Preamble

A Novel Bimodal Radio-frequency Cavity Enabling Independent Tuning and Effective Higher-Order Mode Damping for Advanced Synchrotron Light Sources

Junyu Zhu,¹,²,³,† Xiao Li,¹,²,³,‡ Jiebing Yu,¹,² Zhijun Lu,¹,² Xuerui Hao,¹,² Bin Wu,¹,² Chunlin Zhang,¹,² Wei Long,¹,² Yang Liu,¹,² Shengyi Chen,¹,² Shenghua Liu,¹,² Jian Wu,¹,² Xiang Li,¹,² and Sheng Wang¹,²,³,§

¹Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
²Spallation Neutron Source Science Center, Dongguan 523803, China
³University of Chinese Academy of Sciences, Beijing 100049, China

Double radio-frequency (RF) systems, comprising both fundamental and harmonic cavities, are essential in advanced synchrotron light sources for lengthening beam bunches, thereby increasing the Touschek lifetime and reducing intrabeam scattering. RF cavities must incorporate effective higher-order mode (HOM) damping to mitigate coupled bunch instabilities (CBI). In addition, a compact design is crucial for fitting within the limited straight sections of the storage rings. This paper presents a novel coaxial bimodal cavity that simultaneously delivers fundamental and harmonic voltages, allowing independent operation of both modes and effective HOM damping. It offers a more compact and efficient alternative to conventional separate cavities. A prototype cavity design was developed, featuring resonant frequencies of 166.6 MHz for the fundamental mode and 499.8 MHz for the third-harmonic mode. Simulation results indicate the successful implementation of a bimodal RF cavity, which features independent frequency tuning, separate RF drives, and effective HOM damping. This work offers a compact and efficient solution for implementing double-frequency RF systems in advanced synchrotron light sources.

Keywords: Bimodal RF cavity, Independent tuning, Higher-Order Mode Damping, Synchrotron light sources

INTRODUCTION

Double radio-frequency (RF) systems, consisting of both fundamental and harmonic cavities, have successfully increased Touschek lifetimes by lengthening beam bunches in several third-generation light sources \cite{1,2}. Currently, fourth-generation synchrotron radiation (SR) sources aim to achieve horizontal beam emittances of approximately 100 pm rad or lower and are under construction or active design worldwide, such as HEPS \cite{3}, HALF \cite{4}, APS-U \cite{5}, among others. In these ultralow-emittance storage rings, challenges such as emittance growth due to intrabeam scattering and short Touschek lifetimes become serious concerns, particularly in the low-to-medium energy range \cite{6}. To mitigate these adverse effects, the implementation of double RF systems has become essential in nearly all fourth-generation storage rings.

Many devices, including magnets, RF cavities, and beam position monitors, need to be installed in the storage ring. To maximize the light source's utilization efficiency, it is crucial to install as many insertion devices as possible, making the straight sections of the storage ring particularly valuable. The use of separate fundamental and harmonic cavities simplifies operations in a double RF system but requires multiple cavities, especially for normal conducting (NC) systems, due to limitations such as cavity wall dissipation and the limited capacity of power input couplers \cite{7-9}. The innovative bimodal cavity, which integrates both fundamental and harmonic modes, can reduce the required number of RF cavities, offering space efficiency and economic advantages \cite{10,11}. Reducing the number of cavities can also decrease beam impedance, improving beam quality and stability. Additionally, employing uniform cavity types simplifies maintenance and enhances system stability.

Bimodal cavities have been proposed to enhance the performance of RF electron guns by providing a flat-top-like RF profile, yielding excellent results \cite{12-14}. Shanghai Synchrotron Radiation Facility (SSRF)/Shanghai soft X-ray FEL facility (SXFEL) has developed an advanced two-mode transverse deflecting structure (TTDS) to deflect beams at any angle \cite{15,16}. These bimodal cavities are specifically designed to operate in TM010 + TM011, TM012, TM020 or TM030 modes by modifying the shape of pillbox or elliptical cavities. A similar bimodal cavity design has also been proposed for synchrotron light sources \cite{17,18}. In addition, multi-mode coaxial resonant cavities have been successfully used in beam bunchers \cite{19} and fast kickers \cite{20}. Despite the straightforward concept, bimodal cavities present significant challenges, particularly in achieving independent frequency tuning and efficient higher-order mode (HOM) damping. Existing bimodal RF cavities struggle to address these issues, limiting their application, particularly in synchrotron radiation light sources with high-brightness beams.

