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DESIGN AND STUDY OF CAVITY QUADRUPOLE MOMENT AND
ENERGY SPREAD MONITOR∗
Q. Wang, Q.Y. Dong, L.T. Huang
Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian, China
Q. Luo†
NSRL, University of Science and Technology of China, Hefei, China
Abstract
A nondestructive method to measure beam energy spreadusing the quadrupole modes within a microwave cavity isproposed. Compared with a button beam position monitor(BBPM) or a stripline beam position monitor (SBPM), thecavity monitor is a narrow band pickup and therefore has better signal-to-noise ratio (SNR) and resolution. In this study,a rectangular cavity monitor is designed. TM220 mode operating at 4.76 GHz in the cavity reflects the quadrupole
moment of the beam. The cavity plans to be installed behinda bending magnet in Dalian Coherent Light Source (DCLS),
an extreme ultraviolet FEL facility. In this position, the beam
has a larger dispersion, which is beneficial to measure theenergy spread. A quadrupole magnet, a fluorescent screen,
and a SBPM with eight electrodes is installed near the cavity
for calibration and comparison. The systematic frameworkand simulation results are also discussed in this paper.

This is a preprint — the final version is published with IOP

INTRODUCTION
Energy spread is an important parameter that reflects
the quality of the beam. Especially in linear accelerators
used as storage rings and collider injectors, ensuring agood energy spread of the beam can improve injection efficiency and maintain beam stability. Therefore, real-timeand high-resolution measurement of energy spread needs tobe achieved.
The most widely used method for energy spread measurement at present is to insert a fluorescent screen or an OTR
after bending magnets. By measuring the generated lightspot, the transverse distribution of the beam can be determined, and the energy spread is capable to be obtained basedon the principle that different energy electrons have different turning radius [1]. However, this intercepting method
is unsuitable for real-time measurement or feedback control. Therefore, a nonintercepting energy spread monitor isrequired.
A feasible nondestructive method is using stripline beam
position monitor (SBPM) [2–4]. SBPM is installed at positions with high value of dispersion function, allowing extraction of the beam’s quadrupole moment to obtain the energyspread. The sensitivity of this method is influenced by theangle of electrodes.
∗ Work supported by the National Key Research and Development Proramof China No.2022YFA1602201, the International Partnership Programof the Chinese Academy of Sciences Grant No. 211134KYSB20200057
and the STCF Key Technology Research Project.
† email address: luoqing@ustc.edu.cn

In order to achieve the energy spread measurement withhigh-resolution, this paper proposes the method of RF resonant cavity. Similar to SBPMs, cavities can also measure
the quadrupole moment, thus obtaining the energy spread.
Compared to SBPMs, cavities have higher signal-to-noise
ratio (SNR), enabling better quadrupole moment resolution,
which can also be confirmed in the position measurement.
Additionally, in the direction of the beam, cavities can realizea more compact layout.
This paper introduces a design of cavity-based quadrupole
moment and energy spread monitor for the Dalian Coherent
Light Source (DCLS), an extreme ultra-violet FEL facility
with a length of 100 m, and provides simulation results toverify the feasibility of the proposed approach.

THEORETICAL BASIS
While passing through the cavity, the bunch will excite
different eigenmodes. The signal amplitude of an eigenmodecan be written as∫ 𝑇
𝑉 = 𝑘𝑛𝑞𝑓 (𝑡) exp(𝑖𝜔𝑡) 𝑑𝑡 ,(1)−𝑇

where 𝑞 is bunch charge, the final integral is related to thelongitudinal distribution of the bunch, and 𝑘 𝑛 is the loss
factor, which is related to the mode type. As shown in Fig. 1,
for a rectangular cavity rotated 45◦ about the axis, its TM220
mode amplitude is proportional to the beam quadrupole
moment [5, 6].

Figure 1: Electric field of TM220 mode in rectangular cavity.
Warmer colors represent stronger electric fields.
𝑉𝑇 𝑀220 ∝ 𝑥 2 − 𝑦 2 + 𝜎𝑥2 − 𝜎𝑦2 ,(2)where 𝑥 and 𝑦 are the beam positions in the two directions.
𝜎𝑥 and 𝜎𝑦 are the beam rms sizes in the two directions.
When the bunch length is short and does not change significantly, the quadrupole moment 𝑥 2 − 𝑦 2 + 𝜎𝑥2 − 𝜎𝑦2 can beobtained by extracting the charge and the amplitude of theTM220 mode.

