Bimodal design of 500MHz and 1.5GHz normal conducting RF cavity for advanced synchrotron radiation facilities

The advanced storage ring light source needs to realize ultra-low emissivity beam operation, and improving the Touschek lifetime puts forward higher requirements for the performance of RF cavity. In this paper, a novel bimodal normal conducting RF cavity is proposed. In one cavity, two power sources will be connected at the same time to realize the simultaneous operation of the two frequencies. The TM010 mode with the frequency of 500 MHz is used for acceleration, and the TM020 mode with the frequency of 1.5 GHz is used as the third harmonic to improve the height of the RF bucket and achieve the purpose of lengthening the beam bunch. Two couplers are designed to adapt to the working characteristics of bimodal RF cavity.


Introduction
Higher brightness and better coherence are the development trend of storage ring light source, which needs to further reduce the emittance of the beam cluster.The reduction of the emittance of the beam cluster will aggravate the scattering effect inside the beam cluster, which will cause serious instability of the single beam cluster.The industrial application of synchrotron radiation devices requires compact structural design, which means it is necessary to simplify current double RF systems.The instability of the single beam cluster will directly affect the Touscheck beamlife [1].Particularly, in the medium and low energy storage ring, the Touscheck life is the most important factor determining the beam life.As shown in Formula 1, Touschek lifetime τ t is directly related to the beam bunch size.
Where σ x (s) and σ z (s) describe the transverse beam size, σ s (s) describe the longitudinal beam size.One method that has been proved is to introduce a high-order harmonic cavity to increase potential well to make the beam cluster obtain longitudinal stretch, thus reducing the longitudinal charge density of the electron and obtaining a higher Touschek lifetime [2,3].The advantage of this method is that it will not affect the transverse beam brightness, which is a happy thing for users.In our work, RF cavity is designed to operate in bimodal mode, the higher-order harmonic cavity works in the third-order harmonic (relative to the fundamental mode), which is a choice made after comprehensive consideration of the working effect and system complexity.The bimodal mode operation scheme can lengthen the bunch, and save the space of straight section set up for the RF system, it is significant for building a more compact storage ring light source.

Bimodal cavity and bunch lengthening
In the double RF system composed of RF cavity and high-order harmonic cavity, the voltage can been written as [4]: Where Vrf is the main rf cavity peak voltage, ω rf is the main rf frequency, z is the space variable, ϕ 1 and ϕ n are the phase with respect the main and harmonic voltages, k is the ratio of main and harmonic voltage.The integration of voltage with space variable z is equal to electric potential, the potential seen by the electron is given by : With α c the momentum compaction, E the energy of the particle, T 0 the revolution period.Formula 2 shows the working principle of the double RF system composed of two cavities, the main RF cavity and the high harmonic cavity.The bunch size can be corrected by adjusting the phase and voltage ratio coefficient k.In the resonant cavity structure, multiple modes of electromagnetic fields will be excited at the same time.In other words, it is feasible to operate two frequency modes simultaneously in one cavity, which has been proved by the existing experimental result.In one cavity, the electric field intensity of the composed of fundamental mode and n th order harmonic on the axis can be written as a function of time t and longitudinal coordinate z [5]: Where E 0 and E n respectively the rf accelerating field amplitude of the fundamental mode and n th harmonic mode, It should be noted that in formula 2, z/c = t, indicating that the variables z and t are equivalent when calculating the phase, but in formula 4, the space variable z and the time variable t are separately expressed, because in the case of bimodal operation, the value of z of the electron in the two fields is the same.Assuming that the electric field in the beam tube at both ends of the RF cavity decreases rapidly to 0, the voltage obtained by integrating the electric field with the space variable z can be written as: Considering the transit time of electrons passing through the RF cavity gap, the electric potential expression obtained by integrating the electric field along the axis z is similar to Formula 2, but the difference is the proportional coefficient κ satisfies the following relationship: Where z 0 is only related to the gap of RF cavity.In a bimodal RF cavity, the higher order mode field strength is lower than fundamental mode field strength, so there is κ < k.It shows that selecting phase with stronger fundamental mode field is necessary to obtain the same RF bucket as the double RF system.The common combination of fundamental wave and third harmonic is selected for analysis, as shown in Fig. 1.

