RF design optimization for the SHINE 3.9 GHz cavity

RF design optimization for the SHINE 3.9 GHz cavity 1 2 first-author https://orcid.org/0000-0002-4296-724X 1 3 corresp Corresponding author, chenjinfang@zjlab.org.cn Corresponding author, chenjinfang@zjlab.org.cn chenjinfang@zjlab.org.cn https://orcid.org/0000-0003-2321-0388 1 3 Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201204, China Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201204, China ShanghaiTech University, Shanghai 201210, China ShanghaiTech University, Shanghai 201210, China Shanghai Advanced Research Institute, Chinese Academy of Sciences, Shanghai 201204, China Shanghai Advanced Research Institute, Chinese Academy of Sciences, Shanghai 201204, China The Shanghai High-repetition-rate XFEL and Extreme Light Facility (SHINE) under construction is designed to be one of the most advanced free electron laser (FEL) facilities in the world. The main part of the SHINE facility is an 8 GeV superconducting Linac operating in continuous wave (CW) mode. The Linac consists of seventy-five 1.3 GHz and two 3.9 GHz cryomodules. Based on the design and experience of 3.9 GHz cavities in the European X-ray Free Electron Laser (E-XFEL) and the Linac Coherent Light Source-II (LCLS-II) projects, we optimize the SHINE 3.9 GHz cavity design to adapt it for CW mode operation. In this paper, we present a particular redesign of the end-group for the SHINE 3.9 GHz superconducting cavity that includes a redesign of the end-cell and beam pipe to shift away the potentially troublesome lowest high order modes(HOMs), a modification of the main coupler antenna, and a tuning of the HOM notch filter to meet the cavity requirements. RF losses calculations on the HOM coupler antennas show that the overheating on the inner conductor at the operating mode is diminished significantly. Furthermore, we have also studied the HOMs to ensure there are no dangerously trapped modes in the optimized cavity design. Superconducting cavity Higher order modes RF design 3.9 GHz 1 Introduction 1 Building upon experience in both traditional and newly developed superconducting technology for particle accelerators, the Shanghai High-repetition-rate XFEL and Extreme Light Facility (SHINE) project is expected to be one of the most advanced high repetition-rate X-ray FEL facilities in the world [ 1 1 - 3 3 1-3 ]. The facility will provide hard coherent X-ray radiation over a broad spectrum for basic research in various fields, including biology, nonlinear physics, and material science. The SHINE Linac is equipped with seventy-five 1.3 GHz and two 3.9 GHz cryomodules. The third harmonic section compensates the non-linear longitudinal phase space distortion due to the accelerating field curvature in the 1.3 GHz TESLA cavities prior to bunch compression [ 4 4 , 5 5 ]. This allows the delivery of high current beams with sufficiently low emittance for the production of 0.4 to 25 keV FEL light to the experimental users. The 3.9 GHz cryomodules are installed after the first acceleration section and before the first bunch compressor, as shown in Fig. 1 Fig. 1 . Each 3.9 GHz cryomodule includes eight 9-cell elliptical cavities at 3.9 GHz and a quadruple magnet package. The requirements for the 3.9 GHz cryomodule and cavity in the SHINE Linac are shown in Table 1 Table 1 [ 6 6 ]. 