To meet the stringent requirements of advanced synchrotron radiation light sources, we propose a novel coaxial bimodal cavity design. This structure utilizes its two lowest monopole modes for the fundamental and third harmonic systems, respectively. The distinct spatial distributions of the electromagnetic fields in these modes enable independent frequency tuning. Additionally, the HOMs with high impedance exhibit frequencies above those of the accelerating modes, allowing for effective damping. Furthermore, the coaxial bimodal cavity is ideally suited for low-frequency RF systems that facilitate achieving ultra-low emittances in synchrotron light sources via on-axis injection \cite{21-23}.

In this paper, we introduce the principles of the coaxial bimodal cavity, outline its design requirements, and present a specific prototype design tailored to the South China Advanced Photon Source (SAPS), a fourth-generation light source, planned for construction in China \cite{24,25}. To achieve a natural emittance as low as 33 pm, we plan to adopt an on-axis injection scheme using triple-frequency RF systems \cite{26}. The main parameters and RF requirements of the storage ring are shown in Table 1 [TABLE:1]. In the conventional scheme, six 166.6 MHz HOM-damped cavities are planned for the fundamental system, three active 333.3 MHz HOM-damped cavities for the second harmonic system, and two active 499.8 MHz HOM-damped cavities for the third harmonic system. We propose using three bimodal cavities (BC) to replace the existing three fundamental cavities (FC) and two third harmonic cavities (HC). This approach eliminates the need for two separate third harmonic cavities, thereby saving straight-section space and reducing construction costs. The paper is organized as follows: Section II outlines the principles and design requirements of the coaxial bimodal cavity. Section III provides a comprehensive design of a bimodal cavity prototype, including RF design, multipacting analysis, dual-frequency tuning, dual-power coupling, HOM damping, and thermomechanical simulations. The discussion is presented in Section IV, and the conclusions are summarized in Section V.

[TABLE:1] Main parameters and RF requirements of the SAPS storage ring.

II. CONCEPT OF COAXIAL BIMODAL CAVITY

In an ideal quarter-wave coaxial RF cavity, the length of the accelerating gap is negligible compared to the cavity length, and the resonant frequencies can be calculated using Equations (1) and (2) \cite{27}:

$$
\begin{align}
Z_0 \tan(2\pi f_n L/c) &= 1/(2\pi f_n C_{\text{gap}}) \
Z_0 &= \sqrt{\mu/\epsilon} \cdot 2\pi \log(r_2/r_1)
\end{align}
$$

where $Z_0$ denotes the characteristic impedance, $n$ represents the serial number of the resonant mode, $f_n$ is the resonant frequency, $L$ is the cavity length, $c$ is the speed of light, $C_{\text{gap}}$ is the capacitance of the accelerating gap, and $r_1$ and $r_2$ are the radii of the inner and outer conductors, respectively. Here, $\mu$ and $\epsilon$ represent the permeability and permittivity in vacuum, respectively. According to Eq. (1), when the $C_{\text{gap}}$ is small, the resonant frequencies of HOMs induced by a coaxial structure can be approximated by $f_n = (2n + 1)c/(4L)$ ($n = 0, 1, 2, \ldots$).

The electromagnetic field distribution of these modes at the acceleration gap exhibits monopole characteristics. Consequently, the first two modes, denoted as M0 and M1, have a frequency ratio of 1:3, making them suitable for application as fundamental and third harmonic modes, respectively. However, practical RF cavity designs require a larger accelerating gap to achieve voltages of several hundred kilovolts or more, which results in a very small value of $C_{\text{gap}}$. Solving Eq. (1) illustrates the relationship between $f_0$, $f_1$, and the ratio $f_1/f_0$ with respect to $C_{\text{gap}}$, as depicted in Fig. 1 FIGURE:1, where $Z_0$ is 100 Ω. It is readily observed that reducing $C_{\text{gap}}$ results in a narrower frequency gap between the two modes, falling below the initial 1:3 frequency ratio.

