Preamble

Xuerui Haoa,b, Xiao Lia,b, Junyu Zhua,b, Chunlin Zhanga,b, Wei Longa,b, Shengyi Chena,b, Yang Liua,b, Shenghua Liua,b, Bin Wua,b

aSpallation Neutron Source Science Center, Dongguan, Guangdong 523803, P.R. China
bInstitute of High Energy Physics, 19B Yuquan Road, Beijing 100049, P.R. China

Abstract

The TM020 mode damped cavity enables compact accelerator designs but suffers from severe electromagnetic field leakage caused by coupling-slot misalignment and structural asymmetry. We propose an integrated narrowband filter based on RLC circuit design that selectively suppresses TM020 leakage while preserving higher-order mode damping. The filter reduces electromagnetic field leakage to below 0.6%—more than an order of magnitude improvement over unfiltered cavities (>4%)—and quadruples the positional tolerance from ±0.2 mm to ±0.8 mm. Impedance spectra confirm minimal impact on higher-order mode damping. Under 20-kW operation, thermo-mechanical analysis demonstrates exceptional stability: peak temperature rise ≤21.3 K, frequency detuning ≤0.182 MHz, and leakage increase to only 0.65% (still six times lower than unfiltered baselines). This design resolves the critical trade-off between leakage suppression and operational robustness in next-generation synchrotrons.

Keywords: TM020 mode damped cavity, Electromagnetic field leakage, Integrated narrowband filter, RLC circuit design, Thermo-mechanical stability

1. Introduction

Fourth-generation synchrotron light sources demand ultralow emittance and diffraction-limited performance, which intensifies beam dynamics challenges. Although multi-bend achromat lattices achieve emittances below 100 pm·rad, the resulting high electron density exacerbates intrabeam scattering, degrading both emittance and Touschek lifetime [1, 2]. Double-radiofrequency (RF) systems that employ harmonic cavities to lengthen bunches provide a critical solution to this problem [3].

However, conventional normal-conducting (NC) harmonic cavities operating in TM010 modes suffer from inherent limitations: high shunt impedance per unit length (R/Q) and susceptibility to transient beam-loading during bunch gap modulation compromise voltage stability, particularly at 1.5 GHz [4, 5]. While superconducting cavities mitigate some of these issues, their cryogenic complexity and cost motivate the development of innovative NC alternatives [6].

The TM020 mode damped cavity emerges as a pivotal NC solution that addresses transient beam-loading, coupled-bunch instabilities (CBIs), and compactness requirements. Unlike TM010 mode cavities, the TM020 mode features a unique electromagnetic (EM) field distribution with a radial magnetic node that enables compact embedded damping structures (e.g., coaxial slots, rod-type couplers) to suppress parasitic modes while preserving high Q0 and low R/Q for the accelerating mode [7, 8, 9]. This characteristic is essential for voltage stability in next-generation storage rings [4].

The viability of TM020 mode cavities has been demonstrated across multiple projects. Pioneering work by Ego et al. in the SPring-8-II project demonstrated feasibility, achieving a shunt impedance of 6.8 MΩ and effective parasitic mode damping through circumferential slots lined with microwave absorbers [10]. Subsequent KEK studies optimized cavity geometries, revealing that axial symmetry and precise damping structure placement (e.g., ferrite-loaded slots) are crucial for minimizing TM020 mode power leakage [7].

Low-power measurements confirmed resonant frequency agreement within 270 kHz of simulations and validated the effectiveness of threefold-symmetric tuners in reducing Q0 degradation [11]. High-power prototypes tested at SPring-8 demonstrated stable operation at 120 kW with 900 kV accelerating voltage, though they highlighted challenges in thermal management and frequency stability under beam loading [12]. The Thailand New Light Source (SPS-II) and KEK Light Source (KEK-LS) projects further advanced TM020 applications, achieving shunt impedances of 8.3 MΩ (main cavity) and 2.45 MΩ (harmonic cavity) while suppressing parasitic modes to loaded Q values below 1,000 [7, 13].

A critical challenge persists: EM field leakage from the accelerating TM020 mode into damping structures. As shown in Fig. 1a [FIGURE:1], even minor coupling-slot misalignment (∆Rh = ±0.2 mm) or symmetry-breaking structures (e.g., couplers) shift the sensitive magnetic node position (Fig. 1b), increasing leakage to over 4% (Fig. 1c) [14]. This parasitic power absorption risks exceeding absorber thermal limits (e.g., ferrite) and degrades operational stability [14].

