Comparative Simulation Studies of Multipacting in Higher-Order-Mode Couplers of Superconducting RF Cavities

Multipacting (MP) in higher-order-mode (HOM) couplers of the International Linear Collider (ILC) baseline cavity and the Continuous Electron Beam Accelerator Facility (CEBAF) 12 GeV upgrade cavity is studied by using the ACE3P suites, developed by the Advanced Computations Department at SLAC. For the ILC cavity HOM coupler, the simulation results show that resonant trajectories exist in three zones, corresponding to an accelerating gradient range of 0.6–1.6 MV/m, 21–34 MV/m, 32–35 MV/m, and > 40MV/m, respectively. For the CEBAF 12 GeV upgrade cavity HOM coupler, resonant trajectories exist in one zone, corresponding to an accelerating gradient range of 6–13 MV/m. Potential implications of these MP barriers are discussed in the context of future high energy pulsed as well as medium energy continuous wave (CW) accelerators based on superconducting radio frequency cavities. Frequency scaling of MP’s predicted in HOM couplers of the ILC, CBEAF upgrade, SNS and FLASH third harmonic cavity is given and found to be in good agreement with the analytical result based on the parallel plate model.


I. INTRODUCTION
Multipacting (MP) is a phenomenon of resonant electron multiplication in which some specific inner surface areas of a radio frequency (RF) device are repeatedly bombarded by electrons which gain energy from the RF field contained in the device [1,2]. MP was observed in superconducting radio frequency (SRF) cavities for accelerators in the 1970s and has attracted many studies since then [3,4]. Due to the energy deposition of MP electrons at bombardment sites, the cavity wall temperature will rise. In case the SRF transition temperature is exceeded, this would lead to a thermal run away and ultimately rapid dissipation of the stored energy in the cavity.
As a result, MP used to constitute a limitation to the reachable accelerating gradient in a SRF cavity. Progress has been made over the past decades in understanding and alleviating the MP problem in SRF cavities. For example, by adopting an elliptical profile in an axially symmetric cavity operated in the TM010 mode, "hard" MP's [5] in which electrons may gain energy of hundreds of eV, can be fully avoided [6,7]. This has allowed successful applications of elliptical SRF cavities for electron acceleration (in linacs such as CEBAF, FLASH and storage rings such as TRISTAN, HERA, CESR, KEKB, and LHC etc.) as well as for proton acceleration (in linacs such as SNS).
Higher order modes (HOM) excitation in a RF cavity by charged particles has some adverse effects such as causing emittance growth of the beam under acceleration or resulting in additional heat load to the cryogenics in case of an SRF accelerator. Hence HOM damping and extraction are needed to avoid beam instability and to reduce unwanted heat load to the cryogenic system of an SRF accelerator. Various HOM dampers have been developed and found successful applications. A commonly used HOM damper in an SRF cavity is a coaxial-type coupler, such as the one originally developed for the 1300 MHz TESLA cavity [8]. A sketch is shown in Fig.1.
The TESLA-style HOM coupler has been adopted also in SRF cavities for the SNS (scaled to 805 MHz), the energy doubling of CEBAF from 6 GeV to 12 GeV (scaled to 1497 MHz), the European XFEL, and the proposed International Linear Collider (ILC).
Through systematic numerical simulation studies, the present work deals with the MP characteristics of HOM couplers of the ILC baseline cavity and the CEBAF upgrade cavity. Our studies are motivated by some recently observed experimental phenomena both at Jefferson Lab and elsewhere in TESLA-style 9-cell cavities. Those phenomena are not yet understood but suspected to be related to MP's in HOM couplers. Examples include the baking induced field emission turn-on phenomenon at a gradient of > 23 MV/m [9] and the pass-band mode excitation phenomenon with the detection of only low-energy X-rays [10]. In addition, the ILC baseline design requires an average gradient of 31.5 MV/m for cavities in operation with beam with an allowable gradient spread of 20%. This means that an ILC cavity may need to run at a gradient of up to 38 MV/m. Due to the gradient overhead consideration, a cavity operated with beam at 38 MV/m must be tested up to 42 MV/m during its vertical qualification testing [11]. Previous simulations of MP in HOM couplers of a TESLA-type cavity were limited to a gradient of 35 MV/m [12]. Therefore the present study explores a broader gradient range of up to 47.5 MV/m. Lastly, our studies also concern MP in HOM couplers of a CEBAF upgrade cavity. Eighty of these cavities are to be ultimately installed in CEBAF to double its energy from 6 GeV to 12 GeV [13]. A detailed analysis of MP characteristics of the HOM couplers in a CEBAF upgrade cavity is still lacking. It should be noted that there are differences in the physical configurations between the HOM couplers for an ILC cavity and CEBAF upgrade cavity [14]. Therefore it is of great interest to compare the MP characteristics of the two.