Graphical solutions to transcendental Equation (1) illustrate that an increase in the capacitance ($C_{\text{gap}}$) leads to a notable decrease in the fundamental frequency ($f_0$), while exerting minimal influence on the first harmonic frequency ($f_1$), as demonstrated in Fig. 1(b). Intersection points A and B, where curve $y$ intersects with curve $y_1$, correspond to the solutions for $f_0$ and $f_1$, respectively. These points shift to positions C and D when $C_{\text{gap}}$ increases tenfold. Therefore, a capacitor-loaded plate, as shown in Fig. 2 FIGURE:2, can be utilized to adjust their frequency relationship by increasing the capacitance $C_{\text{gap}}$. This method has been successfully implemented in the 100 MHz RF cavity of the MAX IV facility, enabling the frequency of the first HOM to be increased to four times or more than the fundamental frequency \cite{28}. Additionally, as illustrated in Fig. 2(b), modifying the outer conductor diameter in the accelerating gap region alters the electromagnetic field intensity within the gap, thereby adjusting the $C_{\text{gap}}$ value. Consequently, the two methods of altering $C_{\text{gap}}$ can be employed to design a bimodal cavity with both fundamental and third harmonic modes.

In the double RF system of advanced synchrotron light sources, which is used to lengthen the beam bunch and consists of a fundamental cavity and a high-order harmonic cavity, the voltage can be written as \cite{1}:

$$
V(z) = V_{\text{rf}} \sin(\omega_{\text{rf}}z/c + \Phi_s) + kV_{\text{rf}} \sin(n\omega_{\text{rf}}z/c + n\Phi_h)
$$

where $V_{\text{rf}}$ is the fundamental RF voltage, $\omega_{\text{rf}}$ is the fundamental frequency, $z$ is the longitudinal coordinate of the electron, $k$ is the relative harmonic voltage to the fundamental RF voltage, $\Phi_s$ is the synchronous phase, $\Phi_h$ is the relative harmonic phase, and $n$ is the harmonic number. To lengthen beam bunches, the harmonic amplitude and phase should be adjusted to cancel the slope of the fundamental RF voltage at the bunch center. The harmonic voltage and phase at this condition are given by:

$$
\tan(n\phi_h) = \frac{nU_0/V_{\text{rf}}}{\sqrt{1/n^2 - (U_0/V_{\text{rf}})^2/(n^2 - 1)}}
$$

where $U_0$ is the energy loss per turn.

In a bimodal cavity, the electric fields of both fundamental and harmonic modes exist simultaneously and share the same accelerating gap. Considering the Transit Time Factor (TTF) \cite{29}, the voltage gained by the electrons traveling close to the speed of light as they pass through the bimodal cavity can be expressed as:

$$
V(z) = V_1 T_1 \sin(\omega_{\text{rf}}/cz + \phi_1) + V_n T_n \sin(n\omega_{\text{rf}}/cz + \phi_n)
$$

$$
T_1 = \frac{\sin(\omega_{\text{rf}} d/(2c))}{2c/(\omega_{\text{rf}} d)}
$$

$$
T_n = \frac{\sin(n\omega_{\text{rf}} d/(2c))}{2c/(n\omega_{\text{rf}} d)}
$$

where $V_1$ and $V_n$ are the voltage amplitudes of the fundamental and harmonic modes, respectively; $T_1$ and $T_n$ are the Transit Time Factors for the fundamental and harmonic modes; $\phi_1$ and $\phi_n$ are the phases with respect to the fundamental and harmonic voltages; and $d$ is the length of the accelerating gap.

The relationship between the Transit Time Factors of the fundamental and third harmonic modes and the acceleration gap length in a bimodal cavity is shown in Fig. 3 [FIGURE:3]. The transit time factors decrease significantly with an increasing acceleration gap, especially in the harmonic mode. Coaxial bimodal cavities have a short accelerating gap, less than half of the wavelength for both fundamental and harmonic modes, resulting in high acceleration efficiency.

