Optimization of the S-band side-coupled cavities for proton acceleration

Optimization of the S-band side-coupled cavities for proton acceleration 1 first-author https://orcid.org/0000-0002-3517-2342 1 https://orcid.org/0000-0002-0033-2107 1 https://orcid.org/0000-0002-6465-3030 1 https://orcid.org/0000-0001-6635-7465 1 corresp Corresponding author, Lilizhihui@scu.edu.cn Corresponding author, Lilizhihui@scu.edu.cn Lilizhihui@scu.edu.cn https://orcid.org/0000-0001-7892-8245 The Key Laboratory of Radiation Physics and Technology of Ministry of Education, Institute of Nuclear Science and Technology, Sichuan University, Chengdu 610065, China The Key Laboratory of Radiation Physics and Technology of Ministry of Education, Institute of Nuclear Science and Technology, Sichuan University, Chengdu 610065, China The proton beam with energy around 100 MeV has seen wide applications in modern scientific research and in various fields. However, proton sources in China fall short for meeting experimental needs owing to the vast size and expensive traditional proton accelerators. The Institute of Nuclear Science and Technology of Sichuan University proposed to build a 3 GHz side-coupled cavity linac (SCL) for re-accelerating a 26 MeV proton beam extracted from a CS-30 cyclotron to 120 MeV. We carried out investigations into several vital factors of S-band SCL for proton acceleration, such as optimization of SCL cavity geometry, end cell tuning, and bridge coupler design. Results demonstrated that the effective shunt impedance per unit length ranged from 22.5 to 59.8 MΩ/m throughout the acceleration process, and the acceleration gradient changed from 11.5 to 15.7 MV/m when the maximum surface electric field was equivalent to Kilpatrick electric field. We obtained equivalent circuit parameters of the biperiodic structures and applied them to the end cell tuning; results of the theoretical analysis agreed well with the 3D simulation. We designed and optimized a bridge coupler based on the previously obtained biperiodic structure parameters, and the field distribution un-uniformness was < 1.5% for a two-tank module. The radio frequency (RF) power distribution system of the linac was obtained based on the preliminary beam dynamics design. Proton beam Side-coupled cavity linac Accelerating cavity Biperiodic structure Bridge coupler. 1 Introduction 1 Since the successful first operation of a particle accelerator at the beginning of the last century, particle accelerator physics and technology have made remarkable advances [ 1 1 ]. Energetic particle beams extracted from accelerators have become an indispensable tool for exploring the basic structure and interactions of matter. Applications of particle accelerators have penetrated into almost all fields of the national economy and contributed significantly to the improvement of people’s living standards [ 2 2 ]. However, there are far fewer traditional proton and ion accelerators than there are electron accelerators [ 3 3 ]. Further, there are only a few laboratories able to operate them owing to their vast size and high cost. Therefore, research is ongoing to reduce costs and sizes of proton and ion accelerators. At the beginning of the 1990s, a research group, motivated by the need of a small hadron therapy device suitable for hospitals under the TERA Foundation, proposed to use S-band acceleration structures to accelerate protons [ 4 4 ]. A scheme with the cyclotron as an injector (extraction energy of 30 MeV and 60 MeV) and SCL as the booster was proposed [ 4 4 – 8 8 4–8 ]. The feasibility of an S-band SCL structure for a 60 MeV proton beam accelerator was confirmed experimentally with an effective acceleration gradient of 28.5 MV/m, which was considerably higher than that obtained in conventional proton acceleration structures [ 9 9 – 11 11 9–11 ]. This structure paved the way for compact and cheaper proton facilities. The Institute of Nuclear Science and Technology of Sichuan University is operating a CS-30 cyclotron, which can accelerate protons (extraction energy of 26 MeV), deuterons, and alpha particles. The cyclotron is currently used for studies on radioisotope production and material irradiation [ 12 12 ]. To achieve better utilization of the CS-30 and satisfy the demands of fundamental experimental research, such as the single-event effect and the biological effect of radiation, we proposed to build an S-band SCL booster with a total length of about 10 m to accelerate proton beams to 120 MeV. This paper mainly discussed the optimization design of SCL cavity structures and related work of the RF design. 