
3.1 Slater Perturbation TheoryThe Bead-Pull measurement
technique is typically used to derive the electric and magnetic
field in acavity. It was developed in 1951 by J.C Slater [4]. The
basic concept is to place a perturbing object(bead or needle for
instance) through the longitudinal axis (beam-pipe axis) of the
cavity. A shiftof the resonant frequency is observed while the
object travels the entire length of the cavity. Thisfrequency shift
is proportional to the square of the relative electric field | ~E|2
and magnetic field | ~H|2.This relation is shown by Eq. 11 [7]
[8].
9
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∆f
f0=
t∆V
�0E2dV −
t∆V
µ0H2dV
4U(11)
where U is the total stored energy, ∆f is the frequency shift, E
and H are respectively the electricand magnetic field and f0 is the
initial resonant frequency.It is often easier to work in term of
phase shift instead of frequency shift due to the difficulty to
readit on the VNA [9], the correspondence is given by Eq. 12.
∆f
f0=
tan[φ(f0)]
2QL(12)
where QL is the loaded quality factor ( energy loss relative to
the total stored energy of the cavity )and φ(f0) is the phase shift
of the transmission parameter S21 at f0.
3.2 Introduction to the software used to acquire Bead-Pull
MeasurementsBead-Pull measurements have to be done all along the
beam-pipe axis, an automatic control of thesemeasurements is
therefore a necessity. A brief explanation of the technique used is
presented.The cavity is placed on a base where 3 stepper motors are
installed. Each motor can move in aspecific direction (x̂, ŷ or
ẑ) and is wired to independent controllers. Those controllers and
the VNAare connected and driven by a LabVIEW [11] software (see
Fig. 9).This software is composed of multiples functions that
permit to set all the desired parameters. Thefirst one is the
setting of the VNA to visualize the resonant frequency and save the
initial transmis-sion parameters on a separate file.Then a second
function will be used to set the bead attached to a string at its
starting position (seeFig. 9).Finally, in the main program,
parameters regarding the measurements itself are set. Starting with
thenumber of sweeps at each point (ie. the number of times the bead
will travel in the entire length ofthe longitudinal axis), then the
transverse step between each measure (ie. the number of XY stepthe
motors have to do in the beam-pipe) and the type of movements
(transverse or circular) the beadwill implement in the beam-pipe.
All the measurements are afterwards save in some separated
filesused for the post-process (cf Sec. 3.5).In this configuration,
it is important to notice that the bead will scan the entire length
of the longi-
tudinal axis (beam-pipe axis) first from top to bottom and than
from bottom to top.For more informations regarding the
implementation of the software, please refer to [10].
3.3 The Needle as a Perturbing Object3.3.1 Theory of the Needle
as a Perturbing Object
A needle can be used as a perturbing object, it is usually
placed along the longitudinal axis. Themain advantage is to excite
mainly one component of the electric and magnetic field, in this
case theẑ component (ie. Ez and Hz).The length of the needle must
be chosen such that the phase shift does not exceed ≈ 25
degrees,which is the region where there is a linear relationship
between phase shift and frequency shift (seeEq. 12). In the other
hand, a long needle will have a better sensitivity and therefore
will excite morethe longitudinal component of the fields.
10
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Figure 9: Overview of the system to perform bead-pull
measurements.
Generally, the width of the needle should not exceed 110
of the total length of the needle such that themeasurements
remains exploitable.The last point is that, the obtained curve
should be average all over the needle length (here by creatinga
L-point moving average). It means that the longer the needle is,
the more the averaging error willincrease.From Eq. 11, the
frequency shift and fields relationship for needles can be derived
and is shown inEq. 13.
∆f
f= αE⊥�0
E2⊥U
+ αE‖�0E2‖U
+ αH⊥µ0H2⊥U
(13)
where E⊥ and H⊥ are respectively the transverse electric and
magnetic field, E‖ is the longitudinalelectric field and αE‖ is the
approximated longitudinal form factor calculated with Eq. 14
[12].
αE‖ = πl3
24(log( lr)− 1)
(14)
11
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with l the length of the needle and r the radius of the needle
(assumed to be a cylinder).Note that E⊥ and H⊥ are undesired
contribution in the measurements and need to be suppressed, itis
not necessary to obtain an explicit value for the corresponding
coefficients (see Sec. 3.3.2).
3.3.2 The example of a 30 mm long, 1.2 mm width Needle
Some measurements can show that taking a 30 mm long needle gives
satisfying results [14], it cor-responds to take αE⊥ = 121× 10−8
(using Eq. 14).Even if with a needle, the longitudinal component is
mainly excited as said above, some residual E⊥and H⊥ are still
measured (see Eq. 13). To extract only Ez, an on-axis measurement
was done. Infact for r = 0 mm, the longitudinal contribution of ~E
is zero, so it means that only non-desired fieldsare
measured.Simply by subtracting the off-axis measurement to the
on-axis measurement, a value for E2z is de-rived (see Eq. 15). This
is shown in Fig. 10.