Existing mitigation strategies involve significant trade-offs. Choke structures can recover Q0 (∼30,500) and reduce ferrite power density near safety limits, but they increase cavity length by 80 mm and impair damping efficiency (Fig. 2 [FIGURE:2]) [15]. Coupler compensation topologies reduce leakage to 0.18% but fail to address beam-induced node shifts or transient heating [16]. Achieving a balance between deep leakage suppression, unimpeded higher-order mode (HOM)/lower-order mode (LOM) damping, and mechanical compactness remains an unresolved challenge.

In this Letter, we present a novel approach that resolves this long-standing conflict by integrating a frequency-selective narrowband filter directly into the damping structure. This filter functions as a highly reflective mirror at the precise TM020 accelerating mode frequency (1.5 GHz) while remaining transparent to the spectrum of other LOMs/HOMs that require suppression. Our results demonstrate that this design dramatically reduces EM leakage by more than an order of magnitude, from a baseline exceeding 4% to less than 0.6%. Critically, this is achieved with negligible impact on the impedance of other modes, preserving the cavity's essential damping characteristics across the frequency spectrum. Furthermore, the filter enhances system robustness by quadrupling the positional tolerance for the coupling slot. This integrated filter design overcomes a key obstacle for next-generation high-performance accelerator cavities, ensuring both structural integrity and stable operation under high power.

Fig. 1. Analysis of EM field leakage problems in the TM020 mode. (a) TM020 mode damped cavity structure, where Rh is the distance from the coupling slot center to the axis. (b) Introduction of couplers and other structures breaks the rotational symmetry of the EM field. In cylindrical coordinates, R is the radial coordinate of point (R, 0, 0), while R0 represents the cavity radius. (c) Effect of coupling slot position offset on EM field leakage, where ∆Rh represents the coupling slot position offset.

Fig. 2. Choke structure scheme analysis. (a) Choke structure. (b) Transmission efficiency analysis, where the green curve shows coaxial structure transmission efficiency, the blue curve shows choke structure transmission efficiency, the black square indicates the RF cavity monopole mode, and the red dot indicates the dipole mode.

2. Narrowband Filter Design

This section details the design methodology for the coaxial narrowband filter, beginning with the foundational principles of an RLC equivalent circuit. The process involves deriving the fundamental resonance characteristics from an analytical model, translating the lumped-element parameters into a distributed microwave structure, and performing a multi-objective optimization that considers both electromagnetic performance and practical mechanical constraints.

2.1. RLC Circuit Foundations and Transmission Analysis

The model, shown in Fig. 3a [FIGURE:3], utilizes a hybrid topology combining distributed transmission line sections with a lumped LC tank circuit. Signal transmission through this composite structure is systematically evaluated using ABCD-matrix formulation:

$$
\begin{bmatrix}
A & B \
C & D
\end{bmatrix}
=
\begin{bmatrix}
\cos \beta l_1 & jZ_0 \sin \beta l_1 \
jY_0 \sin \beta l_1 & \cos \beta l_1
\end{bmatrix}
\begin{bmatrix}
1 & 0 \
j\omega C_1 + \frac{1}{j\omega L_1} & 1
\end{bmatrix}
\begin{bmatrix}
\cos \beta l_2 & jZ_0 \sin \beta l_2 \
jY_0 \sin \beta l_2 & \cos \beta l_2
\end{bmatrix}
$$

where β = ω/c is the phase constant and Z0 = 1/Y0 = 50 Ω is the characteristic impedance of the transmission line sections. The forward transmission coefficient S21 is derived by converting the ABCD-matrix:

$$
S_{21} = \frac{2}{A + B/Z_0 + CZ_0 + D}
$$

This model predicts two distinct operational regimes, as illustrated by the RLC curve in Fig. 3d. First, the LC tank creates a resonant suppression band centered at frequency $f_0 = 1/(2\pi\sqrt{L_1C_1})$, producing a sharp rejection notch with over 30 dB attenuation. Second, the transmission lines form a primary passband that acts as a broadband impedance transformer, allowing over 90% power transmission for HOM frequencies. To achieve the desired protection for the TM020 mode at f0 = 1.5 GHz, the inductance-capacitance product is constrained to $L_1C_1 = (2\pi f_0)^{-2} \approx 1.13 \times 10^{-20}$ H·F. The analysis also shows that the 3-dB bandwidth Δf is inversely proportional to the quality factor Q and can be narrowed by increasing capacitance C1 while maintaining the L1C1 product constant.