2. Design and optimization of a single cavity 1 The SCL belongs to the biperiodic structures composed of the accelerating cavities (ACs) and coupling cavities (CCs) [ 3 3 ]. ACs are located on the beam axis, while CCs are located between ACs and off-axis up and down alternately, as shown in Fig. 1 Fig. 1 . When operating at π/2-mode, the electric field distribution in ACs was just as in the π-mode, while the electromagnetic field was approximately zero in CCs for the π/2 phase shifts compared with that in the adjacent ACs [ 13 13 ]. The high-frequency characteristics of the SCL structures were mainly defined by ACs and had both a higher acceleration gradient as π-mode structures and good stability as π/2-mode structures, which afforded the overall structure the advantages of a high acceleration gradient, good stability, insensitivity to machining errors, superior temperature control, and assembly diversity [ 14 14 , 15 15 ]. For easy fabrication, the SCL usually adopts a graded β structure, i.e., AC length is kept constant as β g λ /2 in each tank, where β g is called geometry β . For different tanks, β g increases gradually as the particle energy increases for high acceleration efficiency [ 3 3 ], and it is roughly equal to the average speed of protons through the tank. However, the shape of cavities does not change much with the smoothness of the electric field distribution and convenience for cavity fabrication and post-processes. Referring to the SCL cavities model developed in the Superfish program [ 16 16 ], we designed the initial model of ACs and CCs, as shown in Fig. 2 Fig. 2 . Fig. 2(a) Fig. 2(a) shows a simplified diagram of the AC structure projected in the ( r , z ) plane, with 14 control parameters provided for cavity optimization. Among them, three parameters a , b , and c were used to identify the shape and position of the tuning ring in the tuning procession and would not apply in the initial cavity optimization. The cavity length, L , was determined by the average particle velocity crossing the tank β g . S is the wall thickness and is mainly defined by the requirement of cooling and mechanical stability. D ac is the diameter of the AC and is determined by the resonate frequency. In our optimization, parameters S and D ac were kept constant in all ACs for the convenience of cavity processing and welding. The maximum surface electric field was another crucial factor as it determined the field level at which the cavity could be operated safely. The maximum surface electric field usually locates at the chamfer of the nose cone region, so R 1 and R 0 were the essential parameters in our optimizations. We also had to consider that the linac could accept the beam extracted from the cyclotron, which had a relatively larger emittance than that from linac, so the aperture radius R b required optimization to the largest possible size. Fig. 2 (b) Fig. 2 (b) shows the schematic diagram of the CC structure projected in the ( r , z ) plane. Since in the ideal biperiodic structure, the power level in CC was almost zero, even in real operation, it was also much lower than that in ACs. The only necessary procedure was to keep it resonant at 3 GHz by adjusting the gap width g c . Therefore, our main task concentrated on the optimization of ACs, and all work was based on numerical simulation with CST Studio Suite. The maximum electric field is a function of RF frequency [ 3 3 , 5 5 ]: <math display="block" id="M1"><mrow><mtable><mtr><mtd><mrow><mi>f</mi><mo> </mo><mrow><mo>(</mo><mrow><mtext>MHz</mtext></mrow><mo>)</mo></mrow><mo>=</mo><mn>1.64</mn><msubsup><mi>E</mi><mtext>k</mtext><mn>2</mn></msubsup><msup><mi>e</mi><mrow><mo>−</mo><mn>8.5</mn><mo>/</mo><msub><mi>E</mi><mtext>k</mtext></msub></mrow></msup><mo> </mo><mo>.