HIGHER-ORDER MULTIPOLE ANALYSIS
OF BEAM-INDUCED ELECTROMAGNETIC FIELDSUSING A STRIPLINE-TYPE BEAM POSITION MONITOR
T. Suwada* , KEK, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan
Abstract

2 MULTIPOLE-MOMENT ANALYSIS

A measurement of multipole moments of anelectromagnetic field generated by single-bunch electron
beams with a pulse width of 10 ps was performed using
stripline-type beam-position monitors at the KEKB
injector linac. A theoretical multipole analysis agrees wellwith the experimental results within the measurement
errors. The experiment enables one to measure thetransverse spatial profile of charged beams; especially, thevariations of the higher-order moments were very
consistent with those measured by wire scanners.

1 INTRODUCTION
The KEK B-Factory (KEKB) project[1] is in progress oftesting CP violation in the decay of B mesons. KEKB is
an asymmetric electron-positron collider comprising 3.5GeV positron and 8-GeV electron rings. The KEKB
injector linac[2] injects single-bunch positron and electron
beams directly into the KEKB rings. The beam charges
are designed to be of 0.64 nC/bunch and 1.3 nC/bunch,with a maximum repetition rate of 50 Hz for the positronand electron beams, respectively. High-current primaryelectron beams (~10 nC/bunch) are required to generatesufficient positrons. Stable control of the beam positions
and energies through beam-position and energy-feedback
systems[3,4] using beam-position monitors (BPMs) isessential in daily operation. On the other hand, the spatial
beam profile is generally measured using fluorescentscreen monitors (SCs)[5] and wire scanners (WSs)[6].
However, these monitors have several drawbacks inmeasuring the beam sizes in real time. The SCs destroy
the beam, and the WSs cannot obtain any pulse-by-pulse
beam sizes, although they can measure precise transverse
beam sizes by detecting high-energy γ-rays generated fromthin wires. Miller et al.[7] showed that a stripline-type
BPM with four electromagnetic pickups could be utilized
as a nonintercepting emittance monitor by theoreticallydeveloping a multipole-moment analysis of anelectromagnetic field generated by a charged beam. They
also experimentally demonstrated that the transverse
emittances of electron beams could be derived from thesecond-order moment of the electromagnetic field. In thisreport the author not only presents a clear experimental
verification of this method based on a similar techniqueused by Miller et al., but also demonstrates that thesecond-order and third-order momentsof theelectromagnetic field can be properly measured using two
kinds of stripline-type BPMs.

*

E-mail address: tsuyoshi.suwada@kek.jp

The electromagnetic field generated by relativistic
charged beams inside a conducting duct is predominantly
boosted in the transverse direction to the beam axis due tothe Lorentz contraction. Thus, for a conducting roundduct, the image charges induced by a line charge can besolved as a boundary problem in which the electrostatic
potential is equal on the duct. The formula for the image
charge density (j) according to a similar treatment byMiller[7] is given byj ( r ,φ , R,θ ) =n∞
I ( r ,φ )  r  cos n(θ −φ ),
+
1
2
∑


2πR 
n =1  R (1)where I is the line charge, (r,φ) and (R,θ) are the polar
coordinates of the line charge and the pickup point on theduct, respectively, and R is the duct radius. If a transverse
distribution of a traveling beam obeys a Gaussianfunction inside the duct, the image charge (J) isformulated by integrating the image charge density withthe weight of the Gaussian distribution inside the ductarea. Assuming that the widths of the charge distribution
are sufficiently small compared to the duct radius,J ( R,θ ) ≈

y0
Ib 
 x0

1+ 2  cos θ + sin θ 
R
2πR 
R


 σ x2 −σ y2 x 2 − y 2 
x y+ 2 
+ 0 2 0  cos 2θ + 2 0 20 sin 2θ 
2
R
R
R



2
2

 x  3(σ x2 −σ y2 ) x 2 − 3 y 2 
y  3(σ x −σ y ) 3 x 02 − y02 ++2 0 + 0 2 0  cos 3θ + 0 
sin 3θ 
2
2
R
R
R
R
R 2 
 R 
+ higher orders},(2)

where Ib is the beam charge, σ x and σ y are the horizontal
and vertical root mean square (rms) half widths of thebeam, respectively, and (x0,y0) are the charge center ofgravity of the beam. The first to fifth expanded terms
correspond to the monopole, dipole (first-order),quadrupole (second-order), sextupole (third-order), andhigher order moments, respectively. Four pickups of the
BPM are normally mounted at the polar coordinates (θ =0,
π/2, π, 3π /2) or at the polar coordinates (θ = π /4, 3π /4,
5π/4, 7π /4). Here, for the sake of simplicity, the former
and latter BPMs are called “90o BPM” [see Figs. 1] and“45o BPM”, respectively. A beam-size measurement isperformed to detect the quadrupole moment (Jquad) for the
90o BPM and the sextupole moment (Jsext) for the 45o
BPM at the least orders. These formulas are defined usingthe following four pickup amplitudes [Vi (i=1-4)] of the
BPM:(V + V ) − (V + V )J quad ≡ 1 3 4 2 4 ,
(3)
∑ Vi
i =1