The primary challenge for bimodal cavities in synchrotron light sources is achieving independent tuning of the two operating modes. Variations in $C_{\text{gap}}$ significantly affect $f_0$, whereas their impact on $f_1$ is negligible, as shown in Fig. 1(a). This enables the implementation of an independent tuning system for the fundamental mode by adjusting $C_{\text{gap}}$ via squeezing the 'end plate', a plate adjacent to the accelerating gap. Additionally, in coaxial bimodal cavities, a significant difference in electromagnetic energy density distribution between the two modes near the outer conductor is observed, as illustrated in Fig. 4 [FIGURE:4]. Specifically, the M1 mode exhibits a pronounced magnetic field and a diminished electric field near the 'end plate', whereas the M0 mode displays a comparatively uniform electromagnetic field. A plunger-type tuner can be utilized in this region to fine-tune $f_1$ with negligible influence on $f_0$, by leveraging the principles of cavity perturbation theory as delineated in Equation (6) \cite{27}:

$$
\frac{\Delta f}{f_c} = \frac{\int_V (\mu_0|H|^2 - \epsilon_0|E|^2) dV}{\int_V (\mu_0|H|^2 + \epsilon_0|E|^2) dV}
$$

where $f_c$ represents the resonant frequency of the cavity, and $V_a$ and $V_b$ denote the volumes of the cavity and the perturbation, respectively. Thus, two independent tuning systems for the respective operating modes within the bimodal cavity can be achieved.

In advanced synchrotron light sources, many harmonic cavities operate passively, with the harmonic RF voltages induced by the beam itself \cite{30,31}. However, specific requirements, such as varying beam currents for different operational modes and specific injection schemes, necessitate an active harmonic system in certain facilities, including HEPS \cite{32} and PETRA IV \cite{7}. In these cases, the bimodal cavity requires two input couplers to facilitate active operation for both systems. To avoid mutual interference, these input couplers must be designed with narrow bandwidths of less than tens of MHz. Additionally, HOM damping is crucial for RF cavities in advanced light sources. The coaxial bimodal cavity utilizes the first two monopole modes as accelerating modes, and there are no high-impedance HOMs between them. Waveguide-type couplers (e.g., BESSY 500 MHz HOM-damped cavity \cite{33}), antenna-type HOM absorbers (e.g., MAX IV cavities \cite{34}), and beam-type absorber \cite{35,36} can be used to suppress these HOMs.

With this concept, we designed a coaxial bimodal RF cavity for advanced synchrotron light sources. The specific design objective is to meet the practical application requirements of the SAPS. The detailed design of this bimodal cavity is presented in the following sections.

III. PROTOTYPE DESIGN OF A BIMODAL CAVITY

A. RF Design

The bimodal cavity was designed with fundamental and third harmonic frequencies of 166.6 MHz and 499.8 MHz, respectively. Several constraints were considered in the design process. First, to ensure practicality, the outer conductor diameter and length were kept below 0.8 m and 0.5 m, respectively. Second, the frequencies for the fundamental and third harmonic modes, corresponding to the two lowest monopole modes of the cavity, were set at 166.6 MHz and 499.8 MHz. This design aims to maximize their shunt impedances to ensure that the total cavity wall losses are less than 45 kW at the design voltages ($V_1 = 420$ kV and $V_3 = 160$ kV), reserving some margin for SAPS normal operation. Third, to enable effective HOM damping, the frequency of the next monopole mode with high impedance should exceed 600 MHz, ensuring a significant frequency separation from the operating modes.