1 1 Requirements 1 1 Nominal 1 1 Max 1 1 Beam to RF phase 1 1 -153° 1 1 Number of active cavities 1 1 15 1 1 16 1 1 Cavity Q 0 1 1 2.010 9 1 1 Operating gradient with 15 cavities 1 1 13.1 MV/m 1 1 15.0 MV/m 1 1 HOM Q ext 1 1 < 110 6 1 1   Compared to the mature 1.3 GHz TESLA cavity, the 3.9 GHz cavity is relatively less mature and the geometry is evolving over different projects. The first 3.9 GHz SRF cavity was proposed by DESY in 2002 for the FLASH VUV facility [ 7 7 , 8 8 ], and further optimized by Fermi National Accelerator Laboratory (FNAL) for FLASH. Based on the FLASH 3.9 GHz cavity, INFN-LASA made several mechanical modifications to facilitate production and put the resulting cavity into use in the E-XFEL machine, in which the cavity witnessed its first beam in 2017 [ 9 9 , 10 10 ]. FNAL subsequently reduced the beam pipe size using a tapered transition and shortened the HOM antenna length to adapt the cavity for CW operation in LCLS-II [ 11 11 ]. Based on the above experiences, we optimized the RF design for the SHINE 3.9 GHz cavity. Differing from the modifications for LCLS-II, we modified not only the beam pipe and accessories, but also the shape of the end half-cell. 2 Motivation of design optimization 1 We made some targeted modifications to meet the needs of the SHINE project, based on the previous studies and some measurement results on the prototypical and series 3.9 GHz cavities for E-XFEL at INFN-LASA. During the measurements, it was found that in some cavities, the lowest dipole modes could be shifted down to close to 3.9 GHz after fabrication and chemical treatments, which may lead to difficulties for π -mode operation and notch tuning in the HOM coupler band stop filter [ 12 12 , 13 13 ]. Consequently, the resultant power leakage through the HOM couplers at operating frequency may become considerable. Therefore, it is necessary to optimize the cavity geometry. Simulations show that the lowest dipole mode is located at the region of the end cell, main coupler, and beam pipe (PEC boundary), as shown in Fig.2 Fig.2 . Hence, its frequency can be efficiently shifted up by squeezing the end-cell wall, or by reducing the beam pipe diameter or main coupler port size. Keeping the end cell unchanged from the E-XFEL 3.9 GHz cavity shape, the LCLS-II design shifts the lowest dipole modes up by 100 MHz by reducing the beam pipe diameter from 40 mm to 38 mm with a tapered transition. Additionally, previous vertical tests showed a thermal quench of the HOM coupler antenna at around 20 MV/m in CW mode for the FLASH and E-XFEL cavity designs [ 14 14 ] in which the 3.9 GHz cavity nominally operates in pulse mode. In high-repetition rate machines, such as LCLS-II and SHINE, the cavity will operate in CW mode, which results in higher heat loads. To reduce the risk of cavity quench caused by the HOM antennas, some modifications on the HOM-coupler antennas are necessary to adapt them for CW operation. In this paper, we present our particular modifications on the end-group of the 3.9 GHz cavity for SHINE, including a redesigned end cell and beam pipe, an adjustment of the main coupler antenna, and a tuning of the HOM notch. 3 Redesign of end group 1 To shift the potentially dangerous lowest dipole modes, we redesigned the end half-cell with an iris diameter of 38 mm and a straight beam pipe of the same size. Fig. 3 Fig. 3 shows the end-group shape of the SHINE 3.9 GHz cavity, and compares it to the ones in the E-XFEL and LCLS-II projects. In detail, compared to E-XFEL, we reduced the diameter of the beam tube from 40 mm to 38 mm; compared to LCLS-II, we removed the tapered transition. This modification gives a simpler geometry that is expected to be beneficial to cavity fabrication and surface processing. The cavity shape was designed using the SUPERFISH [ 15 15 ] and BUILDCAVITY codes [ 16 16 , 17 17 ]. Table 2 Table 2 shows the geometric parameters of the 3.9 GHz cavity for SHINE and a comparison to the other two projects. The parameters describing a half cell are shown in Fig. 4 Fig. 4 . In the end-cell redesign, only two parameters were modified, i.e., the iris radius ( R ir ) and the horizontal axis length of the equatorial ellipse ( A ). 