II. COAXIAL-TYPE HOM COUPLER
As shown in Fig. 1, the main parts of a TESLA-style coaxial-type HOM coupler include: (1) A loop antenna for coupling the HOM power; (2) A notch filter for rejection of the power due to the accelerating mode; (3) A HOM pickup probe for extracting HOM's to an external load. The notch filter is tuned to be at resonance with the fundamental mode of the cavity so as to suppress fundamental mode coupling to the HOM pickup probe. For an ILC cavity, two HOM couplers are located with one on each side of the cavity end-group. In case of a CEBAF 12 GeV upgrade cavity, two HOM couplers are located on one side of the cavity end-group [15,16]. The HOM couplers of SNS cavities are also based on the TESLA-type HOM coupler [17].

III. SIMULATION TOOL
The electromagnetic code suites ACE3P developed at SLAC are based on high-order parallel finite element method for geometry fidelity and simulation accuracy. ACE3P includes the eigenmode solver Omega3P and the particle tracking code Track3P [18]. In our simulation studies, Omega3P was used to obtain the field distribution in the cavity and HOM couplers and Track3P was used for calculation of trajectories of MP electrons. These codes are previously used in studying MP in SRF cavities with favorable results [19,20].
Here a brief introduction of the MP algorithm is given [21]. First, electrons are launched from specific surfaces at different RF phases. The initial velocity of the electron is perpendicular to the surface. The initial energy is fixed (typically 2 eV). The initially launched electrons follow trajectories determined by the electromagnetic fields and eventually hit the boundary of the structure. Secondary electrons are then emitted, the number of which depends on the secondary emission yield (SEY) of the boundary material and the impact energy. Tracking of electrons continues until a resonant trajectory is identified. The present study concerns predominantly the onesurface and two-surface MP. In these cases, a resonant trajectory is found when the following two criteria are met: (1) electrons return to a small area repeatedly at a nearly constant phase angle (numerically some spread in phase angle on the order of a few degrees is allowed); (2) the SEY corresponding to the impact energy is greater than unity. Typically the tracking time is long enough to ensure a pre-determined number of impacts are registered. The kinetic energy of electrons upon each impact is also registered for each involved surface. For resonant electrons hitting the same surface, their impact energy is average and plotted against the acceleration gradient. In our simulation, the niobium SEY curve is shown in Fig. 2 [22]. The SEY includes the contribution from true secondary electrons as well as reflected electrons and re-diffused electrons. The effect of SEY enhancement due to glance incidence is presently not considered in the code. A very useful concept of enhanced counter function (ECF) is used for analyzing the MP data. The ECF is a measure of total number of electrons after several RF periods.