The design and optimization of the two schemes outlined in Section II were performed using CST Microwave Studio \cite{37}, and the RF simulation models are illustrated in Fig. 5 [FIGURE:5]. A 30 mm thick space is engineered between the beam pipe and the inner conductor to house cooling pipes, with the cavity's beam pipe having a diameter of 63 mm. For the extraction of HOMs, an enlarged tube is strategically positioned outside the cavity. The main parameters of the schemes are listed in Table 2 [TABLE:2]. Both schemes have distinct advantages. Scheme 1 exhibits a lower $R/Q$ value for the harmonic mode, which is beneficial for mitigating periodic transient beam loading (PTBL) effects in synchrotron light sources with double RF systems \cite{38,39}. In contrast, Scheme 2 achieves higher shunt impedance ($R_{\text{sh}} = V_c^2/P_c$) and quality factor ($Q$) for both operating modes, significantly reducing RF power requirements. For SAPS, both schemes meet critical operational criteria, including requirements for RF voltage, power, and beam stability. Notably, Scheme 2 offers practical advantages by reducing the number of required bimodal cavities and associated costs. Consequently, Scheme 2 was chosen as the baseline for subsequent design.

The electric field distributions of the two operating modes, each with 1 J of stored energy, are depicted in Fig. 6 FIGURE:6 and 6(b). Additionally, the profile of the longitudinal electric field for beam acceleration is presented in Fig. 6(c), where $z = 85$ mm denotes the center of the accelerating gap. The optimized shunt impedances for the two modes are 5.71 MΩ and 1.83 MΩ, respectively.

[TABLE:2] Two options for the bimodal cavity.

B. Multipacting

The multipacting (MP) simulation of the bimodal cavity was conducted using CST Particle Studio and CST Microwave Studio \cite{37}. The argon-discharged copper equivalent to the copper material after high power conditioning \cite{40} was chosen for the cavity's metallic walls. To expedite computations and accurately identify multipacting locations, the cavity's inner surface was divided into several initial particle source regions. The particle sources provided simulations with primary electrons uniformly distributed across the source area and over an energy range of 0–4 eV. The RF electromagnetic field map for both fundamental and harmonic modes, calculated via the eigenmode solver, was imported into the Particle-in-Cell (PIC) solver. Simulations were conducted over twelve RF phases (in 30° increments) and accelerating voltages from 25 kV to 750 kV, tracking particle dynamics over 100 RF periods to ensure steady-state convergence. Emission and collision data from each surface were analyzed to compute the integral secondary emission yield ($\langle\text{SEY}\rangle = \text{Total Secondaries}/\text{Total Impacts}$) \cite{41}. As depicted in Figure 7 [FIGURE:7], the maximum $\langle\text{SEY}\rangle$ for all regions of the fundamental, harmonic, and combined modes consistently remained below the critical threshold of 1.0, indicating the absence of multipacting when the cavity's inner surface is adequately treated.

C. Dual-frequency Tuning

Following the methodology detailed in Section 2, we developed two distinct tuning systems for the bimodal cavity. The fundamental mode tuning is achieved by mechanically squeezing the end plate as illustrated in Fig. 8 FIGURE:8. To minimize effects on the harmonic frequency, precise design of the end plate deformation was necessary, employing cavity perturbation theory and accounting for the electromagnetic field distribution in this area. Figure 8(b) shows the radial distribution of electric and magnetic field energy densities for both operating modes near the end plate. Analysis reveals that the electric field energy of the fundamental mode (M0) in the radial span of 70–188 mm—excluding the 0–70 mm range designated for beam tubes—vastly surpasses the magnetic field energy. In contrast, the electric and magnetic field energies of the third harmonic (M1) are comparable. Consequently, an end plate, designed with a minimum thickness of 5 mm at a radius of 188 mm, aims to enable controlled deformation while ensuring mechanical strength through the tuning mechanism, allowing for independent tuning of the fundamental mode. The achieved frequency tuning range for the fundamental mode is 500 kHz, with an end plate deformation of ±1 mm under a maximum stress of 68.6 MPa. Simulation results from CST multiphysics calculations indicate tuning sensitivities of 252 kHz/mm for the fundamental mode and 1.3 kHz/mm for the third harmonic mode, as illustrated in Fig. 9 FIGURE:9. The impact of fundamental mode tuning on the third harmonic frequency is 0.5%, compared to the initial estimate of 0.1% derived from preliminary calculations (Eq. 6) that assumed uniform end plate displacements.