Cavity design and mode
The RF cavity adopts the conventional standing wave operation mode.The cavity type is cylindrical with nose cone structure, and the adaptive optimization design is carried out on this basis.The plane structure of the cavity is shown in Fig. 2, and the basic structure can be represented by 10 parameters.In the resonator, there will be multiple resonance modes at the same time.Under normal conditions TM010 mode is the most suitable as the acceleration mode of the RF cavity, compared with other modes, TM010 mode is easier to obtain higher shunt impedance and R/Q, which can supplement more energy for the beam and efficiently use the energy supplied by the power source.By adjusting the basic parameters of the cavity, the same high-order resonant frequency can be corrected to different modes under the condition that the fundamental mode frequency is unchanged.
Adjusting parameter R t , that is, modifying the beam tube radius, can effectively improve the shunt impedance and R/Q of TM010 mode.Its frequency is less sensitive to R t , while for higher-order mode, its frequency is easily affected by Gap and Deg.Adjusting the shunt impedance of higher-order mode and R/Q has a significant response to the change of R t and R eq .In this way, by adjusting the parameters of the cavity, the resonance frequency of some specific modes can work on integer harmonics.As Fig. 3 shows, the frequency of fundamental mode TM010 is 500MHz and third harmonic mode TM020 is 1500 MHz, the basic parameters are shown in Table 1.Although there are many resonant modes can be corrected to appropriate frequency, such as the TM021 mode can work at 1.5GHz, there are two peaks in the one cell, and the effect of the electric field of TM021 is negligible when considering the transit time.

Coupler design
The bimodal RF cavity requires two power sources to work together to provide energy with the fundamental frequency of 500 MHz and the third harmonic frequency of 1.5 GHz.The reason why we use feeding of third harmonic power instead of passive power generated by beam excitation is that the third harmonic power generated by beam excitation in the cavity is very low, and the phase and voltage cannot be controlled.The power source is connected from the 14th International Particle Accelerator Conference Journal of Physics: Conference Series 2687 (2024) 082002 two ports of the coupler through metal flanges, and the coupler will be designed together with the cavity body.At present, there are two design schemes for couplers.One is to design couplers suitable for two modes respectively.The couplers with two cylindrical ports are connected with the wide radius of the cavity, as shown in Fig. 4(a), This scheme is convenient to implement and easy to suppress frequency leakage.The other is two-frequency waveguide feed.Based on the principle of directional coupler, this scheme feeds the power of two modes into the main cavity through the same coupling hole with very low reflectivity and crosstalk [6], as shown in the Fig. 4(b), the special design between two waveguides can avoid frequency leakage and only one coupling hole is required.Both methods need to consider the design of isolator and separate tuning.In addition to the fundamental wave TM010 mode and the third harmonic wave TM020 mode, there are many other higher-order resonant modes in the bimodal cavity, it will cause coupling instability between the beam bunch, and need HOM damper to suppress these modes.HOMs are planed to use a waveguide with a cutoff frequency higher than the fundamental mode frequency and lower than the TM020 mode frequency to exit from the cavity, the higher-order mode will be led out through waveguide and finally absorbed by the silicon carbide absorber.Between 500 MHz and 1.5 GHz, there are also some other modes, such as TM011 mode.It is planned to install a band stop filter with a central frequency of 1.5 GHz at the waveguide outlet of the higher-order mode to prevent the third harmonic from passing through.

Conclusion
Based on the principle of the double RF system, we propose a novel bimodal RF cavity design scheme, which choose TM010 mode as the acceleration mode and TM020 mode as the third harmonic.The two modes can work in one cavity at the same time.By adjusting the voltage and phase of the two modes, the slope of the acceleration field can be eliminated, so as to lengthen the beam bunch and improve the Touschek life of the beam.In order to adapt to the working characteristics of the bimodal cavity, we refer to the coupler proposed by Wang Lin and others, and design a simple bimodal coupler.The simulation results show that the scheme is easier to achieve energy isolation between the two modes, and the preliminary optimization results have made the two modes have no reflection problems.In the following work, we will consider using band stop filters to study damping schemes for higher order modes in bimodal cavity.

Figure 1 :
Figure 1: Voltage and phase for lengthening bunch

Table 1 :
Feasible mode for harmonic frequency.