DEVELOPMENT OF A NEW BUNCH-LENGTH MONITOR FOR
DETECTION OF ELECTROMAGNETIC FIELDS IN THE MILLIMETREWAVE REGION
T. Suwada*, KEK, Tsukuba, Ibaraki 305-0801, Japan
Abstract
A new nondestructive bunch-length monitor is underdevelopment at the KEKB injector linac. The monitor
detects electromagnetic radiations generated in a wave
zone through a gap in a vacuum pipe when a relativistic
bunched beam passes through the gap. The frequency
spectra of the emitted radiation encompass the microwave
and the millimetre wave regions for the bunched beamwith a picosecond bunch length. A Fabry-Perot resonator
is a useful device to analyze such frequency spectra. The
characteristic features and responses of the Fabry-Perotresonator are numerically analyzed for single-bunchelectron beams of the KEKB. In this report, numericalresults along with the requirements of a suitable detection
system are discussed.

INTRODUCTION
The bunch-length measurement of a bunched beam is
one of the important beam diagnostics for linear
accelerators. We had proposed a new nondestructive
bunch-length monitor in our previous report [1], wherein
the characteristic features of electromagnetic radiations
emitted from a bunched beam passing through a pipe gap
were described in detail along with some numericalresults and the radiation-detection principle. It has also
been clarified on the basis of numerical analysis that thefrequency spectrum of the emitted radiation strongly
depends on the bunch length if the geometrical structureof the gap is suitably fixed.
Although the results indicate that nondestructive
measurements of the frequency spectrum may provide
information on the bunch length, it is difficult to directlyanalyze the radiation spectrum generated by a shortpulsed beam with a picosecond pulse width. A
commercially available spectrum analyzer may not sweep
frequencies at high speeds; hence, it may not be suitable
for analyzing the frequency spectrum over a wide
frequency range with a suitable narrow bandwidth.
Resonant cavities are useful as one of the devices that
can analyze the frequency spectrum at the microwave
region; however, it is difficult to fabricate conventional
resonant cavities in the millimetre wavelength region
because the spatial dimensions are too small to maintainhigh Q-factors.
A Fabry-Perot (FP) resonator [2] is another device thatcan satisfy these requirements under the abovementioned
conditions for radiation detection. The FP resonator is awell-known open resonator that consists of two
conducting reflectors. It can operate at high Q-factors (> 5× 104) over a wide frequency range over the microwave______________________________________________*E-mail: tsuyoshi.suwada@kek.jp

up to the optical wave regions.
Rf resonances occur when electromagnetic radiations
are transmitted through the resonator in which the tworeflectors are separated by a certain distance such that theresonant condition is satisfied. The resonant frequencies
can be tuned mechanically by controlling the distance
between the two reflectors. Nowadays, FP resonators areused for measuring the dielectric properties of solids andliquids, and magnetic resonance in ferrites, electron spinresonance, etc. Thus, the principle and fabrication
technology of an FP resonator that can be used at themicrowave and the millimetre wave frequencies aredscussed [3].
In the following sections, we shall discuss the use of
the FP resonator in the detection system of the newmonitor by carrying out numerical analysis.

DETECTION SYSTEM OF THE BUNCHLENGTH MONITOR
A schematic diagram of the new bunch-length monitor
system is shown in fig. 1. The detection system of themonitor comprises a receiver antenna, an FP resonator, acrystal detector and a digital oscilloscope.

Figure 1: Schematic diagram of the new bunch-length
monitor system.
The receiver antenna (under design) detects the pulsedrf signals generated from the bunched beam that has apulse width of ~10 ps. The geometrical structure of thepipe gap is described in detail elsewhere [1]. The pulsedrf signals are transmitted to the FP resonator through atransmission line. Under the resonant condition of theresonator, only some frequency components of thetransmitted signal are selected. In other words, theresonator functions as a frequency-tunable bandpass filter
with a high Q-factor. The crystal detector rectifies themodulated pulsed-rf signal for which the pulse height isproportional to the signal power. The digital oscilloscopemeasures the pulse height of the rectified (envelope)signal. On the basis of these measurements and efficiently
sweeping the resonant frequencies of the resonator, thepower spectrum of the radiation emission may be
reproduced.