1 1 Parameters (mm) 1 1 Inner cell 1 1 End cell (E-XFEL, LCLS-II) 1 1 End cell (SHINE) 1 1 R ir (mm) 1 1 15 1 1 20 1 1 19 1 1 L (mm) 1 1 19.22 1 1 19.12 1 1 19.12 1 1 R eq (mm) 1 1 35.79 1 1 35.79 1 1 35.79 1 1 A (mm) 1 1 13.6 1 1 14.4 1 1 14 1 1 B (mm) 1 1 15 1 1 15 1 1 15 1 1 a (mm) 1 1 4.5 1 1 4.5 1 1 4.5 1 1 b (mm) 1 1 6 1 1 6 1 1 6 1 1 d (mm) 1 1 3 1 1 2.4 1 1 2.4 1 1 α (°) 1 1 17.5 1 1 27.3 1 1 26.5 1 1 D bp (mm) 1 1 30 1 1 40, 40 → 38 1 1 38 1 1 L bp (mm) 1 1 1 1 80.11 1 1 80.11 In the cavity design for SHINE, the frequency and field flatness were adjusted by A. We reached the goal frequency and achieved a flat field distribution in the nine cells by setting A at 14 mm. The field flatness of the SHINE 3.9 GHz cavity after modification is shown in Fig. 5 Fig. 5 , in which the accelerating gradient is normalized to 1 MV/m. Table 3 shows the main RF parameters of the SHINE 3.9 GHz cavity design, and a comparison with the parameters in E-XFEL and LCLS-II. Most of the RF parameters are the same, except for the R/Q and the lowest dipole mode frequency. Owing to the smaller iris diameter in the end half-cell, we achieved a slightly higher R/Q , and pushed the lowest dipole modes to 4165 MHz, which is 265 MHz higher than the operating frequency. The electric field distribution of the lowest dipole mode is shown in Fig. 2 Fig. 2 . This larger frequency distance would be sufficient to avoid the downshift of the accelerating mode frequency during the fabrication stage. 4 Coupler modification 1 The 3.9 GHz cavity includes four couplers comprising a fundamental power coupler (FPC), two HOM couplers, and a pick-up probe. Based on the redesign of the end cell and beam pipe, all the couplers need to be modified to meet the requirements of SHINE. In particular, as mentioned above, the HOM coupler antennas need to be optimized to avoid overheating from CW operation. LCLS-II modified the HOM F-part and antenna tip [ 11 11 ] to shift the tip away from the strong magnetic field region and reduce the surface area of the antenna exposed to the magnetic field. For SHINE, we adopted the same idea as LCLS-II in modifying the HOM coupler, and calculated the RF losses on the HOM coupler antennas. 4.1 Main coupler antenna adjustment 2 The FPC in the SHINE 3.9 GHz cavity is a coaxial coupler that transmits the RF power into the cavity. Similar to E-XFEL and LCLS-II, we adopted a hollow inner conductor to reduce its weight. Owing to the modification of the end-group, the length of the FPC antenna has to be adjusted to meet the external Q ( Q ext ) requirement of SHINE. The optimum Q ext is calculated from the following Eqs. (1) and (2) [ 18 18 ]: <math display="block" id="M1"><mrow><msub><mi>β</mi><mrow><mtext>opt</mtext></mrow></msub><mo>=</mo><msup><mrow><mrow><mo>[</mo><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mfrac><mrow><msub><mi>I</mi><mtext>b</mtext></msub><msub><mi>Q</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>R</mi><mo>/</mo><mi>Q</mi><mo stretchy="false">)</mo><mi>cos</mi><msub><mi>φ</mi><mtext>s</mtext></msub></mrow><mrow><msub><mi>V</mi><mtext>c</mtext></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mn>2</mn><msub><mi>Q</mi><mn>0</mn></msub><mi>δ</mi><msub><mi>f</mi><mtext>m</mtext></msub></mrow><mrow><msub><mi>f</mi><mn>0</mn></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mo>]</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mn>,</mn></mrow></math> <math display="block" id="M2"><mrow><msub><mi>Q</mi><mrow><mtext>ext</mtext></mrow></msub><mo>=</mo><msub><mi>Q</mi><mn>0</mn></msub><mo>/</mo><msub><mi>β</mi><mrow><mtext>opt</mtext></mrow></msub><mn>.</mn></mrow></math> For the SHINE 3.9 GHz cavity, using the nominal parameters in Table 1 Table 1 , the cavity RF parameters in Table 3 Table 3 , and the maximum average beam current of 0.3 mA, we obtained the optimum Q ext = 2.2 × 10 7 without microphonics detuning, and Q ext = 2.1 × 10 7 with a maximum estimated microphonics detuning of δf m =30 Hz. The microphonics will be compensated by a piezo-assisted tuner to minimize the peak detuning [ 19 19 ]. 