Fig. 3. Narrowband filter microwave structure design using RLC equivalent circuit. (a) Equivalent circuit model. (b) Equivalent microwave structure. (c) Revised narrowband filtering structure. (d) Transmission characteristic curves: red curve (RLC) shows S21 for equivalent circuit; black curve (Annular) shows S21 for annular microwave structure; blue and green curves (Polygon) show S21 for revised microwave structures with εr = 30 and εr = 40, respectively. (e)-(g) Optimization curves for key parameters (Lc, εr, dc) in revised narrowband filtering structure.

2.2. Realization as a Coaxial Microwave Structure

The microwave implementation (Fig. 3b) employs a ceramic-loaded coaxial structure that transforms lumped elements into distributed parameters. The annular ceramic between inner (radius ri) and outer (radius ro) conductors (gap d0 = ro − ri) provides distributed capacitance (Cdist), while copper rings contribute inductance (Ldist). From transmission line theory:

$$
L_{\text{dist}} = \frac{\mu_r \mu_0}{2\pi} \ln\left(\frac{r_i + d_c + d_1}{r_i + d_c}\right) l_c
$$

$$
C_{\text{dist}} = \frac{2\pi \varepsilon_r \varepsilon_0}{\ln\left(\frac{r_i + d_c + d_1}{r_i + d_c}\right)} l_c
$$

where c = 1/√(μ0ε0) is the speed of light in vacuum, εr is the relative permittivity of the ceramic, lc is the axial length of the ceramic ring, dc is the radial thickness of the ceramic ring, and d1 is the radial thickness of the copper ring. μr is the relative permeability of the material, with copper exhibiting unity relative permeability (μr = 1) due to its non-magnetic properties.

The RLC equivalent circuit assumes lumped parameters (L ≪ λ/4), whereas actual microwave structures exhibit distributed EM fields. In particular, fringing fields and HOM coupling at the interface between the copper ring and inner conductor cause theoretical formulas to underestimate the actual inductance and capacitance. This underestimation is addressed through an enhanced lumped-element modeling approach that incorporates corrections for distributed capacitance and inductance, compensates for edge effects, and establishes a fringing field correction factor:

$$
L'{\text{dist}} \cdot C'}} = k \cdot \frac{\mu_r \mu_0 \varepsilon_r \varepsilon_0 l_c^2}{\alpha \lambda + \beta \tan\delta
$$

where α and β are coefficients calibrated through simulation, and tan(δ) is the dielectric loss tangent of the ceramic ring.

To achieve the TM020 mode resonant frequency of 1.5 GHz, the parameters ri, dc, d1, lc, and εr are adjusted to satisfy the L'dist·C'dist product of 1.13 × 10⁻²⁰ H·F. The inner conductor radius ri is determined by the coupling slot position. The copper ring is fabricated from highly elastic beryllium copper C17200 (Poisson's ratio ∼0.31, electrical conductivity ∼25% IACS), which provides sufficient equivalent inductance while securing the ceramic ring. Given the coupling slot width d0 of only 8 mm, the copper ring thickness d1 is set to 1 mm.

As shown in the corrected equation, for a fixed narrowband center frequency, εr and lc² are inversely proportional—higher εr permits shorter ceramic ring length. Considering the compactness requirements of the TM020 mode damping cavity, the coupling slot length is limited to 40 mm. To ensure adequate installation space for the narrowband filter, the ceramic ring length lc must be constrained to 25 mm, which requires increased ceramic permittivity. Additionally, due to the poor thermal conductivity of ceramic materials, the ceramic ring thickness dc must be limited to 1 mm.

The model shown in Fig. 3b was established and optimized using CST Microwave Studio. With parameters ri = 123 mm, dc = 0.8 mm, d1 = 1 mm, lc = 22.2 mm, and εr = 30, the narrowband filter achieves a center frequency of 1.5 GHz, with the S21 transmission curve shown in Fig. 3d.

Further analysis of Fig. 3d reveals that the S21 transmission curve for the RLC equivalent circuit differs from that of the actual microwave structure. The microwave structure's transmission curve periodically exhibits multiple notches because the ceramic ring and copper ring design can excite HOMs (e.g., third/fifth harmonics) in addition to the TM020 mode frequency (1.5 GHz). The 3.0 GHz operational frequency lies below the beam pipe's TM01 mode cutoff (3.645 GHz for 63 mm diameter) but above the TE11 cutoff (2.791 GHz), requiring careful avoidance of HOM resonant frequencies at this spectral position to ensure unimpeded propagation toward absorbing components.