EQUIVALENT-CIRCUIT ANALYSIS OF
THE FABRY-PEROT RESONATOR
The characteristic features of the FP resonator are
analyzed by using an equivalent resonant-circuit model
adapted to an enclosed resonant cavity that is coupled tothe transmission lines through an rf generator. The
equivalent circuit based on a parallel RLC circuit isshown in fig. 2.

Here, VF(ω) (ω is the angular frequency of the currentgenerator) is the forward-travelling voltage at the input
port of the transmission line, which is coupled to thecavity. The cavity voltage is related to the forwardtravelling voltage in the following manner VC = VF + VR,
where VR is the backward-reflecting voltage. In the casewhere β1 = 1 and β2 = 1 (critical couplings), there is noreflected signal, that is, VR = 0.
The solution to the differential equation is obtained bytaking into account the impulse response of the cavity fora short-pulsed rf signal as follows:
VC (! ) =2 / Qex1!0$
1 " (1 / 2Q )2#00LdVF (! )exp["! 0 (t " # ) / 2QL ]dt% sin &! 0 1 " (1 / 2QL )2 (t " # ) ( d# .
')(5)

Here, the integration is carried out with a range from zeroto the bunch width (τ0). In the integration calculation, thetwo-sigma bunch-width is taken into account. The cavity
power PC and the forward-travelling power PF are related
to the cavity voltage VC as follows:
22PC = ! 2G0 VC , PF = !1G0 VC .

Figure 2: Equivalent resonant circuit of the FPresonator.
The equivalent circuit is represented by lumped circuit
elements in parallel, conductance (G0), inductance (L),and capacitance (C). External coupling between the input(output) port and the cavity is achieved through an input(output) conductance G ex1 (Gex2), which is connected inparallel to the equivalent circuit. The input port isconnected to a current generator (Ig), while the output port
is terminated in a matched load. The basic formulae arefound in several textbooks (for example, see ref. [4]).
These circuit elements can be transformed to the resonantcavity parameters in the following manner:! 0 G0 + Gex1 + Gex 2=,(1)
QL
C! 0 G0 ! 0 Gex1 ! 0G=,=,= ex 2 ,(2)Q0C Qex1C Qex 2
C! 02 = 1 / LC.

(3)

Here, ω0 is the resonant angular frequency of theequivalent circuit, Q0, QL, Qex1 and Qex2 are the unloaded,loaded, input external and output external Q-factors,
respectively. The coupling factors (β1 and β2) are related
to these cavity parameters as follows: β1= Q0/Qex1 and β2=Q0/Qex2.
The cavity voltage (VC(ω)) can be derived by solving
the following well-known differential equation," d2 !0 d%! dVF (! )+ ! 20 ' VC (! ) = 2 0. (4)
$ 2+
# dt

QL dt

&Qex1dt(6)

In order to apply this formula to the FP resonator, thefrequency components contributing to all the resonantmodes of the pulsed-rf signal should be linearly added.
The resonances occur not only at the fundamental
resonant frequency (l = 1) but also at higher resonant
frequencies (l > 1). The resonant condition of theresonator is given by! 0l = " cl / d (l=1, 2, 3, ...),(7)
where c is the speed of light; l an integer and d is thedistance between the two reflectors of the resonator.
Therefore, it should be noted that VC calculated from eq.
(5) must be linearly added for all possible resonant modesof the detection system as follows:nVC (! ) = %i =12 / Qex1

! 0i

$1 " (1 / 2Q )2L#00dVF (! )exp["! 0i (t " # ) / 2QL ]dt& sin '! 0i 1 " (1 / 2QL )2 (t " # ) ) d# .
(*(8)