1 1 Parameters 1 1 E-XFEL 1 1 LCLS-II 1 1 SHINE 1 1 Frequency (MHz) 1 1 3900 1 1 3900 1 1 3900 1 1 Number of cells 1 1 9 1 1 9 1 1 9 1 1 Active length, L cav (m) 1 1 0.346 1 1 0.346 1 1 0.346 1 1 R/Q ( Ω ) 1 1 748 1 1 751 1 1 758 1 1 G ( Ω ) 1 1 277 1 1 275 1 1 276 1 1 E pk / E acc 1 1 2.3 1 1 2.3 1 1 2.3 1 1 B pk / E acc [mT/(MV/m)] 1 1 4.9 1 1 4.9 1 1 4.9 1 1 Lowest dipole mode fre. (MHz) 1 1 3992 1 1 4092 1 1 4165 The target Q ext was achieved by choosing a proper antenna length. Figure 6 Figure 6 shows the relationship between Q ext and the distance from the antenna tip to the inner wall of the beam pipe. In order to ensure HOM safety, we used the conservative value of Q ext = 2.2 × 10 7 for the FPC. Therefore, an indentation of 8.0 mm was chosen for the FPC antenna tip in this paper, as indicated in Fig. 7 Fig. 7 . 4.2 HOM coupler tuning 2 The HOM couplers in the 3.9 GHz cavity are of the coaxial type, and have two basic functions: 1) rejecting the fundamental mode, and 2) extracting the HOM energy over a broad frequency band. The rejection of the accelerating mode can be fulfilled by tuning the HOM notch filter, as illustrated in Fig. 8(a) Fig. 8(a) . We simulated the S-parameters between the 4 coupler ports with CST Microwave Studio [ 20 20 ], as shown in Fig. 8(b) Fig. 8(b) . Fig. 9(a) Fig. 9(a) shows the transmitted signals from the FPC port to the pick-up and from the pick-up to HOM1 (port 2). The notch position for HOM coupler 1 (port 2) is shown in Fig.9(b) Fig.9(b) . Similarly, the notch frequency of HOM coupler 2 (port 3) was tuned the same way. The tuning sensitivity of the HOM notch filter around the goal frequency is about 0.24 MHz/ μ m. Usually, the tuning of the cavity HOM notch filter is completed at room temperature before the vertical test. A slight deviation from the goal notch frequency is common due to the tuning precision limit. We used the S-parameter, which is defined in Eq. (3), to calculate the average RF power leak of the fundamental mode through a HOM coupler, as shown in Fig.9(c) Fig.9(c) . Using 1 W as a reference, the HOM notch frequency of SHINE 3.9 GHz cavity should be tuned to within ± 8 MHz from the goal frequency. <math display="block" id="M3"><mrow><msub><mi>S</mi><mrow><mn>24</mn></mrow></msub><mo>−</mo><msub><mi>S</mi><mrow><mn>14</mn></mrow></msub><mo>=</mo><mn>10</mn><mi>lg</mi><mfrac><mrow><msub><mi>P</mi><mn>2</mn></msub></mrow><mrow><msub><mi>P</mi><mn>4</mn></msub></mrow></mfrac><mo>−</mo><mn>10</mn><mi>lg</mi><mfrac><mrow><msub><mi>P</mi><mn>1</mn></msub></mrow><mrow><msub><mi>P</mi><mn>4</mn></msub></mrow></mfrac><mo>=</mo><mn>10</mn><mi>lg</mi><mfrac><mrow><msub><mi>P</mi><mn>2</mn></msub></mrow><mrow><msub><mi>P</mi><mn>1</mn></msub></mrow></mfrac><mo stretchy="false">(</mo><mtext>dB</mtext><mo stretchy="false">)</mo></mrow></math> where S 24 - S 14 is calculated in Fig.9(b) Fig.9(b) , <math id="M4"><mrow><msub><mi>P</mi><mn>1</mn></msub><mo>=</mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><msub><mi>E</mi><mrow><mtext>acc</mtext></mrow></msub><mo>×</mo><msub><mi>L</mi><mrow><mtext>cav</mtext></mrow></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mrow><mi>R</mi><mo>/</mo><mi>Q</mi><mo>⋅</mo><msub><mi>Q</mi><mrow><mtext>ext</mtext></mrow></msub></mrow></mfrac></mrow></math> is the output power from port 1, P 2 is the radiation power from HOM coupler 1 (port 2), and P 4 is the input power to port 4. 