2.3. Fabrication-Oriented Optimization

To address fabrication challenges, high costs, and structural vulnerabilities associated with solid annular ceramic rings, the design was revised to a polygonal configuration using modular square ceramic elements, as shown in Fig. 3c. This practical approach facilitates manufacturing and assembly. Ceramic pieces are secured within recesses machined into the inner conductor and further stabilized by a corresponding polygonal inner surface on the copper clamp ring. The ceramic piece specifications are: width l0 = 15 mm, thickness dc = 0.8 mm, insertion depth into inner conductor dd = 0.4 mm, and spacing between pieces of 6 mm.

The ceramic ring length lc was subsequently optimized using CST Microwave Studio. To maintain the filter length under the 25 mm constraint, a trade-off between lc and relative permittivity εr was analyzed, as higher εr permits shorter length. For example, increasing εr from 30 to 40 reduces the required lc from 22.5 mm to 19 mm. However, simulations revealed that εr = 40 introduced additional spurious responses in the passband that could interfere with HOM transmission and damping. Therefore, εr = 30 was selected as optimal.

Finally, a parametric scan of key structural dimensions (lc, εr, dc) assessed manufacturing tolerances. The analysis indicated that ceramic ring length lc is the most suitable parameter for fine-tuning during fabrication to precisely correct the filter's center frequency, with machining allowance reserved for this purpose.

3. Analysis of the TM020 Mode Damped Cavity with Narrowband Filter

This section evaluates the performance of the TM020 mode damped cavity after integration of the optimized narrowband filter. The analysis demonstrates the filter's effectiveness in suppressing TM020 mode leakage, confirms its selectivity by assessing impact on other modes, and validates the design's robustness under high-power operational conditions.

3.1. Impact of Narrowband Filter Structures on TM020 mode

Integration of the narrowband filter fundamentally suppresses EM field leakage from the TM020 accelerating mode. As illustrated in Fig. 4a [FIGURE:4], the filter is positioned adjacent to the main cavity to ensure strong coupling. The performance improvement is dramatic: without the filter, the cavity is highly sensitive to manufacturing tolerances, with minimum achievable leakage exceeding 4% and coupling slot deviations of just ±0.2 mm increasing leakage to 6% (Fig. 4b). With the filter loaded, the minimum leakage rate is reduced by nearly an order of magnitude to below 0.6%. Furthermore, the filter enhances robustness against mechanical imperfections, maintaining leakage below 2% even for substantial coupling slot offsets of ±0.8 mm—a fourfold increase in tolerable positional tolerance. Leakage decreases monotonically as the filter is positioned closer to the main cavity wall (Fig. 4c). Table 1 [TABLE:1] summarizes the high-frequency RF parameters under optimal conditions, providing definitive quantitative evidence of the filter's efficacy. The filter reduces the TM020 mode leakage rate (Pm/Pc) from 4.2% to 0.6% while causing negligible changes to cavity resonant frequency and R/Q, establishing it as an essential component for achieving deep leakage suppression and insensitivity to coupling perturbations.

Fig. 4. EM field leakage analysis in TM020 mode damped cavity with narrowband filter structure. (a) Structural diagram of TM020 mode damped cavity loaded with narrowband filter. (b) Impact of coupling slot position offset on EM field leakage with and without narrowband filter. (c) Influence of narrowband filter position on EM field leakage.

Table 1: RF parameters of the TM020 mode damped cavity
without filter with filter
freq.[MHz] 3.58×104 3.60×104
R/Q[Ω] Ra[MΩ] Pm/Pc[%]

3.2. Impact on LOMs and HOMs

A critical requirement is that the filter must suppress the TM020 mode without compromising the cavity's essential damping of other modes. The filter's selectivity is demonstrated through longitudinal and transverse impedance spectra (Fig. 5a [FIGURE:5] and Fig. 5b), which compare cavity impedance with and without the filter.

Analysis reveals that filter loading induces only negligible modifications to LOM/HOM impedance peaks across the relevant frequency range. The absorber structure's characteristic damping performance for these modes remains virtually unaffected, as the magnitude and shape of their impedance peaks are largely preserved. This confirms that the filter operates as a highly selective element: it couples strongly to and suppresses the specific TM020 mode while exhibiting weak coupling to other modes. Consequently, the cavity successfully achieves dual requirements: deep suppression of the leakage-prone TM020 mode while maintaining broadband damping capability for the full spectrum of modes necessary for beam stability.

Fig. 5. Wakefield analysis of TM020 mode damped cavity. Red curve: impedance distribution without narrowband filter; black curve: impedance distribution with narrowband filter. (a) Longitudinal impedance distribution. (b) Transverse impedance distribution.