</mo><mo> </mo></mrow></mtd></mtr></mtable></mrow></math> When the working frequency was 3 GHz, we obtained the maximum electric field E k of 46.8 MV/m from Eq (1). The experimental results demonstrated that in the real case, maximum electric field maintainable was also determined by the pulse length, and for short pulse machine, it could be several times larger than Kilpatrick limits E k . The acceleration gradient was obtained as a function of work conditions and was proportional to the maximum surface electric field. We defined E acc,K as the acceleration gradient when the maximum electric field was E k as in Eq (2): <math display="block" id="M2"><mrow><mtable><mtr><mtd><mrow><msub><mi>E</mi><mrow><mtext>acc</mtext><mo>,</mo><mtext>  K</mtext></mrow></msub><mo> </mo><mo>=</mo><mo> </mo><mfrac><mrow><msubsup><mstyle mathsize="140%" displaystyle="true"><mo>∫</mo></mstyle><mn>0</mn><mi>L</mi></msubsup><mi>E</mi><mrow><mo>(</mo><mrow><mi>s</mi><mo>,</mo><mo> </mo><mo> </mo><mi>β</mi><mo>,</mo><mo> </mo><mo> </mo><mi>t</mi></mrow><mo>)</mo></mrow><mtext>d</mtext><mi>s</mi></mrow><mrow><msub><mi>E</mi><mrow><mtext>peak</mtext></mrow></msub></mrow></mfrac><msub><mi>E</mi><mi>k</mi></msub><mo> </mo><mo>.</mo></mrow></mtd></mtr></mtable></mrow></math> The E peak is the maximum surface electric field, E (s, β, t) is the electric field along the cavity axis, and E k is the Kilpatrick limit. Obviously, the maximum surface electric field is most severe for the smallest cavity. Therefore, our optimization focused on the shortest cavity with length L at 11.5 mm, which corresponded to the CS-30 extraction energy of 26 MeV ( β =0.23). The effective shunt impedance was the ratio of the voltage squared to the power loss in the cavity that described the cavity acceleration efficiency: <math display="block" id="M3"><mrow><mtable><mtr><mtd><mrow><msub><mi>R</mi><mrow><mtext>eff</mtext></mrow></msub><mo> </mo><mo>=</mo><mo> </mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><msubsup><mstyle mathsize="140%" displaystyle="true"><mo>∫</mo></mstyle><mn>0</mn><mi>L</mi></msubsup><mi>E</mi><mrow><mo>(</mo><mrow><mi>s</mi><mo>,</mo><mo> </mo><mi>β</mi><mo>,</mo><mo> </mo><mi>t</mi></mrow><mo>)</mo></mrow><mtext>d</mtext><mi>s</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mi>P</mi></mfrac><mo> </mo><mo>.</mo></mrow></mtd></mtr></mtable></mrow></math> From 3D electromagnetic simulation results, it was evident that the maximum electric field was located at R 1 . Fig. 3 Fig. 3 shows the relationship of the effective shunt impedance R eff and the acceleration gradient E acc,K as a function of parameter R 1 , while keeping the resonant frequency at 3 GHz by adjusting the cavity diameter D ac . It is shown that when R 1 changed from 0.1 to 1.9 mm, the R eff decreased from 23.9 to 21.1 MΩ/m, but the E acc,K increased from 5 to 11.6 MV/m. In the R 1 adjustment process, the D ac varied from 74.5 to 73.4 mm. The value of R 1 = 1.1 mm was finally chosen for a small maximum surface electric field with good acceleration efficiency. Figure 4 Figure 4 shows the relationship between R b / L , R eff, and E acc,K . When R b / L changed from 0 to 0.35 (corresponds R b from 0 to 4 mm), the R eff and E acc,K were both decreasing. When R b / L was < 0.13 ( R b = 1.5 mm), the decreasing trend was relatively flat; however, when R b / L was > 0.13, both R eff and E acc,K decreased more rapidly as R b / L increased. As a trade-off between large acceptance and good RF properties, R b = 3.5 mm was chosen as our research value. The corresponding R eff and E acc,K were 22 MΩ/m and 10.8 MV/m, respectively. Compared with those when R b = 2.5 mm, they decreased by 22.4% and 15.2%, respectively. As energy increased, the cavity length should also be increased. We adjusted the gap width g rather than cavity diameter D ac to keep the eigenfrequency at 3 GHz. Fig. 5 Fig. 5 shows the trend of R eff and E acc,K for the entire accelerating range ( β value ranged from 0.23 to 0.46, corresponding energy changed from 26 to 120 MeV). R eff ranged from 22.5 to 59.8 MΩ/m and E acc,K increased from 11 MV/m at 26 MeV to 14.7 MV/m at 120 MeV. The final cavity geometry parameters are listed in Table 1 Table 1 . 1 1 Name 1 1 Optimized data (mm or angle) 1 1 D ac 1 1 74.64 1 1 S 1 1 3 1 1 R b 1 1 3.5 1 1 R 0 1 1 2.9 1 1 R 1 1 1 1.1 1 1 R 2 1 1 2.5 1 1 F 1 1 0.9 1 1 Α 1 1 45 (angle) 1 1 L 1 1 11–23 1 1 G 1 1 5–14.4 3. Equivalent circuit parameters and end cell tuning 1 In SCL, ACs and CCs were coupled through coupling slots and formed a biperiodic structure with the equivalent circuit shown in Fig. 6 Fig. 6 . Subscript A(C) represents AC(CC). The circuit was characterized by the followed circuit parameters: L A and L C were the circuit inductance, C A and C C were circuit capacitances, and R A and R C represented circuit resistances. The nearest-neighbor coupling was k , while