Model evaluations of multipactor suppression in rectangular waveguides through grooved surfaces and static magnetic field Open Access

Multipactor discharge is a resonant vacuum phenomenon frequently observed in microwave systems, accelerator structures, and high power radio-frequency (RF) satellite components with multiple studies reported in the literature.^1–13 The discharge may occur for a wide range of frequencies (from the MHz range to tens of GHz) in different geometries.^14,15 The underlying mechanism is the continuous generation of secondary electrons upon synchronization with external RF fields.¹⁶ These secondaries, in turn, are accelerated by the RF fields, making them impact again, releasing even more electrons and increasing their population whenever secondary electron yields (SEY) exceed unity. Some of the known harmful effects of multipactor¹⁷ have been the failure of vacuum windows, generation of excessive noise in communication satellites, detuning of resonant cavities, and even increased surface outgassing.¹⁸ 

The harmful nature of multipactor on system reliability has led to numerous studies on suppressing secondary electron emission. Grooves in a rectangular waveguide were shown to suppress multipactor, provided their dimensions satisfied certain geometric conditions.^19,20 However, Chang et al.¹⁹ used a simple analytic model without considering the detailed dynamic evolution of the electron swarm. The simulations by Cai et al.²⁰ ignored the effects of grooves in distorting the local electric field. The use of a triangular groove cross section was reported by Cai and co-workers²¹ and shown to help, provided the angles exceeded some minimum level. However, both a self-consistent inclusion of the electric field in the presence of the modified geometry and the specifics of electronic motion were again excluded. Geometric modifications were also studied by Pivi et al.,²² and Stupakov and Pivi,²³ along with SEY lowering in the presence of a magnetic field. Their simulations simply calculated the probability of emission based on a distribution of energy and angular coordinates.

Geng et al.²⁴ reported benefits from the application of a weak external DC magnetic field along the direction of power flow. A simplified formula relating to the requisite magnetic field for multipactor suppression was derived. Magnetic biasing, when applied to a coaxial coupler (as compared to electric bias methods), would have the advantage of not requiring insulation between the inner and outer conductors. Since an insulator would typically deteriorate over time, the magnetic bias method would perhaps provide better reliability in the longer term. Experimental studies with particle-in-cell (PIC) calculations based on MAGIC, a commercial software tool, have also been reported with grooves, ridges, and/or an external magnetic field.²⁵ 

The concept of micro-porous arrays was examined by Ye et al.²⁶ and Sattler et al.,²⁷ who varied the aspect ratios and porosity. The largest SEY suppression was reported for the highest aspect ratio and porosity. Details, such as the electron swarm dynamics, and field modifications due to the array of holes were ignored. It may be mentioned that other reports of SEY suppression have included the use of copper oxide nanowires,²⁸ free-standing velvet whiskers,²⁹ graphite coatings,^30–32 etc. A drawback to using coatings for SEY suppression is their susceptibility to aging and degradation, especially in high power applications. Engineered surfaces, on the other hand, leverage topographical parameters for enhancing electron absorption and are more robust. Finally, SEY suppression was also shown to be achieved through the application of transverse electric fields.³³ 

A comprehensive analysis should include the appropriate swarm dynamics that account for the six-dimensional phase-space evolution, self-consistent calculations of the internal electric fields taking account of the geometry of the SEY mitigation structure chosen, and details of the energy- and angle-dependent secondary electron emission from the emitting surface. The underlying physical mechanism of secondary emission depends not just on the incident energy of the primaries, but on the impact angles as well.³⁴ Also, the construction of pores or grooves inevitably leads to local field enhancements at corners and edges. Such locally elevated fields potentially have the effect of providing stronger acceleration, raising electron swarm energies, and affecting the secondary electron emission yield.

Here, we first perform simulation studies of multipactor in rectangular copper waveguides and its possible suppression based on surface grooves. In later simulations, a static magnetic field in the axial directions is added to probe the cumulative effects. Changes in the numbers of grooves and their spacings for a specific waveguide structure and dimensions are analyzed. The role of groove placement along either one surface alone or along a pair of opposite surfaces is also evaluated. The simulations are based on the numerical Monte Carlo (MC) method, as discussed in detail elsewhere.¹⁷ Our results without an external magnetic field reveal that multiple grooves with patterns on a pair of parallel faces would mitigate multipactor up to fields of 10⁵ V/m. However, our simulation study was not limited to a pair of parallel groove patterns. A one-sided groove pattern seems to be quite effective as well. With the addition of an external magnetic field, even better suppression is predicted.

Although the concept of using a grooved structure is not new,^19–22,27 here, a dynamic analysis is carried out that includes field distortions due to deviations arising from grooves in the rectangular metal waveguide boundaries. Details of the method are discussed next.