4.3 RF losses on HOM antenna 2 Due to poor cooling of the HOM antenna inner conductor, heating of the niobium antenna tip by the magnetic field may lead to quench at high accelerating gradients, especially in CW mode operation. Figure 10 Figure 10 shows the geometry of the F-part and antenna in the 3.9 GHz HOM coupler. The RF losses can be expressed by the Eq. (4). We calculated the RF losses on the HOM antenna at the maximum operating gradient of 15 MV/m. Table 4 Table 4 shows the RF losses on the HOM antennas of the SHINE 3.9 GHz cavity as well as on the E-XFEL and LCLS-II antennas. The result shows that at the same accelerating gradient of 15 MV/m, the RF losses on the antenna for the SHINE cavity are lower than the other two and are around 3/4 of those for LCLS-II. Further, if the notch filter is not perfectly tuned but has a deviation within ± 8 MHz, the maximum change of the RF losses on the HOM antenna is less than 1 1 1 Parameter 1 1 E-XFEL 1 1 LCLS-II 1 1 SHINE 1 1 H pk (A/m) 1 1 2381 1 1 1222 1 1 1033 1 1 S (mm 2 ) 1 1 249.04 1 1 231.45 1 1 231.45 1 1 RF loss/ R s (A 2 ) 1 1 194.9 1 1 41.4 1 1 30.8 <math display="block" id="M5"><mrow><mtext>RF</mtext><mtext> </mtext><mtext>losses</mtext><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mtext> </mtext><msub><mi>R</mi><mtext>s</mtext></msub><mo>×</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mo>|</mo><mi>H</mi><msup><mo>|</mo><mn>2</mn></msup><mtext>d</mtext><mi>S</mi><mn>.</mn></mrow></math> 5 Dangerous HOMs check 1 HOMs with high loaded quality factor, Q L , may be coherently excited when beams pass through the cavity. Monopole HOMs may be excited when a bunch of charged particles passes through the axis of the cavity, while higher-pole HOMs may be excited if a number of charged particles traverse the cavity with an offset to the axis. The interaction of the HOM fields with the beams may deteriorate the beam quality in terms of the energy spread, emittance growth, or even beam stability. In addition, a significant HOM-induced power loss in the cavity will also add an extra heat load to the cryogenic system, which may lead a cost increase in building and operating the Linac. Generally, most of the HOM power can be extracted out of the cavities through the HOM couplers, the FPC, and the beam pipes. The cut-off frequency of the lowest dipole modes in the 38 mm diameter beam pipe is about 4.6 GHz. Hence, the HOMs above 4.6 GHz can propagate through the beam pipes, and their power can also be extracted by the coupler ports on adjacent cavities, dissipated by the bellows between cavities, or even absorbed by the HOM absorbers located at both ends of the cryomodule. The dangerous HOMs in a cavity are those trapped inside the cavity with high Q L , considerable R/Q , and frequencies close to the harmonics of the beam spectrum. Because the end-groups of the SHINE 3.9 GHz cavity have been redesigned with a smaller beam-pipe diameter, it is necessary to confirm that no dangerously trapped HOMs exist in the cavity. We have simulated the HOM eigenmodes up to 10 GHz with electric boundary conditions. Owing to the existence of hybrid modes, we do not specify the mode types. The Q L , longitudinal, and transverse R/Q of all the modes are displayed in Fig. 11 Fig. 11 . The transverse R/Q was calculated with a radial off-set of 1 mm to the beam axis. We used the following definitions for the longitudinal and transverse shunt impedance [ 21 21 ]: <math display="block" id="M6"><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>R</mi><mo>∥</mo></msub></mrow><mi>Q</mi></mfrac></mrow><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><msup><mrow><mrow><mo>|</mo><mrow><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>z</mi></msubsup></mrow></mstyle><msub><mrow><mrow><mo>[</mo><mrow><msub><mi>E</mi><mi>z</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mn>,</mn><mi>z</mi><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow></mrow><mrow><mi>r</mi><mo>=</mo><mn>0</mn></mrow></msub><msup><mi>e</mi><mrow><mi>i</mi><mi>k</mi><mi>z</mi></mrow></msup><mtext>d</mtext><mi>z</mi></mrow><mo>|</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msub><mi>ω</mi><mn>0</mn></msub><mi>W</mi></mrow></mfrac><mn>,</mn></mrow></math> <math