NUMERICAL ANALYSIS OF A BEAM-ENERGY-SPREAD MONITOR
USING MULTI-STRIPLINE ELECTRODES
T. Suwada* , High Energy Accelerator Research Organization (KEK)
1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan
Abstract
A beam-energy-spread monitor is under development inorder to control and stabilize the energy spread of highcurrent single-bunch electron beams for producing asufficient number of positrons. The author has proposed a
new monitor using multi-stripline electrodes in order toreinforce this purpose. This report describes the basic
design of the monitor based on a numerical analysis. The
analysis result shows that the resolution of an energyspread measurement is expected to be less than 0.3% fornominal operation conditions.

1 INTRODUCTION
The KEK B-Factory (KEKB) project[1] is progressing inorder to test CP violation in the decay of B mesons.
KEKB is an asymmetric electron-positron collider
comprising 3.5-GeV positron and 8-GeV electron rings.
The KEKB injector linac[2] was upgraded in order toinject single-bunch positron and electron beams directly
into the KEKB rings. The beam charges are designed tobe 0.64 nC/bunch and 1.3 nC/bunch with a maximum
repetition rate of 50 Hz for the positron and electron
beams, respectively. High-current primary electron beams(~10 nC/bunch) are required in order to generate asufficient number of positrons. Since the KEKB is afactory machine, a well-controlled operation of theinjector linac is required for minimizing the tuning timeand maximizing stable operation. Stable control of thebeam positions and energies at several sectors throughout
the beam-position and energy feedback systems[3] areessential in daily operation; however, the energy spread of
the primay electron beams is often enlarged due to a longterm phase drift of high-power and booster klystrons.
Thus, beam diagnostic and monitoring tools are required
to cure the beam energy spread; furthermore those are also
expected to control the longitudinal wakefields of thehigh-current primary electron beams pulse-by-pulse,
especially at the 180-degree arc of the injector linac. A
beam-energy-spread monitor (BESM) with multi-striplineelectrodes is one of the very useful monitoring tools forsatisfying such requirements. The monitor has been newly
designed based on a numerical analysis involving a
multipole analysis of the electromagnetic field generated
by charged beams. The BESM detects the spread of beam
sizes at large dispersion secions by detecting any variationof the electromagnetic field distribution induced in themonitor with the multi-stripline electrodes;thus, the*E-mail address: tsuyoshi.suwada@kek.jp

following method can be applied to not only measuring
the energy spread, but also to measuring the transverse
spatial structure of the beam.

2 MULTIPOLE ANALYSISThe electromagnetic field generated by relativistic
charged beams inside a conducting duct is almost boosted
in the transverse direction to the beam axis due to Lorentz
contraction. This phenomenon shows that if the wall lossof the image charges is negligibly small, theelectromagnetic coupling of the inner surface of the duct
to the beams can be well treated as a two-dimentional
electrostatic potential problem. Thus, any derivation of
the image charges induced on the duct is simply attributed
to the electrostatic potential problem on the transverseplane. For a conducting round duct, the image charges
induced by a line charge can be solved as a boundary
problem in which the electrostatic potential is equal on
the duct[4]. The formula for the image charge density (j)is given by
22I (r, φ )R −rj ( r , φ , R, θ ) =,(1)222πR R + r − 2 rR cos(θ − φ )where (r,φ) and (R,θ) are the polar coordinates of the line
charge and the pickup point on the duct, respectively; R isthe duct radius; and I is the line charge (see fig.1). The
formula can also be represented by expanding in power
series of r/R,∞
I (r, φ )  r
1 + 2 k∑=1   cos k (θ − φ ).
2πR Rkj ( r , φ , R, θ ) =Pickup(R,θ)θ=5π/8θ=7π/8(2)yθ=3π/8Line Charge I(r,φ)θ=π/8Rxθ=9π/8θ=15π/8θ=11π/8θ=13π/8

Conducting
Vacuum Duct

Figure 1: Polar coordinates of a line charge and eightpickups defined in a conducting duct.