J sext ≡[(V1 + V2 )−(V3 + V4 )]/ y0 −[(V1 + V4 )−(V2 + V3 )]/ x0
4

. (4)

∑ Vi

i =1

Here, for example, the quadrupole moment for the 90o
BPM is given by σ x2 −σ y2 x 02 − y02 
J quad = 2 +,
2R 2 
 R(5)and the sextupole moment for the 45o BPM is given by
J sext =22
2  6(σ x −σ y ) 4( x 02 − y02 ) 
+

.
R 2
R2
R2


(6)

Normalization by summing the four pickup amplitudes
must cancel out the beam charge variations due to thebeam jitter. It is noted that the sextupole moment cannot
be defined if the beams pass through the center of theBPM, and the absolute beam sizes cannot be
independently obtained by the BPM because the squaredifference (σx2- σy2) of the beam sizes is only related to themultipole moments. This is because the equipotential
lines are invariant under the condition σx2- σy2 = const if
the beam positions do not change. The performance andcharacteristics of the BPMs are described in detailelsewhere[4]; the design parameters are summarized inTable I for both types of BPMs.
V2(φ=π/2)
Stripline-Type Electrodewith a Thickness of t

3 BEAM EXPERIMENT
Beam experiments using single-bunch electron beams
of about 1 nC/bunch were carried out at two locations ofsector B[8]. The nominal beam energy was about 1.7 GeV
at the end of sector B. A set composed of the quadrupole,
the 90o BPM and the WS was used to measure thequadrupole moment, where the beam energy was 1.53
GeV, and another set composed of the quadrupole, the 45o
BPM and the WS was used to measure the sextupole
moment, where the beam energy was 1.7 GeV. The
transverse beam sizes at the locations of these BPMs and
WSs were controlled by changing the quadrupole currents
applied to the quadrupole magnets installed upstream ofthe monitors. Neighboring WSs for each BPM were used
to calibrate the transverse beam sizes at the locations of
the BPMs. Twenty other BPMs located at sectors A and B
monitored the beam positions and charges in order tocontrol the electron beams stably without any beam loss
during the experiment. Beam-orbit feedback using steering
magnets was added instead of that used for the nominal
operation at sector B, particularly to control the beam
positions at the quadrupoles. The beam positions were
controlled within 0.0 ± 0.1 mm at the quadrupole magnet
in order to suppress the dipole moments as much as
possible for the quadrupole-moment measurement, andthey were controlled within 1.0 ± 0.1 mm at thequadrupole magnet in order to keep the term x2 - y2 = 0 for
the sextupole-moment measurement. Figures 2 and 3
show the variations of the beam sizes obtained by the 90o
and 45o BPMs in the quadrupole- and sextupole-momentmeasurements, respectively. The moments were obtainedby averaging twenty BPM data points with an error of one
standard deviation after their beam-position dependence
was corrected by the dipole moments.

4 ANALYSIS
R2
V3(φ=π)V1(φ=0)R1

A beam phase space matrix (Σ1) at location 1 istransformed to that (Σ2) at location 2 by the transport
matrices (M)[9] between locations 1 and 2,tΣ 2 = MΣ1 M ,Angular Width αV4(φ=3π/2)

Figure 1: Schematic cross-sectional drawing of the
stripline-type 90o BPM.
Table 1: Mechanical design parameters of the BPMs.
Parameters
90oBPM 45oBPM
R1 [mm]
13.55
16.0
R2 [mm]18.5
20.0
a [degree]
60
34t [mm]
1.5
1.5
L [mm]
132.5
132.5

(7)

where M t is the transposed matrix of the matrix M .
Assuming that beams are adequately described in linear
optics by two-dimensional ellipsoidal phase spaces
without any x-y couplings, the matrix components, σ11
and σ33, give the squares of the horizontal and vertical rms
half widths of the beam, respectively. They are related byσ 11 = m11σ 11 + 2 m11m12σ 12 + m12σ 22 ,

(8)

(2)
(1)
2 (1)
2 (1)
σ 33
= m33
σ 33 + 2 m33 m34σ 34+ m34
σ 44 ,

(9)

(2)

2

(1)

(1)