display="block" id="M7"><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>R</mi><mo>⊥</mo></msub></mrow><mi>Q</mi></mfrac></mrow><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><msup><mrow><mrow><mo>|</mo><mrow><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>z</mi></msubsup></mrow></mstyle><msub><mrow><mrow><mo>{</mo><mrow><msub><mo>∇</mo><mo>⊥</mo></msub><mrow><mo>[</mo><mrow><msub><mi>E</mi><mi>z</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mn>,</mn><mi>z</mi><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow></mrow><mo>}</mo></mrow></mrow><mrow><mi>r</mi><mo>=</mo><msub><mi>r</mi><mn>0</mn></msub><mn>,</mn></mrow></msub><msup><mi>e</mi><mrow><mi>i</mi><mi>k</mi><mi>z</mi></mrow></msup><mtext>d</mtext><mi>z</mi></mrow><mo>|</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msub><mi>ω</mi><mn>0</mn></msub><mi>W</mi></mrow></mfrac><mo>×</mo><mfrac><mn>1</mn><mrow><msup><mi>k</mi><mn>2</mn></msup></mrow></mfrac><mn>,</mn></mrow></math> where E z ( r , z ) is the longitudinal component of the electric field along the beam trajectory, r is the radial offset, and W is the stored energy in the cavity. ω 0 is the resonant angular frequency, k = ω 0 / c , and r 0 = 1 mm. A general case of HOM excitation can be expressed by the Eqs. (7) and (8) [ 21 21 ], representing the HOM-induced maximum power loss and maximum transverse kick, respectively: <math display="block" id="M8"><mrow><msub><mrow><mo>〈</mo><mi>P</mi><mo>〉</mo></mrow><mrow><mi>max</mi></mrow></msub><mo>=</mo><mfrac><mrow><mrow><mo>(</mo><mrow><msub><mi>R</mi><mo>∥</mo></msub><mo>/</mo><mi>Q</mi></mrow><mo>)</mo></mrow><msub><mi>ω</mi><mn>0</mn></msub><msubsup><mi>q</mi><mn>0</mn><mn>2</mn></msubsup></mrow><mrow><mn>4</mn><msub><mi>t</mi><mtext>b</mtext></msub></mrow></mfrac><mrow><mo>(</mo><mrow><mfrac><mrow><msup><mi>e</mi><mi>a</mi></msup><mo>+</mo><mn>1</mn></mrow><mrow><msup><mi>e</mi><mi>α</mi></msup><mo>−</mo><mn>1</mn></mrow></mfrac></mrow><mo>)</mo></mrow><mn>,</mn></mrow></math> <math display="block" id="M9"><mrow><msubsup><mrow><mrow><mo>|</mo><mrow><msub><mi>V</mi><mo>⊥</mo></msub></mrow><mo>|</mo></mrow></mrow><mrow><mi>max</mi></mrow><mrow><mi>r</mi><mo>=</mo><msub><mi>r</mi><mn>0</mn></msub></mrow></msubsup><mo>=</mo><mfrac><mrow><mrow><mo>(</mo><mrow><msub><mi>R</mi><mo>⊥</mo></msub><mo>/</mo><mi>Q</mi></mrow><mo>)</mo></mrow><msub><mi>ω</mi><mn>0</mn></msub><msub><mi>q</mi><mn>0</mn></msub><msub><mi>r</mi><mn>0</mn></msub><mi>k</mi></mrow><mn>4</mn></mfrac><mrow><mo>(</mo><mrow><mfrac><mrow><msup><mi>e</mi><mi>α</mi></msup><mo>+</mo><mn>1</mn></mrow><mrow><msup><mi>e</mi><mi>α</mi></msup><mo>−</mo><mn>1</mn></mrow></mfrac></mrow><mo>)</mo></mrow><mn>,</mn></mrow></math> where α = t b / τ , t b is the bunch spacing, τ =2 Q L / ω 0 is the HOM signal decay time, and q 0 is the individual bunch charge. In the SHINE linac, the minimum t b = 10 -6 s, and the maximum bunch charge q 0 = 300 pC. In general, SHINE requires the Q L of each HOM to be less than 10 6 to ensure negligible power loss due to these HOMs [ 6 6 ]. For the maximum repetition rate of 1 MHz, a HOM with Q L > 10 6 leads to t b τ , and the equations can be simplified to P max ( R ‖ / Q ) Q L I 0 2 and V ⊥ I 0 ( R ⊥ / Q ) Q L r 0 ω 0 / c . Using the maximum average beam current of 0.3 mA in the SHINE Linac, and Q 0 = 2 ×10 9 for the 3.9 GHz cavity, the sum of the static and dynamic heat loads at the fundamental mode is about 20 W per cavity. The maximum power loss calculated by equation 7 for all the modes (except accelerating mode) under 10 GHz is shown in the