NUMERICAL STUDY OF A NEW BUNCH LENGTH MONITORUTILIZING A DETECTION OF ELECTROMAGNETIC FIELDS INMILLIMETER-WAVE REGION
T. Suwada* and M. Satoh, KEK, Tsukuba, Ibaraki 305-0801, Japan
Abstract
A new nondestructive bunch-length monitor has been
numerically investigated in high-energy electron/positronlinear accelerators. The monitor detects electromagnetic
radiation emission generated in wave zone through a gapof a vacuum pipe when a relativistic bunched beam passes
across the pipe gap. The frequency spectrum of theradiation emission spreads over microwave to amillimeter-wave region for the bunched beam with abunch length of a picosecond region. The frequency
spectrum strongly depends on the bunch length if thegeometrical structure of the pipe gap is suitably fixed.
The detection principle of the new bunch-length monitor
and some numerical analysis results applied to a singlebunch electron beam at the KEKB injector linac aredescribed in this report.

INTRODUCTION
A bunch-length measurement of a bunched beam is oneof important beam diagnostics in linear accelerators.
There are several methods to measure the bunch length[1]. One is a standard method with a streak camera. It
measures a pulse width of electromagnetic radiations intime domain, such as Cherenkov radiation and opticaltransition radiation generated through electromagnetic
interactions between materials and the bunched beam.
This method comes in useful for the bunch-length
measurement in a picosecond region. Recently, a faststreak camera can makes the bunch-length measurement
possible with a time resolution of sub-picoseconds [2].
These are well-known methods to measure the bunch
length in a pico (or sub-pico) second region, and however,they have several drawbacks because they are basically
based on destructive diagnostics, and they need expensive
instrumentations with complicated systems.
Here, we propose an alternative new method tomeasure the bunch length in frequency domain bynondestructively detecting electromagnetic radiation
emissions. They are shaken off the bunched beam bydiffraction (or scattering) at a pipe gap of a beam line.
This method requires only a pipe gap normally vacuumsealed with a ceramic. It does not require any otherdevices in the beam line.
There are many pipe gaps vacuum-sealed with aconventional ceramic along the beam line of the KEKB
injector linac in order to install other beam diagnostic
devices. Such pipe gaps are available for the bunch-length
measurement based on this new method at any locationsof the entire beam line
In this report, the basic principle on this new method______________________________________________*E-mail address: tsuyoshi.suwada@kek.jp.

are presented. The characteristics of the electromagnetic
radiation emitted from the pipe gap are also given, andsome numerical results analyzed for a single-bunchelectron beam at the KEKB injector linac are summarized.

CHARACTERISTICS OF THE
RADIATION EMISSION
When a charged particle with a relativistic energy
passes through a vacuum pipe in linear accelerators,
image charges with the inverse sign are induced on theinner surface of the pipe, and they also flow
simultaneously through it. Electromagnetic fields (or selffields) induced between the charged particle and theimage charges are relativistically boosted in thelongitudinal direction. When such pancake-like self-fields
pass across the pipe gap, a part of the self-fields are
emitted out of the pipe gap in wave zone because the selffields must be met with the electromagnetic boundary
conditions under which the charged particle is surroundedwith both the pipe gap and the metal pipe.
The characteristics of the electromagnetic radiation
emission from the charged particle are determined bydepending on the geometrical structure of the pipe gap(see fig. 1(a)) if the charge and the energy of the charged
particle are constant.

Figure 1: (a) Geometrical structure of the vacuum pipe
and the pipe gap and (b) a schematic drawing of theelectromagnetic radiation emission through the pipegap.