The numerical multipactor calculations were carried out based on the Monte Carlo technique^35,36 with charge dynamical evolution based on Newtonian motion in a collisionless regime. In cases where a magnetic field was considered, the Lorentz force was appropriately added. For the simulations, electrons were generated from the top plate of the waveguide with energies chosen from a semi-hemispherical thermalized Boltzmann distribution following the approach discussed elsewhere.¹⁷ Practically, the particle release can be attained by seeding electrons through the illumination of the copper waveguide walls.³⁷ Due to the presence of grooves in the metallic waveguide surface, the driving field F(x, y, z) was modified (see Sec. II B) from the standard F₀ sin(πx/L_x)exp[−jk_zz] a_y expression, where k_z = (ω²ε₀μ₀ − (π/L_x)²)^½, f is the operating frequency, F₀ is the peak amplitude, and ε₀ and μ₀ represent the permittivity and permeability of free space, respectively. The waveguide dimension L_x along the x-axis was taken to be 72.1 mm, while the shorter distance between a set of parallel plates L_y was set at 2.1 mm. The present choice of 2.1 mm for the waveguide height was based on practical consideration tied to the experimental effort within our group.³⁸ The focus is on moderate power, and levels of about 5 kW are available for the RF sources. Incidentally, the average power P is related to the field intensity of the signal (F₀) as P = F₀² L_x · L_y/(4η_TE), where η_TE = 120π/[1-(f_c/f)²]^½, with L_x and L_y being the dimensions of the rectangular waveguide, f the signal frequency, and f_c = c/(2 L_x), with c the speed of light. The relatively small separation chosen here allows the attainment of the requisite high electric fields. The dimension along the axial z-direction was assumed to be infinitely long.

Space–charge effects and the development of internal fields leading to possible charge bunching or mutual repulsion based de-focusing were neglected. This was done since this analysis is geared simply toward gauging the possibility of population build-up for a given geometry and applied electric field magnitude. Hence, the system is assumed to be far from breakdown. So, in the present situation simulated, a Monte Carlo scheme without inclusion of a Poisson solver or the many-body interactions as achieved through the fast multipole method^39,40 would deemed to be sufficient. It may also be mentioned that multipactor resonant trajectories are tolerant to errors, as reported by Chojnacki.⁴¹ 

Most reports^42–47 have tended to incorporate a simple approach, such as the Vaughan model⁴⁸ with some modifications for treating secondary electron emission. Vaughan’s original treatment is empirical and does not incorporate an angular dependence that is expected to vary with incident particle energy. Analytic expressions for SEY including an angular dependence have been proposed,⁴⁹ although these were for gold electrodes. While physics-based numerical SEY calculations for copper have recently been reported,³⁴ a convenient parameterized form is not yet available. Here, as a compromise between accuracy and computational simplicity, the empirical model⁵⁰ based on data from Shih and Hor⁵¹ was used for the yield (σ),

and

with w = (E_i − E₀)/(E_{max_θ} − E₀), σ_{max_θ} = σ_{max_0} [1 + k_sθ²/π], E_{max_θ} = E_{max_0} [1 + k_sθ²/(2π)], and E_i denotes the incident electron energy. For incident energies below the workfunction, the SEY was set to zero. In the above, k_s is a surface factor (0 < k_s < 2) and has been taken to equal unity here. σ_{smax_0} and σ_{smax_θ} denotes the maximum yield at normal incidence and at an angle θ, respectively. Finally, E₀ represents the workfunction, and E_{max_θ} represents the energy at maximum yield σ_{max_θ}. Based on the above model, the SEY dependence on energy and incident angles shown in Fig. 1 is obtained. In the calculations, the workfunction value was set to 4.7 eV (a representative average value over the [100], [110], and [111] surfaces¹⁶) and E_{max_0} taken as 198 eV, with σ_{max_0} = 2.25. The values yielded fairly good agreement with data being generated within our group (though not shown here) and were guided by findings from Cimino et al.,⁵² suggesting σ_{max_0} > 2 and energy E_{max_0} less than 200 eV. Similarly, Furman and Pivi⁵³ had also shown σ_{max_0} > 2.

For completeness, it may be mentioned that more rigorously, the use of energy-dependent SEY model formulations need to combine a Monte Carlo technique that incorporates electron transport within the target material with the associated electron energy losses and mean-free paths obtained from Density Function Theory (DFT) evaluations. We have already reported such results that are more physically complete.^16,34 However, such a DFT-based analysis is beyond the present scope and would require specification of the surface orientation for concreteness.