left picture of Fig. 12 Fig. 12 . The largest power loss appears at 7.33 GHz. Its E-field distribution is shown in Fig. 13(a) Fig. 13(a) , from which we can see that it is not a trapped mode. Some other modes with P max at around 0.1 W, such as the one at 7.5 GHz that belongs to the second monopole passband, are similar to the modes in cavities with the same inner cells in E-XFEL and LCLS-II [ 22 22 ]. The maximum power loss of these modes is small compared with the fundamental mode. Therefore, we do not regard them to be dangerous in the cavity. Similarly, we calculated the transverse voltages using Eq. (8) for all the modes (except accelerating mode) under 10 GHz in the SHINE 3.9 GHz cavity, as shown in the right picture of Fig. 12 Fig. 12 . One passband at around 9.6 GHz shows a relatively high transverse kick voltage with a maximum voltage of about 52 V. Because the RMS transverse momentum spread of the bunch in the SHINE 3.9 GHz section is similar to that in LCLS-II, which is around 1.3 kV [ 23 23 ] with the voltage less than 5%, the emittance growth caused by HOMs is expected to be small. The E-field distribution of the mode with the maximum transverse voltage ratio is described in Fig. 13(b) Fig. 13(b) . The mode is an octupole TE-like mode with R ⊥/ Q & lt;10 -3 Ω. In consideration of the above facts, we do not expect dangerous HOMs in the SHINE 3.9 GHz 6 Discussion 1 According to the FNAL studies, multipacting is not an issue in this HOM coupler design [ 24 24 ]. Multipacting did not appear in all the vertical test results of the E-XFEL 3.9 GHz cavities at LASA [ 9 9 ]. Therefore, we do not expect any surprises from multipacting in our cavity design. In this paper, we only describe the HOMs in one 3.9 GHz cavity. Some more HOM studies for the cavity string in a cryomodule are expected to be performed in the near future to assess the full HOM damping. Mechanical analysis of the cavity, and the cryogenic design of the helium tank and chimney are underway as well. Figure 14 Figure 14 shows a drawing of the bare SHINE 3.9 GHz cavity. 7 Conclusion 1 An optimized RF design for a 3.9 GHz cavity has been proposed for the SHINE project based on the existing E-XFEL and LCLS-II designs. We particularly redesigned the end groups of the cavity with a simple geometry with the aim of improving its performance for CW operation. A systematic study on the cavity RF design has been carried out, including a careful check of potentially dangerous HOMs. The simulation results show that the optimized design meets the SHINE requirements. Based on this design, we plan to fabricate some prototype cavities to verify their performance in the near future. References Z.Y. Zhu , Z.T. Zhao , D. 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TD-17-002 . https://web.fnal.gov/organization/TDNotes/Shared%20Documents/2017%20Tech%20Notes/TD-17-002.pdf https://web.fnal.gov/organization/TDNotes/Shared%20Documents/2017%20Tech%20Notes/TD-17-002.pdf Multipacting in the LCLS-II 3.9 GHz HOM coupler. TD-17-002 The authors would like to thank Paolo Pierini at ESS and Carlo Pagani at INFN-LASA for pre-reviewing the RF design. We are also grateful to Meng Zhang at SARI for discussing the physics parameters in the SHINE linac, and to Hongjuan Zheng at IHEP for discussing the HOM-related issues. 10.1007/s41365-020-00772-z This work was partially supported by the National Key Research and Development Program of China (No. 2016YFA0401900) and the Shanghai Municipal Science and Technology Major Project (No. 2017SHZDZX02). 2020-01-02 2020-04-17 2020-04-21 2020-07-05 2020-07-01 2021-04-10 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2020 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-020-00772-z http://dx.doi.org/10.1007/s41365-020-00772-z . 2020 Nuclear Science and Techniques 7 31