A. ILC cavity HOM couplers
For the purpose of MP simulations in HOM couplers, we adopt a simplified model as shown in Fig. 3. It consists of a one-cell TESLA cavity (end half-cell shape) and a HOM coupler. The validity of this model is justified in viewing that the electromagnetic field configuration in the HOM coupler region is uniquely proportional to the field in the end cell of a 9-cell cavity. The subtle difference between the center half-cell and the end half-cells is expected to introduce negligible effect for the MP characteristics in HOM couplers. The notch filter RF behavior is very sensitive to the notch gap. Before the MP simulation, the notch frequency has to be tuned to the cavity resonance frequency. Details about tuning the notch gap can be found in Ref. [23].
The notch filter gap in our model is tuned to 1.41 mm.
The RF electric field distribution of the fundamental mode in the HOM can is shown in Fig. 4. The ratio of the local peak electric field pt E at the tip of the HOM antenna to the accelerating gradient pt acc E E is ~ 27%. At an accelerating gradient of 42 MV/m, the peak electric field in the HOM coupler region amounts to > 11 MV/m, which is clearly not a trivial field in terms of MP or even field emission consideration. In Zone 2, the gap between the inner conductor and the HOM can is 8.2 mm (Gap2 in Fig.1) by design.
Resonant trajectories exist in Zone 2 for an accelerating gradient range of 21-35 MV/m. The electron impact energy for the sustained impacting of 10, 20, and 30 times is given in Fig. 6(a). As can be seen, sustained impacting for 10 times is possible over a broad gradient range of 21-35 MV/m with an impacting energy in the range of 40-2500 eV. However, sustained impacting for 20 times is only possible for a reduced gradient range of 21-25 MV/m, around 28 MV/m and 32-34 MV/m with an impacting energy in the range of 40-1100 eV. The gradient range for which sustained impacting for 30 times is further narrowed to 32-34 MV/m with an impacting energy in the range of 400-800 eV. The enhanced counter function (for 20 RF periods) for different SEY's is given in Fig. 6(b). Detailed analysis of simulation results shows that Zone 2 MP can be divided into two groups.
The first MP is from 21-30 MV/m with typical electron trajectories as shown in Fig. 6(c). Due to its complex nature of the trajectories, we term this type of MP as "corner MP", in which more than two points are involved and perhaps involves some level of hybrid between a one-point MP and a two-point MP. As a result, the MP electron will ultimately slip out of the resonant phase after some number of impacts and ultimately get lost. This kind of "quasi resonant" barrier typically is not expected to be observable for a clean surface with a low SEY because the growth in electron density is limited by multiplication time. Even if the surface cleanliness is less ideal, such a barrier is expected to be only observable at the initial power rise during RF testing. The second MP is from 32-34 MV/m and the typical electron trajectories are shown in Fig. 6(d). This is a 1 st -order one-point MP with resonant trajectories that sustain even after 30 impacts. In addition, as shown in Fig. 6(a), the impact energy for this MP is in the range of 400-800eV. As a result, this kind of "hard" barrier can persist even after extended "conditioning" by deliberately parking the RF field at the barrier gradients. The electron impact energy for sustained impacting of 10, 20, and 30 times is given in Fig. 7 (a). The gradient range for sustained MP becomes narrowed as the impact number increases. The ECF after 20 RF periods are shown in Fig. 7(b). Through detailed analysis, the resonant electrons are found to impact surfaces in two regions shown in Fig. 7 (c). The electron impact energy in the upper region tends to be less than 100 eV. This barrier is a two-surface 3 rd -order MP which involves the inner surface of the HOM can and the outer surface of the HOM loop antenna. Typical upper region electron (around 50eV) trajectories (which can persist even after 30 RF cycles) are shown in Fig. 7(d).

B. CEBAF 12 GeV Upgrade Cavity HOM Coupler
The HOM couplers in a CEBAF 12 GeV upgrade cavity, as shown in Fig. 8, are based on the TESLA-type HOM coupler design. In comparison to the ILC cavity, the arrangement of the HOM couplers is different: (1) Two HOM couplers are placed on one side of the cavity; (2) The HOM antenna tip is farther away from the endcell cavity. As a result, the coupling of fundamental mode between the HOM couplers and the end cell is reduced. This reduces the HOM heating and is understood to be an important change for the CW operation of these cavities [14]. The ratio of the local peak electric field at the tip of the HOM antenna to the accelerating gradient pt acc E E is ~ 8% (27% for the ILC cavity). The ratio of the peak electric field across the gap in Zone 1 to the accelerating gradient pg acc E E is ~ 4.1% (19% for the ILC cavity). Simulation results show that resonant trajectories exist in Zone 1 for an accelerating gradient in the range of 6-13 MV/m. In this region, the gap between the HOM antenna and HOM can is tuned to 1.79 mm. The electron impact energy is shown in Fig.9. This simulation result agrees well with the analytical prediction for a twosurface MP across a pair of parallel plates [24]. This MP barrier corresponds to the one for the gradient range of 0.6-1.6 MV/m in an ILC cavity.