NUMERICAL ANALYSISDetails of the numerical analysis of the power spectraof the radiation emission from the KEKB beams is
described elsewhere [1]. The results were obtained with abunch-length parameter in one sigma (σt = 3 to10 ps) in
the frequency range of 1-100 GHz. By using the results
obtained previously, the impulse responses of the FP
resonator are numerically analyzed from eq. (8). For
simplicity, detection-efficiency reduction (or frequencydependence) of the receiver antenna and high-frequency
transmission losses are not taken into account.
When the pulsed-rf signal with the radiation-emission
power spectrum is transmitted through into the resonator,some frequency components of the signal may resonate at
a discrete resonant frequency depending on theircorresponding Q-factors. They are interferentially built up

in the resonator and the crystal detector outputs the signalenvelope. The rise time and the peak power of the signal
envelope depend on the Q-factor and coupling factors.
For simplicity, the numerical calculations are carried out
by assuming the Q-factors to be constant and the coupling
factors β1 = 1 and β2 = 1 to be in the frequency range of 1100 GHz.
Figure 3 shows a typical example of the calculation
results obtained with the fundamental resonant frequency
alone f0 = 10 GHz as the only parameter of the loaded Qfactor, wherein the peak powers are normalized. It isinferred that the rise time of the output pulse ranges from
~1 to ~4 µs for the input pulsed-rf signal with a pulse
width of σt = 5 ps.

minimizing the high-frequency transmission losses over
the wide frequency range required for this monitor.

Figure 3: Impulse responses of the pulsed-rf signal of
the FP resonator corresponding to the loaded Q-factors.
The bunch length is fixed at σt = 5 ps.
Owing to the characteristic impulse response of theresonator, the rise time of the signal envelope strongly
depends on the Q-factor. Thus, if the optimal Q-factor isaccurately determined, the rise time and the power levelof the signal envelope would depend on the bunch length.
Figure 4 shows the numerical results for the calculated
power (a.u.) as functions of the bunch length, the rise time(TP) and the fundamental resonant frequency (Rf). The QL
values considered are (a) 104 and (b) 5 × 104. From thefigures, it can be inferred that the power is strongly
correlated in three-dimensional spaces, the rise time, thefundamental resonant frequency and the bunch length at ahigher Q-factor. Hence, a higher Q-factor results in abetter resolution in the bunch-length measurement.
The results show that when QL = 5 × 104, the power
difference measured at Rf = 1 GHz for each 1-ps bunchlength difference in one sigma is greater than 50%
depending on the bunch length. This difference issufficiently large and can be easily detected by a digital
oscilloscope with an 8-bit resolution. This implies that forachieving a 1-ps resolution QL should be greater than to 5
× 104 while the detection limit is ~2 ps in one sigma. Sucha high Q-factor can be easily achieved using a suitably
fabricated FP-resonator. However, it should be noted thatthe entire detection system should be constructed with

Figure 4: Calculated power (a.u.) obtained in the caseof (a) QL = 104 and (b) QL = 5 × 104 as functions of therise time of the signal envelope, the bunch length andthe fundamental resonant frequency. The solid lines
over the data points are drawn to guide the eyes.

SUMMARY
The impulse response of the Fabry-Perot resonator,
which is used in the new bunch-length monitor, has been
numerically investigated for the single-bunch electron
beams of the KEKB. The results show that with the FPresonator, a resolution of ~ 1 ps can be achieved in thebunch-length measurements when the loaded Q-factor isgreater than 5 × 104.

REFERENCES[1] T.Suwada and M.Satoh, Procs. of the XXIV International
Linac Conference (LINAC’08), Victoria, British, Columbia,
Canada, 2008; KEK Preprint 2008-35 (2008).
[2] R.E.Collin, Foundations for microwave engineering(McGraw-Hill, New York, 1966), p. 337.
[3] R.N.Clarke and C.B.Rosenberg, J. Phys. E: Sci. Instrum., 15,1982, p. 9.
[4] D.H.Whittum, SLAC-PUB-7802 (1998).