The frequency of the harmonic mode is adjusted by employing a tuner plunger, which has a diameter of 120 mm, as illustrated in Fig. 8(a), strategically positioned at a location where the third harmonic exhibits a strong magnetic field and a weak electric field, while the distribution of the fundamental electromagnetic field remains relatively uniform, as illustrated in Fig. 4. The tuner has a travel range of 40 mm (−20 mm to +20 mm), with a maximum insertion depth into the cavity of 20 mm. The simulation results indicate that the tuning range for the third harmonic is ±500 kHz, with tuning sensitivities of approximately 25.6 kHz/mm for the third harmonic mode and 0.7 kHz/mm for the fundamental mode, as illustrated in Fig. 9(b). The impact of harmonic tuning on the fundamental frequency is less than 2%, which is sufficient to meet the requirements for independent tuning of bimodal cavities. This is because the normal conducting RF cavity has a bandwidth of several kHz, and the frequency change due to temperature shifts (after reaching thermal equilibrium) and beam loading (8.7 kHz @ SAPS 500 mA) is typically within a few kHz. It should be noted that during the initial stage of powering the cavity, the maximum frequency shift caused by changes in cavity temperature can reach up to 100 kHz. In such cases, both the fundamental and harmonic tuners need to be adjusted simultaneously. Additionally, since designing a choke structure for the two operating modes is challenging, spring fingers are installed near the inner wall of the ports to reduce the electromagnetic field entering the tuner. When the tuner is inserted 20 mm into the cavity, the power of the fundamental and third harmonic modes entering the tuner is approximately 90 W and 175 W, respectively, under normal operation. A water cooling system with a flow rate of 1.0 L/min is necessary to ensure that the temperature rise of the tuner does not exceed 15°C during operation.

D. Dual-power Coupling

In this study, we designed two input power couplers to enable a double active RF system. The input power coupler for the fundamental mode was designed with a structure similar to the one developed for the 500 MHz/5-cell copper cavity \cite{42}, as depicted in Fig. 10 FIGURE:10. To prevent mutual interference between the modes, it was engineered with narrow bandwidths of less than tens of MHz by incorporating a T-piece as a filter. This T-piece, featuring an electrical short in one arm, allows for optimization of RF performance by adjusting the electrical length of the short circuit. Additionally, water cooling is integrated with the high-power coupler's inner conductor to manage thermal loads. As illustrated by its $S_{11}$ scattering parameters in Fig. 10(b), the coupler exhibits excellent transmission characteristics at the fundamental frequency, but approaches total reflection at the third harmonic frequency, demonstrating the fundamental input coupler's negligible impact on the third harmonic mode. To handle a power capacity greater than 125 kW, both the ceramic window and the inner conductor are water-cooled. Additionally, the ceramic window is coated with Titanium Nitride (TiN) to suppress multipacting. The power coupling coefficient is adjustable between 1.0 and 5.0 by altering the coupling loop, with the target coupling coefficient for SAPS at nominal current being approximately 3.2.

The power of the third harmonic mode is delivered through a WR-1500 waveguide coupler, as illustrated in Fig. 11 FIGURE:11. The input coupler port is rectangular, measuring 381 mm by 190.5 mm, and incorporates a rectangular iris measuring 160 mm by 40 mm in the cavity's common wall. To accommodate different beam currents, the power coupling coefficient is adjustable between 1.0 and 5.0 by altering the length of the coupling tuner post, which has a diameter of 70 mm and is positioned 70 mm above the coupling slot \cite{43}. The target coupling coefficient of the third harmonic coupler is approximately 4.2. The $S_{21}$ parameters of the coupler, as shown in Fig. 11(b), indicate its ability to block the transmission of fundamental power while allowing effective propagation of third harmonic power. As a result, the two input couplers, each designed for a specific operating mode, can independently transmit power to the cavity without interference.

E. HOM Damping

In this study, to achieve stronger HOM damping, we adopted a beam absorber. This approach is rarely used in normal-conducting (NC) cavities due to its significant impact on accelerating modes. However, in a coaxial resonator cavity, the impedance is primarily determined by the inner and outer radii and length, with negligible influence from enlarging the external beam tube. Consequently, a beam absorber can be used in NC coaxial cavities to effectively damp HOMs with minimal performance impact. Additionally, NC cavities can withstand power losses of several hundred watts with minimal performance impact, enabling the absorber to be positioned closer to the cavity for high absorption efficiency \cite{35,36}.