E2z =1
αE‖�0(∆f
f(offaxis)− ∆f
f(onaxis)) (15)
Figure 10: On and Off-axis (x = 36 mm) measurements with a 15 mm
long needle, E2z profile (blackcurve) obtained by subtraction.
3.4 The Sphere as a Perturbing ObjectTaking a bead as a
perturbing object is an interesting case too. In fact the relation
between thefrequency shift and the fields is simplified as shown in
Eq. 16.
∆f
f0=−πr3
U(�0.
�r − 1�r + 2
E2 + µ0.µr − 1µr + 2
H2) [13] (16)
where U is the total stored energy in the cavity, r is the
radius of the bead, �r and µr are respectivelythe relative
permittivity and permeability, �0 and µ0 are the permittivity and
permeability of vacuum
12
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and finally f0 is the initial resonant frequency.More
specifically, if a dielectric bead is used, the contribution of ~H
becomes zero (µr = 1) andtherefore the electric field is directly
obtained according to Eq. 17.
∆f
f0=−πr3�0(�r − 1)U(�r + 2)
E2 (17)
If a metallic bead is used (perfect electric conductor implying
�r =∞, µr = 0 ), the frequency shiftis hence expressed as a
combination of the electric and magnetic field as shown in Eq.
18.
∆f
f0=−πr3
U(�0E
2 − µ0H2
2) (18)
The derivation of the magnetic field is therefore possible with
some mathematical manipulationswith Eq. 17 and Eq. 18.Using Eq. 16,
Fig. 11 was obtained from the data of ~H and ~E previously
calculated with the HFSSsimulation considering first a metallic
bead and then a dielectric one .
(a) Metallic bead ( �r =∞, µr = 0 ). (b) Dielectric bead ( �r =
9.22, µr = 1 ).
Figure 11: Relative shift frequency according to the electric
and magnetic fields along the longitudi-nal axis.
3.5 Post Process ProcedureA post-processing procedure with
Matlab [15] must be realized on the raw data before the
analysis(see Fig 12(a)). In fact, the LabVIEW software is saving
only for each scan on the longitudinal axisthe real and imaginary
part of the S21 parameter. From it, the first step is to find the
correspondingphase ∠S21. Then a few post-process step on the phase
must be done, this procedure is described asfollow.The first point
of all the curves is set to have a zero angle at the beginning of
the measure (ie.∠S21 = 0). Then due to some temperature drift in
the cavity who distorts the data, a linear correctionand adjustment
is made such that all the curves end with a zero angle too.As said
previously in Sec. 3.2, data are acquired in both direction (from
top to bottom of the cavityand vice-versa). So, to counterbalance
an eventual lack of symmetry in the cavity or in the way thebead
was attached and placed, all the odd curves are flipped.Finally, as
said in Sec. 3.3.1, a L-point-moving average filter over the entire
length of the perturbingobject (typically from 10 mm to 30 mm) is
created to smooth the data (see Fig. 12).
A time-distance domain transformation can be performed using the
linear speed of the motor used
13
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(a) Raw data standard deviation σ = 9.75× 10−3 rad. (b)
Post-processed data standard deviation σ = 2.56 ×10−3 rad.
Figure 12: Phase of the S21 parameter versus time from 4
measurements made with an offset ofx = 36 mm, using a 15 mm needle
length.
to move the bead as shown in Eq. 19.
x =t
vlin(19)
Usually, the longitudinal motor speed used is 30 Hz that
corresponds to vlin = 5.7mm.s−1.When the phase is post-processed,
the linear relationship between phase shift and frequency shift
isused first (see Eq. 12). Then by taking the advantage of the
particular symmetry of the cavity (seeSec. 3.3.2), a value for Ez
can be derived.
3.6 Potential Vz obtained with the measurements and
identification of theHigher Order Components
Measurements in the entire beam-pipe were acquired like done
previously for the simulation. Ex-tracted points were spaced with a
radius ∆r = 6 mm in a [−36,+36] mm range. Azimuth angles of0 to 180
degrees with ∆φ = 30 degrees step was taken.The post-processed
procedure explained in Sec. 3.5 was applied to the measurements.
From it, Ezover the distance for all the different offsets and
azimuth angles were obtained (see Fig. 13).Eq. 20 gives the formula
to compute Vz for a defined offset and azimuth angle.
Even if the longitudinal electric field at r = 0 mm is in theory
zero, some residuals componentsare still measured and needs to be
suppressed from the off-axis measurements. As explained inSec.
2.3.2, the azimuth dependence of the potential is reduced to a
factor cos(φ). It explains thedifferences between Eq. 20 and Eq.
8.
Vz =1
cos(φ)
L∫0
Ez(r, φ)sin(ω0z
c)− Ez(r = 0, φ)sin(ω0z
c) (20)
The values obtained for Vz for different azimuth angles are
afterwards averaged to increase theprecision of the results (see
Fig. 14).Note that the standard deviation for all the measurements
and the post processed data were calculatedbut it was only
displayed for the plot of the potential Vz over the beam-pipe cross
section.As done previously, a 4th order fit of the obtained results
was done. The coefficients are presented inTab. 3, as a reminder
the simulated coefficients are also displayed here. V (1)z fits
really well betweenthe simulation and the measurements. The order
for the other coefficients was found to be same too.