3.3. High-Power Robustness: Thermal and Mechanical Stability

The viability of the filter-enhanced cavity for practical accelerator applications requires validation of thermal and mechanical stability under high-power operation. An integrated thermo-mechanical analysis was performed on the structure shown in Fig. 6 [FIGURE:6].

Thermal analysis (Fig. 7 [FIGURE:7]) assumed 20 kW power input and 20°C ambient temperature. Results demonstrate effective heat management, with maximum temperature rise limited to 21.3 K at the main cavity nose cone (a high field loss region). Critically, temperature rise in the microwave absorber and ceramic filter material (dielectric constant = 20, tan δ = 5 × 10⁻⁵, thermal conductivity = 26.8 W/mK) remains below 8 K, preventing excessive thermal stress and preserving material properties.

Mechanical analysis assessed combined thermal load and vacuum stress (1.2 atm). As shown in Fig. 8 [FIGURE:8], von Mises stress in copper components and first principal stress in ceramic remain well below material yield strengths, ensuring structural integrity. Mechanical deformation is limited to the micrometer scale.

Thermo-mechanical deformation effects on RF performance were evaluated in Table 2 [TABLE:2]. Deformations induce negligible frequency detuning of only −182 kHz. While EM field leakage increases slightly from 0.60% to 0.65% under deformed conditions, this remains more than six times lower than the minimum leakage achievable without the filter and is operationally acceptable.

Given that mechanical deformation is primarily on the micrometer scale and the filter was intentionally designed for weak coupling with lower- and higher-order modes to preserve their damping, these minute structural changes will have even less impact on broadband impedance of other modes. Therefore, parasitic mode damping efficiency remains robust and negligibly affected by deformation. These results conclusively demonstrate that the integrated filter design maintains exceptional electromagnetic performance under 20 kW operation while exhibiting robust thermal and mechanical stability.

Fig. 6. Mechanical structure design of narrowband filter type TM020 mode damped cavity.

Fig. 7. Thermal analysis of TM020 mode damped cavity, showing temperature distribution of damped cavity, microwave absorber, and narrowband filter structure under 20 kW input power.

Fig. 8. Mechanical stress and deformation analysis of TM020 mode damped cavity. (a) Initial condition settings. (b) Vacuum and thermal stress analysis schematic: ceramic material analyzed using First Principal Stress, copper material using Von Mises Stress. (c) Mechanical deformation from vacuum and thermal stress.

Table 2: Changes in RF parameters caused by mechanical deformation
freq.[MHz] ∆f [MHz] + The EM field leakage rate after mechanical deformation.
5.98×106 -0.91×105 Pm/Pc[%] Pm/Pc[%]+

4. Conclusion

We demonstrate a narrowband filter structure that fundamentally solves EM field leakage in TM020 mode damped cavities. By integrating an RLC-designed coaxial filter, TM020 leakage is suppressed to <0.6%—more than a sixfold improvement over conventional designs—while maintaining broadband HOM damping (verified via longitudinal/transverse impedance spectra). The filter provides unprecedented robustness, tolerating ±0.8 mm coupling-slot offsets while maintaining leakage below 2%. High-power validation (20 kW) confirms exceptional thermo-mechanical-electromagnetic stability: peak temperature rise ≤21.3 K, mechanically-induced frequency shift ≤0.182 MHz, and leakage increase to merely 0.65%. This performance, achieved without cryogenics or significant cavity lengthening, satisfies dual requirements of leakage immunity and power resilience for fourth-generation light sources. The design methodology is applicable to superconducting cavities and multi-frequency systems, enabling compact, high-intensity accelerators.

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Author Contributions

Xuerui Hao: Conceptualization (lead), Data curation (lead), Investigation (lead), Writing, original draft (lead).
Xiao Li: Project administration, Supervision; Writing, review, editing.
Junyu Zhu: Data curation, Software.
Wei Long: Data curation, Software.
Shengyi Chen: Data curation, Software.
Yang Liu: Data curation, Software.
Shenghua Liu: Data curation, Software.
Bin Wu: Data curation, Software.
Chunlin Zhang: Data curation, Software.
All authors reviewed the manuscript.

Data Availability

The datasets used and/or analysed during the current study available from the author (Hao Xuerui) on reasonable request.

Ethics Statement

This research adheres to all applicable ethical standards, including but not limited to the welfare of experimental animals and informed consent for human studies.

Conflict of Interest Statement

The authors declare that there are no financial or personal conflicts of interest that could influence the interpretation of the results of this study.