The beam-line absorber, designed with a 380 mm diameter, damps HOMs, while a 200 mm transition section attenuates the accelerating mode, as illustrated in Fig. 12 [FIGURE:12]. The absorbing material used here is ferrite-C48 with a thickness of 4 mm \cite{44}. The cavity iris, with a 140 mm aperture diameter, facilitates HOM field propagation and accelerating mode rejection. Additionally, to reduce the loss factor, a taper measuring 100 mm in length and 63 mm in diameter at the exit is employed. The impedance spectrum of HOMs in the cavity was calculated using CST Microwave Studio and Particle Studio. Initially, a wakefield solver with a wake length of 450 meters was employed to identify high-impedance HOMs. Subsequent eigenmode simulations were performed to verify the peak impedance values of critical modes, both with and without the absorber. The HOM impedance spectrum is illustrated in Fig. 13 FIGURE:13 and Fig. 13(b). The results indicate that longitudinal HOM impedances are below 4.0 kΩ and transverse HOM impedances are below 70 kΩ/m, and both are below the required impedance threshold of the SAPS. The absorber reduces the impedance of the fundamental and harmonic modes by 0.1% and 2%, respectively, which are levels deemed acceptable.

F. Thermomechanical Simulation and Design

The mechanical design focus was on the fundamental frequency tuning mechanism and the thermomechanical characteristics of the main structure. Fundamental frequency tuning is achieved by squeezing the cavity's end plate using a mechanical tuner. To achieve a frequency tuning range of 166.6 ± 0.25 MHz, a deformation of 1 mm is required at the cavity end plate. To minimize the deformation force, an arc groove was designed with a minimum wall thickness of 5 mm, as shown in Fig. 8(a). Simulation results indicate that a maximum deformation of 1 mm can be achieved by applying a pressure of 7.2 kN or a tension of 8.5 kN. The combined mechanical and thermal stresses at the end plate result in a maximum stress of approximately 70 MPa, as shown in Fig. 14 FIGURE:14. To maintain tuning within the elastic limit, the end plate must be made from forged copper, which possesses a yield strength exceeding 100 MPa. Additionally, to retain the properties of the forged copper, electron beam welding with minimal thermal impact should be employed for affixing the end plate to the cavity body \cite{28}. The fatigue evaluation of the fundamental tuner was performed using finite element analysis. As the deformation of the tuner remains within the elastic range, the analysis focuses on high-cycle fatigue. The fatigue endurance curve (S–N curve) of oxygen-free copper was chosen for the cavity's metallic walls \cite{45}. The simulation results demonstrate that in the extreme tuning case (with a single maximum deformation of 1 mm), the cumulative damage is less than 1.0 and the safety factor of the end plate is greater than 1.5, indicating a safe design. The minimum lifespan is near the groove, with a predicted lifespan exceeding 2 million cycles, as depicted in Fig. 14(b). If the deformation of the end plate exceeds 1.25 mm, plastic deformation will occur, and the lifespan will significantly decline. Transient structural simulation results indicate that when the deformation reaches 1 mm, frequencies exceeding 25 Hz will impact the lifespan. In practice, cavity tuning is primarily induced by temperature fluctuations, and the tuner's operating frequency is much less than 25 Hz.

The cooling system is designed to accommodate a maximum cavity power of 45 kW. Based on the average power loss density across the inner cavity surfaces, the design includes 12 cooling pipes for the cavity mantle and 2 spiral-shaped cooling pipes for the inner conductor. Thermal analysis simulation results indicate that the maximum cavity temperature remains below 55°C, as depicted in Fig. 15 [FIGURE:15]. The temperature distribution results in a maximum thermal stress of 45 MPa at the inner conductor. The thermal deformation is 0.12 mm, leading to frequency changes of 69 kHz for the fundamental mode and 129 kHz for the third harmonic mode. Considering the adjustment ranges of 166.6 ± 0.25 MHz for the fundamental mode and 499.8 ± 0.5 MHz for the third harmonic mode, the thermal deformation and frequency changes are within acceptable limits.