14
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Figure 13: Ez profile obtained from the measurements along the
longitudinal axis on x-axis. Fourmeasurements were acquired at each
point in a [−36,+36] mm range.
Figure 14: Vz profile obtained from the measurements along the
beam-pipe axis. Four measurementswere acquired at each point.
An other approach to identify the higher order terms could have
been to decompose the longi-tudinal electric field E‖ into a sum of
higher order contributions. It corresponds to take the
discreteFourier transform (DFT) of E‖ for different azimuth angles
at one defined radius [6].
15
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Order Simulated Coefficients Measured Coefficients
Offset 9.1× 10−7 5.7× 10−7
1st 3.8× 10−2 3.7× 10−2
2nd 4.0× 10−1 3.0× 10−1
3rd 2.0× 102 1.5× 102
4th 3.0× 103 2.0× 103
Table 3: 4th order fitting coefficients of the potential Vz in
the beam-pipe obtained from the simulationand the measurements.
4 ConclusionIn this report, Radio Frequency properties of the
DQW cavity were studied, particularly the electro-magnetic fields
on and off-axis. The cavity and the interaction occurring into it
were simulatedwith HFSS. The bead-pull measurement technique
assisted by a LabVIEW software was used toacquire the data. From
it, higher order terms of the main deflecting mode were identified.
A correctagreement was find between the simulation and the
measurements.All the raw data, post-process functions and LabVIEW
software used to obtain those results werestored in the directory
:cern.ch/dfs/Departments/AB/Groups/RF/Sections/BR/Crab Cavity
BeadPull Measurements.
5 AcknowledgementsI would like to particularly thank my
supervisors Maria Navarro-Tapia and Rama Calaga for theirreally
helpful comments and conversations about my work during my project
at CERN.
References[1] I. Ben-Zvi et al., Quarter Wave Resonator, LHC CC
workshop, Geneva, 2011
[2] http://hilumilhc.web.cern.ch/ /Projects
[3] High Frequency Structure Simulator (HFSS), version 15.0,
ANSYS Inc., Canonsburg, PA.
[4] L. C. Maier and J. C. Slater, Field Strength Measurements in
Resonant Cavities, Journal ofapplied physics, Volume 23, Number 1,
January 1951
[5] Graeme Burt and Philippe Goudket, Crab cavity system design,
Lancaster University,ASTeC,2005
[6] Study of Multipolar RF Kicks from the Main Deflecting Mode
in Compact Crab Cavity forLHC, J. Barranco Garcia, R. Calaga, R. De
Maria, M. Giovannozzi, A. Grudiev, R. TomsCERN, Geneva,
Switzerland
[7] Sudeshna Seth, Sumit Som, Aditya Mandal, P.R.Raj, S. Saha,
R.K. Bhandari, RF Character-ization and Measurement of a Full Scale
Copper Prototype of 5-Cell Elliptical Shape Super-conducting RF
Linac Cavity, Variable Energy Cyclotron Centre, Kolkata
[8] Stephen Bauman, SPX Copper Prototype Cavity Testing
[9] Sung-Woo Lee, Bead-Pulling Measurement (Multi-cell Cavity
Field Flatness), Oak Ridge Na-tional Laboratory, Department of
Energy
16
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[10] Instruction Manual for the MAIN Bead Pull Measurements
Software, M. Navarro, G. JaumeCERN, Geneva, Switzerland
[11] LabVIEW National Instrument, 2013
[12] HOM identification by bead pulling in the Brookhaven ERL
cavity, H. Hahna, R. Calaga a,b,Puneet Jainc,E. C. Johnsonc, W. Xu,
Daresbury, UK June 25-27, 2012
[13] Hasan Padamsee, Jens Knobloch, Tom Hays, RF
Superconductivity for Accelerators, John Wi-ley and sons inc, 1998,
Chap 2.2
[14] M. Navarro-Tapia, R. Calaga, Bead-Pull Measurements on the
Fundamental Mode of theDouble-Quarter-Wave Crab Cavity, CERN,
Geneva, Switzerland
[15] MATLAB R2014a, The MathWorks Inc., Natick, MA
17
IntroductionRF Model of the DQW cavityResonant Frequencies of
the DQW CavitySimulated Resonant FrequenciesMeasured Resonant
Frequencies
Electromagnetic Field and Force in the DQW CavityTransverse
Momentum and Higher Order Modes IdentificationTransverse Momentum
TheoryTransverse Momentum Calculation from the Simulation
Bead-Pull MeasurementsSlater Perturbation TheoryIntroduction to
the software used to acquire Bead-Pull Measurements The Needle as a
Perturbing ObjectTheory of the Needle as a Perturbing ObjectThe
example of a 30 mm long, 1.2 mm width Needle
The Sphere as a Perturbing ObjectPost Process ProcedurePotential
Vz obtained with the measurements and identification of the Higher
Order Components
ConclusionAcknowledgements