If the transverse distribution of a traveling beam isaccording to a Gaussian function inside the duct, the total
image charge (J) is formulated by integrating the image
charge density with a weight of the Gaussian distribution
inside the duct area,J ( R, θ ) =3 NUMERICAL METHOD

Ibj ( r , φ , R, θ ) ×
2πR ∫∫

 − ( x − x 0 ) 2   − ( y − y0 ) 2  dxdy, exp 22 2σ x   2σ y 

exp (3)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) is the charge center of thegravity of the beam. Assuming that the widths of thecharge distribution are sufficiently small compared to theduct radius, σx,σy<<R, integration is easily performed byextending the integration area to x,y→±∞, as follows:J ( R, θ ) ≈Ib y0
 x01 + 2  cos θ + sin θ 2πR R
R σ x2 − σ y2+2 

R2 x 0 y0
 cos 2θ + 2 R 2 sin 2θ  , ( 4 ) x 0 − y0 2+2R2

where the higher orders are neglected; the first to third
expanded terms correspond to the monopole, dipole,quadrupole moments, respectively. The beam sizes arerelated to the quadrupole moment at the least order. Thus,a beam-size measurement can be performed to detect thequadrupole moment (Jquad). The formula normalized to thetotal image charges is given by
J quad ≡obtained, because the square difference (σx2- σy2) of thebeam sizes is only related to the quarupole moment. Thisis because the equipotential lines are invariant under thecondition σx2- σy2=const if the beam positions do notchange.

2π2π00∫ J ( R,θ ) cos 2θdθ / ∫ J ( R,θ )dθ , σ x2 − σ y2 x02 − y02 =+.
22R R(5)(6)

The quadrupole moment is dependent upon both of thebeam positions and the sizes from the above formulas;
however, the position dependence can be corrected byusing the dipole moments. It is also approximately given
by using the n-pickup amplitudes (Vi [i=1-n]),nnk =1k =1

J quad ≈ ∑ Vk cos 2θ k / ∑ Vk .

(7)

Normalization by summing the n-pickup amplitudes
needs to cancel out the beam charge variation due to thebeam jitter and the sensitivity dependence of the pickups
due to the angular width of the electrodes. It is noticed
that the absolute beam sizes can not be independently

A numerical analysis was carried out based on a chargesimulation method[5] in order to calculate the voltages
induced on the electrodes dependent on the beam positionand sizes. Here, only a brief overview of this method isgiven (see ref.6 in detail). The method is based on theboundary element method for analyzing a two-dimentional
electrostatic potential problem. In this method someboundary elements and imaginary charges are introduced inorder to analyze an electrostatic-field system (see fig.2).
All of the conductor surfaces in the system are divided
into many domains, which are called “boundaryelements”; imaginary charges are also arranged near to theboundary elements in a one-to-one manner. The
electrostatic potential of each conductor can be calculated
so as to satisfy the boundary conditions of the system,that is, so as that the calculated equipotential surfaces
correspond to the conductor surfaces by using the linear
superposition of the electrostatic field contributed by allof the real (beam) and imaginary charges. The goodness ofthe numerical analysis was investigated by analyzing theconvergency of the electrostatic potentials calculated foreach electrode in various segmentation numbers, while thetransverse charge distribution obeyed a Gaussian function
with a total horizontal (vertical) width of 6σx(σy), which
was also segmented by using finite number of linecharges with infinite length. The parameter ε /δ was tuned
so as to produce good symmetrical and constantelectrostatic potentials on the electrode surfaces. The
calculation convergency was deduced to be better than 1%.
The obtained parameters are summarized in Table 1.
Outline point on the duct surface
Imaginary charge
εδbtaSegmentation (m)Outline point on
the electrode surface

2π/n
Segmentation (n)Angular width(∆θ)
Duct center

Figure 2: Segmentation of the beam duct and electrodesurface based on the charge-simulation method.