For accurate multipactor analysis, it is important to determine the electromagnetic field distribution inside the waveguide structure since it dictates the driving force on the electron system. Field analysis of corrugated metallic structures or the presence of holes in metals is not new and has been discussed in several texts^54,55 and in the context of engineered structures supporting plasmon-like waves.^56,57

For the present simulations, the Ansys HFSS software⁵⁸ was used to obtain the frequency-domain electric field vector components in each waveguide at the operating frequency of 2.85 GHz. A two-port waveguide with the groove geometry was simulated. Only the transverse electric field components were needed since the waveguide supports the TE₁₀ mode only at this frequency. The electric field transverse components in the waveguide’s cross section were then exported to an ASCII file using a fine rectangular grid. A Matlab script was used to convert the frequency domain values to the time domain and obtain the electric field values at any desired point in the waveguide’s cross section using cubic interpolation. It may be mentioned for completeness that magnetic fields associated with RF signal in the waveguide had been included in a few initial evaluations but were seen to have negligible effects. Besides, other reports in the literature (e.g., Refs. 35 and 36) have also ignored magnetic fields in their kinetic simulation.

For the present simulation work, three different groove geometries in a rectangular waveguide were chosen, as shown in Fig. 2. The three-dimensional view in Fig. 2(a) shows a case with grooves only on one plate. The groove width is denoted as “W,” the spacing between grooves by “S,” and the depth as “D.” Planar view in Fig. 2(b) represents grooves on a pair of parallel surfaces with the bottom set directly below the one on top. Finally, grooves on two parallel sides (planar view), but with the top set shifted to the right by “W/2” are shown in Fig. 2(c). In all simulations presented and discussed here, the value of depth “D” was kept fixed at 1 mm. The power flow is axial, as shown by the arrow in Fig. 2(a).

Secondary electron yield (SEY) predictions were obtained from Monte Carlo based particle-in-cell (PIC) simulations. For each run, an electron swarm was released from the broad waveguide wall for a given applied field profile, as obtained from the HFSS simulator. A recent analysis had indicated the formation of columnar electron populations located symmetrically on either side of the center, arising from the SEY characteristic.^48,53 This suggests placing grooves on either side of the center. However, since the energy gain from the field depends on its magnitude, the separation between columnar structures could be a function of the applied field and waveguide dimensions. Thus, an array of closely spaced grooves could provide a more appropriate option, as was utilized here.

The electric field profiles were obtained for 17.16 kW input power and 2.85 GHz frequency. The computed electric field (F₀) at the waveguide center was ∼5 × 10⁵ V/m. Figures 3(a)–3(f) show the electric field profiles for different values of width “W” and groove spacing “S.” The depth for all cases was kept fixed at 1 mm (=“D”). Cases (3a)–(3d) are for six grooves on the top plate only, while Fig. 3(e) is for six matching grooves on both the top and bottom surfaces. Case (3f) is for six grooves on both the top and bottom surfaces, but with the bottom set displaced by 1.5 mm as was shown in Fig. 2(c). The dimensions chosen for the various cases were: (a) W = 3 mm, S = 4 mm, (b) W = 1 mm, S = 6 mm, (c) W = 3 mm, S = 6 mm, (d) W = 4 mm, S = 6 mm, (e) W = 3 mm, S = 4 mm, matching grooves on top and bottom, and (f) W = 3 mm, S = 4 mm, with bottom grooves offset from those at the top by 1.5 mm. A comparison of cases (3a)–(3d), all representing grooves on one side only, shows Fig. 3(d) with W = 4 mm and S = 6 mm to have the lowest field with the smallest areal spread. Although the peak electric field magnitude for the W = 1 mm and S = 6 mm case [in Fig. 3(b)] is not very high, it does include large regions of moderately high electric-fields.

The difference between the two cases of six matching grooves on both the top and bottom waveguide faces arises from the geometric variation. For the offset structure, electrons emerging from (say) the bottom surfaces lying on either side of a groove would have a finite probability of reaching a groove on the top face due to the overlap and likely be absorbed. The same argument would apply for electrons emerging from top surfaces lying on either side of a groove and moving downwards. However, without an offset structure, electrons emerging from surfaces lying on either side of a groove would be less likely to reach a groove on the opposite face. Without an offset, the overlap between an emitting surface and a groove on the opposite face is reduced. Hence, the matched groove geometry would not be as efficient at multipactor suppression. The scattering parameters for the simulated copper rectangular waveguide with six grooves on the top and bottom walls were |S₁₁| = −35.04 dB and |S₂₁| = −0.0488 dB without offset, and |S₁₁| = −37.536 dB and |S₂₁| = −0.0481 dB for the case of grooves with an offset. In both cases, the waveguide is well-matched to the feed ports, which are rectangular to match the dimensions of a waveguide with no grooves. For comparison, the scattering parameters for the simulated rectangular waveguide with no grooves were |S₁₁| = −75.39 dB and |S₂₁| = −0.046 dB. Although the presence of grooves increases the return loss, it is still extremely small. Using the simulation S-parameters and the waveguide length, the attenuation constants due to conductor losses obtained for the three cases were as follows: α_c = 0.0265 Np/m for the waveguide with no grooves (matching the theoretical α_c for the TE10 mode), α_c = 0.0273 Np/m for the waveguide with no-offset grooves, and α_c = 0.0272 Np/m for the waveguide with offset grooves. Thus, the grooves contribute a mostly negligible increase in conductor losses.