2

(1)

where superscripts (1) and (2) of the beam phase spaces
show locations 1 and 2, respectively. If the beam sizes(σ112 and σ332) are measured at location 2, other unknown

2

Experimental data by the BPM
Fitting curve by the WS data

0.07

0.06

0.05

0.04

0.03

J quad + b,

122

J sext + b,

6.5

(10)

7

7.5

8

8.5

9

9.5

Quadrupole Current [A]Figure 2: Variations of the quadrupole moment at the 90o
BPM by changing the quadrupole current. The solid line
shows the fitting curve measured by the WS.

and for the sextupole-moment analysis,
3
Reff
2
2
−
=
σx σy

0.08

(11)

where parameter b is the offset due to the gain imbalance
and the geometrical position errors of the four pickups,
and parameter Reff is the effective radius of the BPM. The
effective radius is required to parametrize in this analysis
because the simple formulation described in section 2 isinsufficient because the BPM radius is not clearly definedfor the real configuration of the electrodes.
Parametrization using an effective radius means that theelectromagnetic-field lines are slightly deformed near theelectrodes, and that the BPM radius viewed from theelectromagnetic fields is shortened due to this effect, that
is, Reff < R2. By using the least-squares fitting procedures,the effective radii and offsets of the BPMs are obtained as
R eff = 17.3 ± 0.78 mm and b = 0.0478 ± 9 x 10-4 mm 2,
respectively, for the quadrupole-moment measurement,
and R eff = 21 ± 8 mm and b = -0.067 ± 0.005 mm 2,
respectively, for the sextupole-moment measurement.
They are in good agreement with the experimental results
obtained by a bench[10] within the permissible error
specifications.

5 CONCLUSIONSA transverse beam-size measurement using two kindsof stripline-type BPMs has been carried out on the basisof a multipole-moment analysis for the electromagnetic
field induced by single-bunch electron beams. The secondorder and third-order moments of the electromagnetic field
were accurately measured using the 90o and 45o BPMs,
respectively. The experimental results show that theanalyzed beam sizes obtained by both types of BPMs

-0.04
-0.05
-0.06

Jsext

2
Reff
2
2
σx − σy =agree with those measured by the WSs within theestimated errors. The result also shows that the multipole
moments must be corrected in this analysis because of thedeformation of the electromagnetic-field lines due to thegeometrically complex configuration of the stripline-type
electrodes, and that the correction can be performed byusing the effective BPM radius which was measured by
the test-bench calibration for the BPMs.

Jquadparameters on the beam phase spaces at location 1 can besolved by a least-squares fitting procedure. Thus, thehorizontal and vertical beam phase spaces at the BPMs
can be transformed by using the beam sizes measured by
the WS and the transport matrices. The transport matrices(M) were calculated according to the optics database of thelinac. Figure 2 shows the variations of the quadrupole
moments for the 90o BPM(solid points) and the WS(solidline) by changing the quadrupole current. Figure 3 shows
the variations of the sextupole moments for the 45o
BPM(solid points) and the WS(solid line) by changing thequadrupole current. Here, for deriving the multipole
moments, first, the square differences of the beam sizes(σ112 - σ332) by the WS are fitted by a parabolic function.
The square differences of the beam sizes at the BPM
locations are calculated using eqs. (8) and (9). The BPM
and WS data are analyzed using the following formulas forthe quadrupole-moment analysis,

-0.07
-0.08

Fitting curve by the WS
Experimental data by the BPM

-0.09
-0.1
6

7

8

9

10

11

12

13

Quadrupole Current [A]Figure 3: Variations of the sextupole moment at the 45o
BPM by changing the quadrupole current. The solid line
shows the fitting curve measured by the WS.

REFERENCES[1] KEKB B-Factory Design Rep., KEK Rep. 95-7 1995.
[2] Design Report on PF Injector Linac Upgrade for KEKB,KEK Rep. 95-18, 1996.
[3] K.Furukawa, et al., ICALEPCS’99, Trieste, 1999, p.248.
[4] T.Suwada, et al., NIM A396 (1997) p.1.
[5] T.Suwada, et al., Proc. the 20th Linear Accelerator
Meeting in Japan, Osaka, 1995, p.245.
[6] N.Iida, et al., PAC’99, New York City, 1999, p.2108.
[7] R.H.Miller, et al., HEAC’83, Fermilab, 1983, p.602.
[8] T. Suwada, Jpn.J.Appl.Phys. Vol.40 Part 1, No.2A(2001) p.890.
[9] K. L.Brown, et al., SLAC-R-0091-REV-2 (1977).
[10] T. Suwada, et al., Proc. the 23rd Linear Accelerator
Meeting in Japan, Tsukuba, 1998, p.175.