the second nearest-neighbor couplings were k A and k C for AC-AC and CC-CC, respectively. According to the cavity equivalent circuit model, we wrote the resonance equations of the infinite biperiodic structure with non-lossy [ 17 17 - 19 19 17-19 ]: <math display="block" id="M4"><mrow><mtable><mtr><mtd><mrow><mrow><mrow><mtable><mtr><mtd><mrow><msub><mi>U</mi><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>I</mi><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mrow><mo>(</mo><mrow><mn>2</mn><mi>j</mi><mi>ω</mi><msub><mi>L</mi><mtext>A</mtext></msub><mo>+</mo><mfrac><mn>1</mn><mrow><mi>j</mi><mi>ω</mi><msub><mi>C</mi><mtext>A</mtext></msub></mrow></mfrac></mrow><mo>)</mo></mrow><mo>+</mo><mi>j</mi><mi>ω</mi><mi>k</mi><msqrt><mrow><msub><mi>L</mi><mtext>A</mtext></msub><msub><mi>L</mi><mtext>C</mtext></msub></mrow></msqrt><mrow><mo>(</mo><mrow><msub><mi>I</mi><mrow><mn>2</mn><mi>n</mi></mrow></msub><mo>+</mo><msub><mi>I</mi><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub></mrow><mo>)</mo></mrow><mo>+</mo><mi>j</mi><mi>ω</mi><msub><mi>k</mi><mtext>A</mtext></msub><msub><mi>L</mi><mtext>A</mtext></msub><mrow><mo>(</mo><mrow><msub><mi>I</mi><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>I</mi><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>3</mn></mrow></msub></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>U</mi><mrow><mn>2</mn><mi>n</mi></mrow></msub><mo>=</mo><msub><mi>I</mi><mrow><mn>2</mn><mi>n</mi></mrow></msub><mrow><mo>(</mo><mrow><mn>2</mn><mi>j</mi><mi>ω</mi><msub><mi>L</mi><mtext>C</mtext></msub><mo>+</mo><mfrac><mn>1</mn><mrow><mi>j</mi><mi>ω</mi><msub><mi>C</mi><mtext>C</mtext></msub></mrow></mfrac></mrow><mo>)</mo></mrow><mo>+</mo><mi>j</mi><mi>ω</mi><mi>k</mi><msqrt><mrow><msub><mi>L</mi><mtext>A</mtext></msub><msub><mi>L</mi><mtext>C</mtext></msub></mrow></msqrt><mrow><mo>(</mo><mrow><msub><mi>I</mi><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>I</mi><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><mo>)</mo></mrow><mo>+</mo><mi>j</mi><mi>ω</mi><msub><mi>k</mi><mtext>C</mtext></msub><msub><mi>L</mi><mtext>C</mtext></msub><mrow><mo>(</mo><mrow><msub><mi>I</mi><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>2</mn></mrow></msub><mo>+</mo><msub><mi>I</mi><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub></mrow><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mo>.</mo></mrow></mtd></mtr></mtable></mrow></math> U n corresponds to a voltage driven in the n th circuit and I n is the circulating current in the n th circuit. In the RCL resonant circuit, cavity resonant frequency is <math id="M5"><mrow><msup><mtext>ω</mtext><mrow><mtext>-2</mtext></mrow></msup><mtext> = 2LC</mtext></mrow></math> [ 13 13 ]; in the equivalent circuit, the frequency of AC and CC are equal to <math id="M6"><mrow><mn>1</mn><mo>/</mo><msqrt><mrow><mn>2</mn><msub><mi>L</mi><mtext>A</mtext></msub><msub><mi>C</mi><mtext>A</mtext></msub></mrow></msqrt></mrow></math> and <math id="M7"><mrow><mn>1</mn><mo>/</mo><msqrt><mrow><mn>2</mn><msub><mi>L</mi><mtext>C</mtext></msub><msub><mi>C</mi><mtext>C</mtext></msub></mrow></msqrt></mrow></math> , respectively. Moreover, we set <math id="M8"><mrow><msub><mtext>X</mtext><mrow><mtext>2n-1</mtext></mrow></msub><msub><mrow><mtext> = I</mtext></mrow><mrow><mtext>2n-1</mtext></mrow></msub><msqrt><mrow><msub><mrow><mtext>2L</mtext></mrow><mtext>A</mtext></msub></mrow></msqrt><mtext> </mtext></mrow></math> and <math id="M9"><mrow><msub><mtext>X</mtext><mrow><mtext>2n</mtext></mrow></msub><msub><mrow><mtext> = I</mtext></mrow><mrow><mtext>2n</mtext></mrow></msub><msqrt><mrow><msub><mrow><mtext>2L</mtext></mrow><mtext>C</mtext></msub></mrow></msqrt></mrow></math> <math id="M10"><mrow><mo stretchy="false">(</mo><msubsup><mtext>X</mtext><mtext>n</mtext><mtext>2</mtext></msubsup><mtext>/2</mtext></mrow></math> is energy stored in a single cavity) and substituted them into Eq. (4), yielding: <math