Heat generation in HOM absorbers arises from RF leakage heating and the HOM power induced by the beam. At the design operational levels of 420 kV for the fundamental mode and 160 kV for the harmonic mode, RF leakage heating from the two modes is less than 400 W. With a natural bunch length of 5.0 mm, the loss factor of the HOMs in the cavity is approximately 1.8 V/pC. The HOM power induced by the beam does not exceed 3 kW at a beam current of 500 mA. To ensure redundancy, the maximum heat load for the absorber was set at 5 kW in the design. Simulation results indicate that the maximum temperature of the ferrite remains below 55°C, with the temperature increase from inlet to outlet water not exceeding 8°C, as illustrated in Figs. 16(a) and 16(b) [FIGURE:16].

Comprehensive theoretical analysis and detailed simulations have established the feasibility of the bimodal cavity scheme. The frequency tuning devices, power input couplers and HOM absorber all utilize common techniques, facilitating ease of implementation. Based on the presented design, fabrication of the bimodal cavity with resonant frequencies of 166.6 MHz and 499.8 MHz has commenced.

IV. DISCUSSION

The bimodal cavity repurposes the previously problematic monopole HOM as an operating mode. This approach not only reduces the difficulty of suppressing HOMs but also enhances the fundamental performance. The bimodal cavity exhibits high fundamental shunt impedance, approximately 5.7 MΩ, significantly exceeding the traditional coaxial RF cavities' shunt impedance of 3.4 MΩ \cite{9,28}. In conventional cavities, the first monopole HOM often resonates at a frequency close to that of the fundamental mode, and suppressing it typically compromises accelerating performance. Although the bimodal cavity's harmonic impedance is lower than that of a conventional single-mode harmonic cavity, its design allows for harmonic voltage distribution across multiple cavities. As a result, the cumulative harmonic impedance can match that of conventional separated cavities. The bimodal cavity includes two input couplers, facilitating independent control of the amplitude and phase of both voltages. Its harmonic mode can also operate in passive mode or as a single-mode cavity through harmonic frequency detuning, offering flexibility for beam commissioning and operation. To achieve deep suppression of HOMs, a beam-line absorber scheme was implemented for the coaxial RF cavity, effectively damping HOMs while maintaining high shunt impedance \cite{32,36}, albeit requiring some longitudinal space. If HOM damping requirements are less stringent, more compact antenna-type HOM absorbers can be employed \cite{34}. In modern synchrotron radiation facilities with double RF systems, it is paramount that RF systems meet requirements for RF voltage, power, and HOM damping, while also considering instabilities excited by harmonic modes, particularly the PTBL effect \cite{38,39}. The harmonic mode of the bimodal cavity features a lower $R/Q$ compared to that of a traditional NC cavity, offering a significant advantage that can enhance the PTBL instability threshold. Consequently, this bimodal cavity scheme is well-suited for advanced synchrotron radiation light sources.

V. SUMMARY

In this paper, a novel coaxial bimodal cavity, comprising both fundamental and third harmonic modes, has been proposed for advanced synchrotron light sources. This work addresses the longstanding challenges of achieving independent frequency tuning and effective HOM damping, thereby meeting the stringent requirements of advanced synchrotron radiation light sources. Compared to traditional separate fundamental and harmonic cavities, such bimodal cavities offer significant advantages in space efficiency and economic benefits.

This paper elucidates the principles of the coaxial bimodal cavity and presents a detailed prototype design with resonant frequencies of 166.6 MHz and 499.8 MHz for the defined application. Simulation results indicate that the cavity successfully achieved two accelerating modes for the fundamental and third harmonic systems, enabling independent frequency tuning, separate RF drives, and effective HOM damping. This research provides a compact and efficient solution for implementing double-frequency RF systems in advanced synchrotron light sources.

ACKNOWLEDGMENTS

The authors would like to thank Weihang Liu and Yu Zhao for helpful discussion and advice.

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