On the other hand, when a bunched beam goes acrossthe pipe gap, the characteristics of the electromagnetic
radiation emission may be modified depending not only
on the geometrical structure of the pipe gap, but also thegeometrical charge distributions of the bunched beam.
Based on the Bethe’s diffraction theory by small holes[3], a flow of electromagnetic radiations through suchsmall holes may induce electric and magnetic dipole
moments on the surface of the small holes. The vector
directions of these dipoles correspond to those of the field
vectors induced on the surface of the closed holes. The
time oscillation of the induced dipoles can emit theelectromagnetic radiations in wave zone out of the smallholes. A schematic drawing on the radiation emission
based on the dipole oscillations is shown in fig. 1(b). Oneof the characteristic features of the radiation emission isthat the emission intensity becomes a maximum(minimum) in the direction of the beam axis (r axis).
On the other hand, an emission of rf waves out of afinite open waveguide with an rf oscillator at one end wasexactly investigated by Weinstein [4]. Based on the
Weinstein’s theory, when the relativistic bunch passes
across the pipe gap, a large part of the self-fields transmits
across the pipe gap, and however, a part of the self-fields
reflects at both open ends of the pipe gap to be met withthe electromagnetic boundary conditions. Thus, themultiple reflections of the self-fields occur between boththe open ends, and a part of the interfered self-fields isemitted into free space out of the pipe gap due todiffraction (or scattering) (see fig. 1(b)).
It is generally very difficult to exactly solve theseelectromagnetic radiation emission problems. Palumbo[5] and Kheifets [6] give an analytical formula on anenergy loss of the charged particle generated at such apipe gap by approximately working out these problems
based on the Weinstein’s theory. The energy loss (orradiation power) per a solid angle d Ω is given bydW (! ) # h$ (q / e)2 sin 2 ! % J 02 (ka sin ! )=)d"& (1 ' $ cos! )2 I 02 (ka / $( )L' (* / v) 1 ' $L' (k cos! ) 1 ' cos!

e' jkl (1' $ cos ! )/ $ ' j

L+ (* / v) 1 + $L+ (k cos! ) 1 + cos!

on the basis of the Wiener-Hopf factorization method [4]by
L+ (! ) = j" + (! ) ! + k(2)L# (! ) = # j k # ! / " # (! ).

(3)

The functions ! ± are specified [7] by+- " a ka ln %$#1 J 0 (#1 )H 0(1) (#1 ) '(! ± (" ) = [2#I 0 (#)K 0 (#)]±1/2 exp , * dt &" 2a2 ) t 2-. j$ 0/ln [ 2# 2 I 0 (# 2 )K 0 (# 2 )] 0
"a+jPV * dt(4)
1.
$t 2 ) " 2a2ka2(1)Here, H 0 is the Hankel function of the first kind, K 0 is
the modified Bessel function of the second kind, and theother parameters are given by! 2 = a 2 (" 2 # k 2 ), !12 = k 2 a 2 # t 2 , ! 22 = t 2 # k 2 a 2 . (5)It should be noted that Eq. (1) gives the relativistic
radiation emission formula in frequency domain.

NUMERICAL STUDIES OF THE
RADIATION EMISSION
In general, the radiation emission spectra from thebunched beam may be derived from the convolution of Eq.
(1) with the spatial charge distributions. It is easy tonumerically solve the integrations rather than to calculatethem analytically. Assuming the longitudinal charge
distribution of the bunched beam being expressed by aGaussian function, several number of macro particles arearranged discretely at equal distance along thelongitudinal direction.
Therefore, the radiation emission spectra can be
numerically investigated by summing the imaginary
amplitude contributions from all the macro particles
following Eq. (1). Here, the number of macro particles
was assumed to be 6 arranged within the total width of±3σt (σt, one-sigma bunch length) of the longitudinalcharge distribution. The gap length and the pipe diameter
were fixed to be 2l=12mm and 2a=60mmφ, respectively.

2e jkl (1' $ cos ! )/ $ (1).

Here, α the fine structure constant, h the reduced plankconstant, q the particle charge, e the electron charge
magnitude, β=v/c, v the velocity of the charged particle, θthe spatial emission angle, ω the angular frequency of theelectromagnetic radiation, k= ω /c, c the velocity of light,
γ the relativistic Lorentz factor, J0 the Bessel function of
the first kind, I0 the modified Bessel function of the firstkind. The other parameters are indicated in fig. 1(a). It
should be noted that the higher-order multiple-reflection
terms are neglected in Eq. (1) due to the smaller
contribution. The right-hand side of Eq. (1) shows that theradiation spectrum mainly comprises two terms for which
the first (second) term makes a contribution from theradiation emission generated at the left (right)-side end of
the pipe gap (see fig. 1(b)). The functions L± are derived

Figure 2: Radiation emission spectra as functions of
the frequency and the bunch length.