MOP00405 Longitudinal Diagnostics and Synchronization

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12th Int. Beam Instrum. Conf.
ISBN: 9 7 8 - 3 - 9 5 4 5 0 - 2 3 6 - 3

IBIC2023, Saskatoon, CanadaJACoW PublishingISSN: 2 6 7 3 - 5 3 5 0d o i : 1 0 . 1 8 4 2 9 / J A C o W - I B I C 2 0 2 3 - M O P 0 0 4

Assuming that the vertical dispersion, 𝜂 𝑦 , is zero, therelationship between energy spread and quadrupole moment
can be written as
Δ𝐸 2𝜎𝑥2 − 𝜎𝑦2 = 𝛽 𝑥 𝜖 𝑥 + (𝜂 𝑦) − 𝛽𝑦 𝜖 𝑦 + 𝑔 ,(3)𝐸
where 𝛽 𝑥 and 𝛽 𝑦 are 𝛽 functions, 𝜂 𝑥 is the horizontal dispersion, 𝜖 𝑥 is the horizontal emittance, and 𝑔 is the correction
factor, which can be determined by calibration. The two
directions x and y are interchangeable. Therefore, after excluding the influence of the beam position, the relationshipbetween the TM220 amplitude and the energy spread canbe established.
As indicated in Eq. (3), placing the monitor at a position with high value of dispersion function is beneficial toimprove the sensitivity and resolution of the quadrupole
moment measurement. For a linear accelerator, it is moresuitable to place the monitor after the bending magnets.

Figure 2: Layout diagram.

LAYOUT AND FRAMEWORKThe scheme for measuring beam energy spread using acavity will be carried out in the undulator section of DCLS.
The beam parameters in this section are shown in Table 1.
Table 1: Beam Parameters in the Undulator Section of DCLS
Parameter
Beam energy
ChargeBunch length (rms)Transverse dimension (rms)
𝛽 function (mean)
Normalized emittance

Value
300 MeV
500 pC~0.12 mm~0.1 mm6-29 m
1–2 mm · mrad

The cavity will be installed after the bending magnet atthe end of the undulator section, as shown in Fig. 2. The dispersion at this position is significant, which is advantageousfor energy spread measurement. A fluorescent screen is set
downstream to get information of beam position and provide
online calibration, and the upstream ICT on the bending
magnet can synchronously provide absolute measurementof the beam charge. Additionally, an eight-electrode SBPM
will be installed for energy spread measurement as a reference, which can also provide information of beam charge
and beam position.
The systematic framework is shown in Fig. 3. When thebeam passes through the cavity, the signal generated by theexcited eigenmodes will be coupled out via feedthroughs. Inorder to overcome cable attenuation, the signal first enters anamplifier, then passes through long cables to the accelerator
tunnel and connects to the analog front end for down conversion. Afterwards, it is fed into the electronics for amplitude
extraction and energy spread calculation.

PHYSICAL DESIGNThe TM220 mode of the rectangular cavity can indicate
the magnitude of the quadrupole moment. In order to share

Figure 3: Systematic framework of the cavity quadrupolemoment and energy spread monitor.
the analog front end with other cavity monitors on the DCLS,
the operating frequency of the TM220 mode is set to be
4.76 GHz.
The relationship between the frequency of the TMmn0
mode and the dimensions of the cavity can be expressed as𝑐𝑓𝑇 𝑀220 =2√︂(𝑚 2𝑛) + ( )2 ,𝑎𝑎(4)where 𝑐 is speed of light in vacuum, and a represents thelength of one side of the square cavity. By calculation, we
have 𝑎 = 89.1 mm. In order to reduce the impact on beam,
the longitudinal length 𝑙 of the cavity is set to 10 mm.
The TM220 mode signal is coupled out by feedthroughs.
The feedthroughs are set at the position with the strongest
electric field of the TM220 mode, and are inserted along
the electric field direction for optimal coupling. Four feedthroughs are used to maintain electric field symmetry. The
inner conductors of the feedthroughs should not be inserted
too deeply to avoid distorting the electric field.
The diameter of the beam pipe at the installation location
is 35 mm. The cutoff frequency of it is 5.023 GHz, which
ensures that the TM220 mode signal operating at 4.7 GHz
does not leak out. The beam pipe will affect the frequenciesof all eigenmodes inside the rectangular cavity.
When machining a rectangular cavity, the milling cutter
cannot process the edge of the beam direction to 90◦ , sochamfering must be considered. A square groove is milled
on a cubic metal plate to serve as the cavity. Due to the lengthof the cavity in the 𝑧-direction (beam direction) reaching
10 mm, a milling cutter with a blade length of 15 mm anda blade diameter of 6 mm is chosen to enter along the 𝑧-MOP004
38