Table 1: Parameters used in the numerical method

REFERENCES

4 APPLICATION TO THE BESM
The number of the electrodes was decided by analyzing
the convergency of σx2- σ y2 using eq. (7), while the total
angular width (2/3×2π) of the electrodes was constant.
Figure 3 shows that good convergency was obtained ateight electrodes. The angular width should be carefully
designed from a point of view in terms of the signal-tonoise ratio and the coupling strength between theelectrodes[7] under the condition that the outer radius (b)and the electrode length (L) are fixed by replacing of thepresent beam-position monitor installed in the arc. An
electrode with an angular width of 15 degree satisfies theabove requirement and, furthermore, it is almost free froma direct strike on the electrode surface of which
synchrotron radiations are generated at the entrance of thearc. The total angular width is 1/3×2 π , viewed from thebeam. For this angular width, the minimum detectablebeam charge is estimated to be ~0.6 nC/bunch, analyzed
based on the data of the present beam-position detection
system[8]. The inner raidus (a) should be determined so as
to comprise a 50-Ω-transmission line. Table 2 shows thedesign paramters of the BESM. Based on the designparameters, Fig. 4 shows that using the design parameters
the variation of the quadrupole moment to the energy
spread (σE/E) expected at the 180-degree arc, where thehorizontal dispersion is ηx=7.2 mm/% and the beamenergy is 1.7 GeV, with various vertical beam sizes. The
beam-position dependence was neglected in this analysis
for the sake of simplicity. Assuming that the noise levelof the pickup amplitudes is expected to be less than 0.5%,
which determines the minimum detectable charge, thedetection error of the energy spread is expected to be about
0.3%.
Table 2: Design parameters of the BESM
Inner radius a [mm]Outer radius b [mm]Angular width ∆θ [deg.]
Electrode thickness t [mm]Stripline length L [mm]Number of electrodes

20.6
23.4
15
1.5
132.5
85 CONCLUSIONS
A new beam-energy-spread monitor with eight-striplineelectrodes was designed on the basis of the multipole
anaysis for charged beams. Eight electrodes with eachangular width of 15 degree are required to detect the precise
quadrupole moment for a beam charge greater than 0.6
nC/bunch. The rms resolution of the energy-spreadmeasurement is expected to be 0.3% for nominal
operation conditions. This is small enough to stablytransport high-current primary electron beams at the 180degree arc in the injector linac.

[1] KEKB B-Factory Design Report, KEK Report 95-7
(1995).
[2] Design Report on PF Injector Linac Upgrade for
KEKB, KEK Report 95-18, 1996.
[3] K.Furukawa, et al., Proc. the Int’l Conf. on
Accelerator and Large Experimental Physics Control
Systems (ICALEPCS’99), Trieste, Italy, 1999,
pp.248-250; KEK-Preprint 99-139 (1999).
[4] J.D. Jackson, Classical Electrodynamics-2nd ed. (JohnWiley & Sons, New York,1975), p.54.
[5] T. Kohno and T. Takuma, Numerical Calculation
Method of Electric Field (CORONA, Tokyo, 1980).
[6] T. Suwada, Proc. the 11th Symposium on Accelerator
Science and Technology, Akou-gun, Hyougo, Japan,
1997, pp.264-266; KEK Preprint 97-183 (1997).
[7] T. Suwada, Proc. the 10th Symposium on Accelerator
Science and Technology, Hitachinaka, Japan, 1995,
pp.269-271; KEK Preprint 95-136 (1995).
[8] T. Suwada, N.Kamikubota, H.Fukuma, N.Akasaka,and H.Kobayashi, Nucl. Instrum. & Methods. A 440
No.2 February (2000) pp.307-319.
109σx2-σy2 [mm2]4240
0.6~0.7

8765051015202530

Number of Electrodes
Figure 3: Variation of the square difference of the beam
sizes depending upon the number of electrodes. The
horizontal and vertical beam sizes are assumed to be σy=3
and σy=1 mm, respectively.
0.3
σy=0.1mm
σy=2.1mm

0.2

σy=4.1mm

Jquad

Radial segementation mAzimuthal segmentation nδ/ε0.1
0-0.1
-0.2
00.51

Energy Spread (σE/E) [%]1.5

Figure 4: Variation of the quadrupole moment to theenergy spread expected at the 180-degree arc.