display="block" id="M11"><mrow><mtable><mtr><mtd><mrow><mrow><mrow><mtable><mtr><mtd><mrow><mfrac><mrow><msub><mi>U</mi><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><mrow><mi>j</mi><mi>ω</mi><msqrt><mrow><mn>2</mn><msub><mi>L</mi><mtext>A</mtext></msub></mrow></msqrt></mrow></mfrac><mo>=</mo><msub><mi>X</mi><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mfrac><mrow><msubsup><mi>ω</mi><mtext>A</mtext><mn>2</mn></msubsup></mrow><mrow><msup><mi>ω</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow><mo>+</mo><mi>k</mi><mrow><mo>(</mo><mrow><msub><mi>X</mi><mrow><mn>2</mn><mi>n</mi></mrow></msub><mo>+</mo><msub><mi>X</mi><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub></mrow><mo>)</mo></mrow><mo>+</mo><msub><mi>k</mi><mtext>A</mtext></msub><mrow><mo>(</mo><mrow><msub><mi>X</mi><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>X</mi><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>3</mn></mrow></msub></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><msub><mi>U</mi><mrow><mn>2</mn><mi>n</mi></mrow></msub></mrow><mrow><mi>j</mi><mi>ω</mi><msqrt><mrow><mn>2</mn><msub><mi>L</mi><mtext>C</mtext></msub></mrow></msqrt></mrow></mfrac><mo>=</mo><msub><mi>X</mi><mrow><mn>2</mn><mi>n</mi></mrow></msub><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mfrac><mrow><msubsup><mi>ω</mi><mtext>C</mtext><mn>2</mn></msubsup></mrow><mrow><msup><mi>ω</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow><mo>+</mo><mi>k</mi><mrow><mo>(</mo><mrow><msub><mi>X</mi><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>X</mi><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><mo>)</mo></mrow><mo>+</mo><msub><mi>k</mi><mtext>C</mtext></msub><mrow><mo>(</mo><mrow><msub><mi>X</mi><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>2</mn></mrow></msub><mo>+</mo><msub><mi>X</mi><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub></mrow><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mo>.</mo><mo> </mo></mrow></mtd></mtr></mtable></mrow></math> We solved the homogeneous equation by assuming that: <math display="block" id="M12"><mrow><mtable><mtr><mtd><mrow><mrow><mrow><mtable><mtr><mtd><mrow><msub><mi>X</mi><mrow><mn>2</mn><mi>n</mi></mrow></msub><mo>=</mo><mo> </mo><mi>A</mi><mtext>cos</mtext><mn>2</mn><mi>n</mi><mi>φ</mi></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>X</mi><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo> </mo><mo>=</mo><mo> </mo><mi>B</mi><mtext>cos</mtext><mrow><mo>(</mo><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mtext>φ</mtext></mrow></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mo> </mo><mo>,</mo><mo> </mo></mrow></mtd></mtr></mtable></mrow></math> where φ = π q /2 N is the phase shift between adjacent cavities; N is the number of ACs, q = 0, 1, 2, …; and 2N corresponds to different eigenfrequencies of different eigenmodes. Substituting Eq (6) into Eq (5), the following relationship is obtained: <math display="block" id="M13"><mrow><mtable><mtr><mtd><mrow><mrow><mrow><mtable><mtr><mtd><mrow><mo>−</mo><mi>B</mi><mi>k</mi><mtext>cos</mtext><mi>φ</mi><mo> </mo><mo>=</mo><mtext> </mtext><mi>A</mi><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msubsup><mi>ω</mi><mtext>A</mtext><mn>2</mn></msubsup><mo>/</mo><msup><mi>ω</mi><mn>2</mn></msup><mo>+</mo><msub><mi>k</mi><mtext>A</mtext></msub><mtext>cos</mtext><mn>2</mn><mi>φ</mi></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mo>−</mo><mi>B</mi><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msubsup><mi>ω</mi><mtext>C</mtext><mn>2</mn></msubsup><mo>/</mo><msup><mi>ω</mi><mn>2</mn></msup><mo>+</mo><msub><mi>k</mi><mtext>C</mtext></msub><mtext>cos</mtext><mn>2</mn><mi>φ</mi></mrow><mo>)</mo></mrow><mo> </mo><mo>=</mo><mo> </mo><mi>A</mi><mi>k</mi><mtext>cos</mtext><mi>φ</mi></mrow></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mo>.</mo></mrow></mtd></mtr></mtable></mrow></math> Eliminating A and B, the dispersion relation is found [ 18 18 - 20 20 18-20 ]: <math display="block" id="M14"><mrow><mtable><mtr><mtd><mrow><msup><mi>k</mi><mn>2</mn></msup><mi>c</mi><mi>o</mi><mi>s</mi><msup><mo> </mo><mn>2</mn></msup><mi>φ</mi><mo> </mo><mo>=</mo><mo> </mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mfrac><mrow><msubsup><mi>ω</mi><mtext>A</mtext><mn>2</mn></msubsup></mrow><mrow><msup><mi>ω</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><msub><mi>k</mi><mtext>A</mtext></msub><mtext>cos</mtext><mn>2</mn><mi>φ</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mfrac><mrow><msubsup><mi>ω</mi><mtext>C</mtext><mn>2</mn></msubsup></mrow><mrow><msup><mi>ω</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><msub><mi>k</mi><mtext>C</mtext></msub><mtext>cos</mtext><mn>2</mn><mi>φ</mi></mrow><mo>)</mo></mrow><mo>.