05 Longitudinal Diagnostics and Synchronization

This is a preprint — the final version is published with IOP

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12th Int. Beam Instrum. Conf.
ISBN: 9 7 8 - 3 - 9 5 4 5 0 - 2 3 6 - 3

12th Int. Beam Instrum. Conf.
ISBN: 9 7 8 - 3 - 9 5 4 5 0 - 2 3 6 - 3

direction for cutting the groove. This will leave chamferswith a radius of 3 mm at the four corners. This also affects
the frequencies of the eigenmodes. Therefore, numericalcalculations are needed to correct the cavity dimensions
obtained from Eq. (4). CST Microwave Studio [7] is used
for frequency simulation calculation. The final structure anddimensions of the cavity are determined as shown in Fig. 4
and Table 2.

Figure 4: The structure of the rectangular cavity.

Figure 5: The output signal of the cavity when 𝑥 = 10, 𝑦 =0, 𝜎𝑥 = 0, and 𝜎𝑦 = 0. (a) is the time-domain signal. (b) is
the frequency-domain signal.

Table 2: Optimized Dimensions of the Cavity
Parameter

Value𝑎𝑙𝑝𝑦𝑑Chamfer86.2 mm
10 mm
61.8 mm
1.4 mmR3

This is a preprint — the final version is published with IOP

SIMULATIONThe cavity model is established in CST, and virtual beams
are loaded for simulation. The beam simulation is divided
into two cases. In the first case, the transverse sizes of thebeam are zero (𝜎𝑥 = 0, 𝜎𝑦 = 0), and the beam position
is changed. In the second case, the beam is located at theaxis (𝑥 = 0, 𝑦 = 0), and the transverse sizes are varied. The
time-domain and frequency-domain output signals of thecavity are shown in Fig. 5.
The results show that there is a good linear relationship between the amplitude of the TM220 mode and the quadrupolemoments. The quadrupole moment sensitivity of the cavity
monitor is better than 0.03 mm2 in the frequency domain.
This parameter of the eight-electrode SBPM is generally
only on the order of 10−3 . The cavity monitor is more likely
to achieve better measurement resolution.
The simulation results show that the TM220 mode is successfully excited when the quadrupole moment is present.
The variations of quadrupole moments 𝑥 2 − 𝑦 2 + 𝜎𝑥2 − 𝜎𝑦2 andthe amplitude of TM220 modes obtained through simulation
at different beam positions and beam transverse sizes areshown in Fig. 6.

Figure 6: Variation of TM220 mode amplitude with beamquadrupole moment. (a) is 𝑥 varying from 1 to 10 mm when𝜎𝑥 = 0, 𝜎𝑦 = 0, and 𝑦 = 0. (b) is 𝜎𝑦 varying from 0.03 to8.66 mm when 𝑥 = 0, 𝑦 = 0, and 𝜎𝑥 = 1 mm.

MOP00405 Longitudinal Diagnostics and Synchronization

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IBIC2023, Saskatoon, Canada
JACoW Publishing
ISSN: 2 6 7 3 - 5 3 5 0
d o i : 1 0 . 1 8 4 2 9 / J A C o W - I B I C 2 0 2 3 - M O P 0 0 4

IBIC2023, Saskatoon, Canada
JACoW Publishing
ISSN: 2 6 7 3 - 5 3 5 0
d o i : 1 0 . 1 8 4 2 9 / J A C o W - I B I C 2 0 2 3 - M O P 0 0 4

Content from this work may be used under the terms of the CC-BY-4.0 licence (© 2023). Any distribution of this work must maintain attribution to the author(s), title of the work, publisher, and DOI

SUMMARY
In this paper, the design of the cavity quadrupole momentand energy spread monitor is shown, and the simulation
results indicate a good linear relationship between its TM220
mode output signal and the beam quadrupole moment. The
beam experiment will be carried out at DCLS soon. This
study provides a new approach for nondestructive and highresolution measurement of beam energy spread.

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05 Longitudinal Diagnostics and Synchronization

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12th Int. Beam Instrum. Conf.
ISBN: 9 7 8 - 3 - 9 5 4 5 0 - 2 3 6 - 3