The results at the beam energy of 1 GeV are shown infig. 2. Here, for the sake of simplicity, the rf loss at aceramic seal of the pipe gap and the energy spread of thesingle-bunch beam were neglected. The radiation
emission angle was fixed to be θ=π/2. As shown in fig. 2,
the radiation emission powers are larger at the millimeterwave region over the microwave region for the shorterbunch length in a picosecond region. The spectra also
indicate periodical dip structures due to diffraction (orscattering). The frequency and the depth at the dip
location are determined by the geometrical structure ofthe pipe gap. The solid curves in the figure are envelopes
without the dip structures. The result (blue line, σt=5 ps)corresponds to that for the KEKB injection beam since
the bunch length is ~10 ps in FHWM. The difference ofeach emission spectrum is so remarkable in millimeterwave region (>~20 GHz) even for the bunch length
difference of a few picoseconds.
The radiation emission spectra based on the envelopes
depending on the beam energy are shown in fig. 3 while
the bunch length is fixed to be σt=5 ps.

Assuming that the frequency resolution in the criticalfrequency measurement is attainable within δf ~1 GHzover the millimetre-wave region (f=10-30GHz), theresolution of δσt~1 ps may be attainable enough although
it is not easy to measure the precise beam energy with thismethod at highly relativistic energies.

Figure 4: Relations (mapping curves) of the criticalfrequency to the radiation power as functions of thebunch length and the beam energy.
An example of the detection system is simply presented
here. It comprises a receiver antenna, a tunable band-passfilter with a narrow bandwidth, and a peak-power meter.
The millimetre waves are directly detected with thereceiver antenna installed near the pipe gap. The power
spectra are measured through the band-pass filter bysweeping the centre frequency over the frequency rangeof 5-40 GHz. The peak power obtained is ~54 dBm/nC, if
the bandwidth of the detectable frequency is Δf~1 MHz. It
seems to be possible by using a commercially available
peak-power meter at millimetre-wave region.
Figure 3: Radiation emission spectra as functions of
the frequency and the beam energy.
It should be noted that the radiation power spectra arenormalized with each power at the frequency of 1 GHz. It
is understood that the energy dependence of the criticalfrequency is not very large at highly relativistic energies
while the critical frequency giving the maximal power of
each distribution increases with the increase of the beamenergy. This may come from the relativistic Lorentz boostof the self-fields.
Figure 4 shows the variations in the normalizedradiation power spectra as functions of the criticalfrequency, the bunch length, and the beam energy. The
results indicate that the radiation power at the criticalfrequency increases with the increase of the beam energy,
and the critical frequency largely shifts towards the higher
frequency region for the shorter bunch length. It is ofgreat benefit to this method for the shorter bunch-length
measurement. The results also give the mapping curves asfunctions of the bunch length and the beam energy.

SUMMARY
A new nondestructive bunch-length monitor has been
numerically investigated for a single-bunch beam of theKEKB injector linac. The numerical results show that thebunch length with a resolution of δσt~1 ps may bemeasured based on this new method with a suitabledetection system.

REFERENCES[1] A.W.Chao and M.Tigner (eds.), Handbook of Accelerator
Physics and Engineering (World Scientific, Singapore,1999), p.547.
[2] http://jp.hamamatsu.com/en/index.html.
[3] H.A.Bethe, Phys. Rev. 66, 163 (1944).
[4] L.A.Weinstein, Theory of Diffraction and the Factorization
Method (The Golem Press, Colorado, 1969).
[5] L.Palumbo, CERN LEP-TH/84-4.
[6] S.Kheifets, et al., CERN LEP-TH/85-23.
[7] B.W.Zotter and S.A.Kheifets, Impedance and Wakes in
High-Energy Particle Accelerators (World Scientific,Singapore, 1998), p.267.