</mo></mrow></mtd></mtr></mtable></mrow></math> Five periodic parameters k , ω A , ω C , k A , and k C in Eq (8) were critical for studying the properties of the biperiodic structure. Therefore, according to Eq (8), five different modes corresponding to five different phase shifts ( φ 1 ~ φ 5 ) were needed to calculate these five parameters. The shortest period structures in our research model of CST simulations were two complete ACs, one complete CC, and two half CCs. Four eigenmodes were obtained for CST calculation when the periodic boundary conditions were applied in the longitudinal direction with phase shift at γ for each per period: <math display="block" id="M15"><mrow><mtable><mtr><mtd><mrow><msub><mi>φ</mi><mi>n</mi></msub><mo>=</mo><mo> </mo><mfrac><mrow><mi>γ</mi><mo>+</mo><mn>2</mn><mi>n</mi><mi>π</mi></mrow><mi>m</mi></mfrac><mo>,</mo><mo> </mo></mrow></mtd></mtr></mtable></mrow></math> where m is the number of cavities in the model and n ranges from 0 to m -1. In our case, m = 4. When n equals to 0, 1, 2, and 3, the corresponding φ 1 ~ φ 4 are γ/4, π/2+γ/4, π+γ/4, and 3π/2+γ/4, respectively. When we limited the phase shift within 0 to π, the phase shifts of the four modes were γ/4, π/2+γ/4, π-γ/4, and π/2-γ/4. We could easily to get parts 1 and 2 in Fig. 7 Fig. 7 corresponding to phases φ 1 and φ 3 as the 0- and π-mode electric field distributions were calculated when setting γ = 0. Furthermore, according to the symmetry of the dispersion curve, when setting the ranges of γ /4 from 0 to π/2, we could achieve a full dispersion curve as seen in Fig. 7 Fig. 7 . It was easy to understand parts 3 and 4 corresponding to φ 2 and φ 4 , respectively. Based on the phase relation deduced above, five different phase shifts between adjacent cavities could be calculated by setting two different phases in the shortest periodic structure. First, we calculated φ 1 = 0, φ 2 = π/2 (the electric field distribution existed in CC or AC), φ 3 = π/2 (the electric field distribution existed in AC or CC), and φ 4 = π with corresponding frequencies ω 1, ω 2, ω 3, and ω 4 by setting the structure period phase to 0. Second, we set the periodic structure phase shift to π and calculated φ 5 = π/4, φ 6 = 3π/4 with corresponding frequencies of ω 5 and ω 6 . Finally, we applied phases φ 1 = 0, φ 2 = φ 3 = π/2, φ 4 = π, φ 5 = π/4, φ 6 = 3π/4, and ω 1 , ω 2 = ω 3 , ω 4 , ω 5 , and ω 6 to Eq. (8), and it was easy to obtain five circuit parameters. As shown in Fig. 7 Fig. 7 , the "stopband" would appear for the untuned biperiodic structure, and this situation broke the stability at π/2-mode operation. By adjusting the gap capacitance of both AC and CC ( g and g c ), the stopband was eliminated. Nine situations would appear during the stopband adjustment: ω 2 could have been > , < , or = 3 GHz, and so could ω 3 . ω 2 and ω 3 were independent of each other during the adjustment process. We only needed to find the frequency corresponding to cavity (AC or CC) and adjusted the cavity gap width ( g or g c ) to meet a frequency at 3 GHz. Usually, the difference between ω 2 and ω 3 would be < 1 MHz to maintain stability under π/2-mode operation [ 18 18 ]. The illustration in Fig. 8 Fig. 8 is the shortest period structure in our research. In order to more intuitively show the change of the coupling coefficient k with the size of the coupling slot and ensure the D ac and the relative position between ACs and CCs kept constant for all tanks, we applied the cube coupling slot for simulation. As shown in Fig. 8 Fig. 8 , coupling slots are established between ACs and CCs. The coupling coefficient could be changed by adjusting the slot parameter L . Further, the graph shows the relationship among parameters L , k , ω a , and ω c . The range of k was between 0.0034 and 0.097 when L changed from 7.5 mm to 48 mm, and the effective shunt impedance R eff decreased from 31 to 22 MΩ/m as k increased. The coupling coefficient k was proportional to the slope of the dispersion curve at the π/2-mode. The larger the slope, the larger the frequency difference was between adjacent modes and the more stable was the systems. In the SCL structure, when the ACs length was the same as our model, the R eff was usually > 25 MΩ/m. L = 32 mm was chosen with a corresponding k at 0.04. At this time, R eff was about 29.6 MΩ/m; five parameters are shown in Table 2 Table 2 . 