A. Overview and comparison of MP in HOM coupler of ILC and CEBAF upgrade cavity
The MP characteristics of the TESLA cavity HOM couplers were numerically studied previously by Gonin et al [12]. Those work predicted two barriers, one at acc E A summary of those prior results and our new simulation results is given in Fig.10. It should be mentioned that there is a slight difference in the Zone 1 gap length as used by different workers. In our model, the nominal notch filter gap 1.73 mm is tuned to 1.41 mm for reasons given in Sec. IV A. No tuning attempt was reported in Ref. [12,25] where the nominal gap length 1.73 mm was used instead. This results in a higher E acc (more than ~ 50%) for the same Zone 1 MP barrier by Gonin and Kostin as compared to the present work. During the commissioning and operation of the SNS superconducting linac, the high-beta 805 MHz cavities exhibited MP from 9.5 to 18.6 MV/m and abnormal signals originated from the HOM coupler were observed.
Simulations done by Gonin and coworker using ANALYST showed MP from 10 to 26 MV/m in Zone 3 [12]. Our simulation results predict two barriers in Zone 3 at gradient 11-19 MV/m (3 rd -order MP) and 25-35MV/m (1 storder MP), respectively. There is a reasonably good agreement between the first barrier we predict and previous experiment and simulation results. The second barrier we predict is above the experimentally achieved gradient, preventing direct comparison.
MP barriers in Zone 1 and Zone 3 were observed also in HOM couplers of the original 3.9 GHz 9-cell cavity.
During the high power test of the #2 cavity, thermal breakdown was observed in the HOM coupler with the first barrier occurring at about 1-2 MV/m and the second wider and stronger barrier from 12 to 19 MV/m [27]. The electric field dominates in Zone 1 of the ILC cavity. As can be seen in the fllowing, the two-surface MP model is suitable for analyzing Zone 1 over a broad frequency range. To study the frequency scaling, a series of cavities were scaled from a 1.3GHz TESLA-type one-cell cavity. The notch filter is confirmed to work still for the scaled cavities. The frequency scaling of the Zone 1 MP f is shown in Fig.11. The frequency scaling of the    Fig. 12. Also shown in Fig. 12    For the ILC, CEBAF upgrade and SNS HOM coupler, the cross section of the HOM loop antenna has a race-track shape. For the original 3.9 GHz HOM coupler, the cross section has a circular shape (see Fig. 13). The inner surface of the HOM can and the outer surface of the HOM loop antenna forms a two-surface region where MP is possible. From simulations, it is understood that the space occupied by resonant trajectories depends on the gradient. MP starts from region A at the MP onset gradient and moves toward region B with an increasing gradient (Fig.14).  The frequency scaling of the Zone 3 MP for scaled TESLA-shape cavities is shown in Fig.15. Table 3 gives parameters of Zone 3 1 st -order MP for the scaled TESLA-shape cavities. It should be noted that the scaled cavities are only for the purpose of analyzing the frequency scaling of Zone 3 MP. HOM damping is not considered. It should be also noted that the broad gradient range covered by our simulation concerns only the MP dynamics for the completeness of the understanding. In reality, there are physical limits (such as the critical RF field of the superconductor) prohibiting a cavity from reaching a field beyond a threshold. From Fig.15, we can see that the simulation results agreed fairly well with the two-surface MP analytical result over a broad frequency range. Fig.15 Frequency scaling of Zone 3 MP for scaled TESLA-type cavities.

( )
MV m GHz . It should be pointed out that the lower region (above 100 eV impact energy, see Fig.7 (a) & (c)) MP is only possible in the space between line A and B (Fig. 13) where the gap separation is fixed in case of ILC, CEBAF upgrade and SNS cavities; whereas in case of the 3.9 GHz cavity, the MP develops not only in the above region but also in regions beyond line B where the gap separation varies as the gradient is further raised. A summary of the Zone 3 MP parameters including the local electric field and the local gap separation is given in table 4.  From simulation result, "corner MP" is found in Zone 2 from 133 MV/m to 155 MV/m (for theoretical MP study) in CEBAF upgrade cavity. No MP is found in Zone 2 for 3.9GHz cavity. For ILC cavity, the MP is "corner MP" or one point MP. No scaling law is found for MP in Zone 2.

E. Discussion of MP modeling
The initial energy of the emitted electrons can be varied changed (the default value is 2 eV). In our simulations, 3 rd -order MP is found in Zone 3 in the ILC cavity, CEBAF upgrade cavity and SNS cavity with impact energy of around 50 eV. The impact energy increases to 80 eV when the initial energy is set to 5 eV.
Due to the fact that the initial energy of MP electrons is not very high, the magnetic force is much less compared to the electric force ( qv B qE  ). Therefore any initial angle deviation from the normal direction immediately disappears. In our simulation, the electric field is around several hundred KV/m in zone1, zone2 and zone3 and the magnetic field magnitude is 8 orders less than the electric field magnitude. So, the electric field will play a dominant role during the initial emission stage. Because of this, it is expected that our simulation results do reflect the true physical process despite our limited selection of perpendicular emission. However, in case of MP in a high magnetic field region, the initial angle effect is important and must be considered.
The secondary emission yield can be enhanced due to the glancing incident. This effect should be included in the analysis if measured SEY curves with glancing incident angles are available. The impact angle effect is not included in ACE3P at the moment.