1 1 1 1 k 1 1 ω A (MHz) 1 1 ω C (MHz) 1 1 k A 1 1 k C 1 1 Cavity parameters 1 1 0.0414 1 1 3012.73 1 1 2997.68 1 1 -0.0087 1 1 0.0008 In the practical application, two end cells broke the original biperiodic properties [ 17 17 ], as shown by the dashed line in Fig. 10 Fig. 10 . This situation led to the electric field distributes un-uniformness. We needed to choose the right frequency for the end cell to compensate for field distribution caused by the loss of periodic properties. As shown in Fig. 9 Fig. 9 , with a complete AC (No. 0 cavity) as the end cell, we filled the missing cavity (No. -1 CC and No. -2 AC) by the dotted line in the biperiodic structure as there was only the secondary adjacent coupling between the AC and the CC in our calculation. The boundary condition was antisymmetric as X - 1 -n = -X - 1 +n for the electromagnetic level at zero in No. -1 CC. The end cell circuit equation is as follows [ 21 21 – 23 23 21–23 ]: <math display="block" id="M16"><mrow><mtable><mtr><mtd><mrow><msub><mi>X</mi><mn>0</mn></msub><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mfrac><mrow><msubsup><mi>ω</mi><mtext>A</mtext><mn>2</mn></msubsup></mrow><mrow><msup><mi>ω</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow><mo>+</mo><msub><mi>X</mi><mn>1</mn></msub><mfrac><mi>k</mi><mn>2</mn></mfrac><mo>+</mo><msub><mi>X</mi><mn>2</mn></msub><mfrac><mrow><msub><mi>k</mi><mtext>A</mtext></msub></mrow><mn>2</mn></mfrac><mo>+</mo><msub><mi>X</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub><mfrac><mrow><msub><mi>k</mi><mtext>A</mtext></msub></mrow><mn>2</mn></mfrac><mo>=</mo><mn>0.</mn></mrow></mtd></mtr></mtable></mrow></math> Taking the boundary condition X 0 = -X - 2 into Eq. (10), <math display="block" id="M17"><mrow><mtable><mtr><mtd><mrow><msub><mi>X</mi><mn>0</mn></msub><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mfrac><mrow><msubsup><mi>ω</mi><mtext>A</mtext><mn>2</mn></msubsup></mrow><mrow><msup><mi>ω</mi><mn>2</mn></msup></mrow></mfrac><mo>−</mo><mfrac><mrow><msub><mi>k</mi><mtext>A</mtext></msub></mrow><mn>2</mn></mfrac></mrow><mo>)</mo></mrow><mo>+</mo><msub><mi>X</mi><mn>1</mn></msub><mfrac><mi>k</mi><mn>2</mn></mfrac><mo>+</mo><msub><mi>X</mi><mn>2</mn></msub><mfrac><mrow><msub><mi>k</mi><mtext>A</mtext></msub></mrow><mn>2</mn></mfrac><mo>=</mo><mn>0.</mn></mrow></mtd></mtr></mtable></mrow></math> The end cell should be satisfied with <math display="block" id="M18"><mrow><mtable><mtr><mtd><mrow><mfrac><mrow><msubsup><mi>ω</mi><mtext>A</mtext><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>ω</mi><mo> </mo><mn>2</mn></msubsup></mrow></mfrac><mo>+</mo><mfrac><mrow><msub><mi>k</mi><mtext>A</mtext></msub></mrow><mn>2</mn></mfrac><mo>=</mo><mfrac><mrow><msubsup><mi>ω</mi><mi>e</mi><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>ω</mi><mo> </mo><mn>2</mn></msubsup></mrow></mfrac><mo>.</mo></mrow></mtd></mtr></mtable></mrow></math> This condition only needs to be satisfied with the π/2-mode. Taking <math id="M19"><mrow><msub><mtext>ω</mtext><mrow><mtext>π/2</mtext></mrow></msub><msub><mrow><mtext> = ω</mtext></mrow><mtext>A</mtext></msub><mtext>/</mtext><msqrt><mrow><msub><mrow><mtext>1-k</mtext></mrow><mtext>A</mtext></msub></mrow></msqrt></mrow></math> into Eq. (12): <math display="block" id="M20"><mrow><mtable><mtr><mtd><mrow><msubsup><mi>ω</mi><mi>e</mi><mn>2</mn></msubsup><mo>=</mo><mfrac><mrow><msubsup><mi>ω</mi><mtext>A</mtext><mn>2</mn></msubsup><mrow><mo>(</mo><mrow><mn>2</mn><mo>−</mo><msub><mi>k</mi><mtext>A</mtext></msub></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msub><mi>k</mi><mtext>A</mtext></msub></mrow><mo>)</mo></mrow></mrow></mfrac><mo>=</mo><mfrac><mrow><msubsup><mi>ω</mi><mrow><mi>π</mi><mo>/</mo><mn>2</mn></mrow><mn>2</mn></msubsup><mrow><mo>(</mo><mrow><mn>2</mn><mo>−</mo><msub><mi>k</mi><mtext>A</mtext></msub></mrow><mo>)</mo></mrow></mrow><mn>2</mn></mfrac><mo>.</mo></mrow></mtd></mtr></mtable></mrow></math> We then obtained ω e = 3006.3 MHz by substituting the circuit parameters in Table 2 Table 2 into Eq. (13). That is, the end cell needed to satisfy their frequency at <math id="M21"><mrow><msubsup><mtext>ω</mtext><mtext>A</mtext><mtext>,</mtext></msubsup></mrow></math> =3006.3 MHz in the biperiodic structure with other conditions, such as coupling slot and CC cavity, remaining constant. By optimizing AC control parameters a , b , and c link to the tuning ring, we achieved the final result of <math id="M22"><mrow><msubsup><mtext>ω</mtext><mtext>A</mtext><mtext>,</mtext></msubsup></mrow></math> = ω e = 3006.3 MHz, which corresponded to a single cavity frequency equal to 3051.5 MHz. We employed six ACs models with end cells at both ends in our investigation to verify the final tuning result. The electric field distribution shows a solid line in Fig. 10 Fig. 10 , and, compared with the field distribution without tuning (dashed line in Fig. 10 Fig. 10 ), the field un-uniformness was < 1%. 4. Design of the bridge coupler 1 A bridge coupler (BC) was applied to connect two tanks in a module and maintain the possibility of having enough space to install a quadrupole magnet; these are usually comprised of three parts: two half cavities (BCC) and a complete central cavity (BCA) [ 9 9 , 24 24 - 26 26 24-26 ]. Each BCC combines with a half CC from a single side of the tank to form a new complete cavity; meanwhile, the BCA couples with the waveguide to provide an RF source for the accelerator. Fig. 11(a) Fig. 11(a) is a cross-section of the BC, wherein each BCC is connected to the BCA through two coupling slots. Usually, two pairs of coupling slots are adjusted 90° to each other to reduce the second nearest-neighbor coupling. As shown in Fig. 11 (b) Fig. 11 (b) , the coupling coefficient between BCA and BCC can be adjusted by changing the angle or width of the slot. In order to keep the uniformness distribution of the electric field in the two tanks, BC periodic parameters must satisfy the periodic conditions when only AC and CC are existing. In this paper, the BCA played the role of AC in the periodic structure but did not provide the acceleration electric field for the particles; meanwhile, the BCC played a role as CC. The steps of the BC design could be divided into the following steps: (1) design of the BCA and BCC, (2) connection of three elements by coupling slots, and (3) adjustment of the coupling coefficient and calculate biperiodic circuit parameters. At the same time, the following two conditions were required: (1) selection of the suitable coupling slot to reduce the energy reflection into the waveguide and (2) ensuring that the resonance frequencies of BCA and BCC were equal to ω A and ω C , respectively [ 17 17 ]. We used the same biperiodic structure model and parameter calculation method as AC-CC. Five periodic parameters were ω BCA , ω BCC , k BCA-BCC , k BCA-BCA , and k BCC-BCC , where ω BCA and ω BCC were the frequencies of BCA and BCC in the periodic structure, respectively; k BCA-BCC was the coupling coefficient between BCA and BCC; and the second nearest-neighbor coupling was k BCA-BCA and k BCC-BCC . By adjusting the coupling slot and cavity capacitance, we obtained the basic parameters of BC, as shown in Table. 3 Table. 3 . 1 1 1 1 k BCA-BCC 1 1 ω BCA (MHz) 1 1 ω BCC (MHz) 1 1 k BCA-BCA 1 1 k BCC-BCC 1 1 Parameters of cavities 1 1 0.049 1 1 3002.83 1 1 2997.68 1 1 -0.002 1 1 0.015 Note that we only discuss the first three parameters in Table 3 as the second nearest-neighbor coupling for BCC-BCC and BCA-BCA would not exist in one BC. The k BCA-BCC value usually chosen for reducing store energy in BAC is 0.05 [ 17 17 ]. However, in cavity simulation processes, it is impossible to tune ( ω BCA , ω BCC ) to be precisely equal to ( ω A , ω C ) for the different shapes and the coupling method between BCA-BCC and AC-CC. As shown in Fig. 12 Fig. 12 , a BC was applied to connect the two tanks, where each tank contained five ACs with end cells at both ends to verify the final electric field distribution. The peak electric field unevenness was controlled below 1.5% after end cell and BC were added in the biperiodic structure. 5. Conclusion 1 Single-cell cavity optimization was performed for the β g range of 0.238–0.467. The effective shunt impedance ranged from 22. 5 to 59. 8 MΩ/m; the corresponding acceleration gradient was 11 MV/m at 26 MeV and increased to 14. 7 MV/m at 120 MeV when the maximum surface electric field was equal to the Kilpartic field. 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Tech . 42 ( 2 ), 020201 ( 2019 ). https://doi.org/10.11889/j.0253-3219.2019.hjs.42.020201 https://doi.org/10.11889/j.0253-3219.2019.hjs.42.020201 Development of RF superconducting cavity forward power coupler 10.1007/s41365-020-0735-7 This work was supported by the National Natural Science Foundation of China (Nos. 11375122 and 11875197). 2019-12-10 2020-01-19 2020-01-20 2020-02-22 2020-03-01 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/s41365-020-0735-7 http://dx.doi.org/10.1007/s41365-020-0735-7 . 2020 Nuclear Science and Techniques 3 31