In order to gauge the effect on electron growth evolution, MC simulations were first performed for the waveguide without any grooves without an external magnetic field to serve as the baseline. Different electric field values for F₀ were chosen (corresponding to time-averaged powers of 513 W, 619 W, 686 W, 6.17 kW, and 17.16 kW), as given in Fig. 4. The lowest power levels used in experiments in our laboratory start from about 1.5 kW. The operating frequency was 2.85 GHz. The main takeaway for the smooth-wall base case is that the electron swarm is predicted to grow for all evaluated power levels, thus promoting multipactor. The growth rate increases with electric field values and points to a need for engineered techniques to mitigate multipactor.

Next, MC simulations for two different cases were carried out to compare the effect of grooves. In one scenario, smooth rectangular waveguide walls were assumed with no grooves (N), while a second configuration was taken to have a single set of six grooves on the bottom plate (S) with dimensions of W = 3 mm, S = 4 mm, and D = 1 mm. For each case, the responses at four different electric fields (F₀) of ∼8.7 × 10⁴ V/m, 9.5 × 10⁴ V/m, 10⁵ V/m, and 3 × 10⁵ V/m were gauged. The responses obtained are shown in Fig. 5. For the no groove (N) cases, the electron populations are predicted to grow monotonically, as with the previous trends of Fig. 4. However, for the S cases, the swarm population is predicted to reduce for the three fields of ∼8.7 × 10⁴ V/m, 9.5 × 10⁴ V/m, and 10⁵ V/m. However, at the highest 3 × 10⁵ V/m field, an increase is predicted, though the population rise is lower than the corresponding N case at the same field. Thus, the presence of grooves does seem to have a mitigating effect on multipactor.

Different groove placements were probed next through MC simulations and four cases were modeled: (i) a waveguide with no grooves (N), (ii) six grooves (W = 3 mm, S = 4 mm) on the bottom plate denoted as S, (iii) six matching grooves on both the bottom and top plates (each with W = 3 mm, S = 4 mm) denoted as SB, and (iv) six matching grooves on the bottom and top plates with a 1.5 mm offset at the top (SBO). The results obtained are depicted in Fig. 6 at an electric field (F₀) of 10⁵ V/m (i.e., time averaged 686 W power). With no grooves, the electron population is predicted to monotonically increase over time. With a matching pair of six grooves on both the parallel (top and bottom) surfaces, it is seen to arrest the rise but does not lead to suppression. Ultimately, a slow and gradual rise is predicted. Having grooves at the top and bottom (with no offset) is not seen to be as helpful as having grooves only on one side, since the former structure gives rise to larger number of corner points with high local fields. For the other two scenarios in Fig. 6 of six grooves at only the bottom surfaces or a pair of six grooves on the top and bottom surfaces but with an offset, the electron population is predicted to decrease. Hence, either structure would mitigate multipactor at the chosen fields.

Since the results do depend on groove dimensions, simulations were next carried out to probe possible changes with groove separation (S) and width (W) one at a time. The time-dependent electron populations shown in Fig. (7(a) were obtained for different W values of 1 mm, 3 mm, and 4 mm at a constant groove spacing of 6 mm at a 10⁵ V/m applied field. Rapid growth is predicted without any grooves. A reduction in population results for all three groove widths. The wider grooves (hence greater porosity) are a better option, qualitatively in keeping with the observations by Sattler et al.²⁷ However, effective mitigation depends on both the porosity and aspect ratio of the groove. If the width is increased at constant depth, the aspect ratio changes, in addition to an increase in porosity. Furthermore, the mutual coupling between adjacent ridges falls, and so does the screening effect; a phenomenon is known to also occur in multi-emitter arrays.⁵⁹ Consequently, the electric fields at the groove edges are likely to increase as the grooves are widened. In addition, a purely geometric effect will also be in play, with a narrower groove more likely to promote higher absorption at the sidewalls. Thus, increasing porosity, edge field enhancements at larger widths, and geometric effects produce somewhat complex influences, and their interplay decides the overall outcome. In Fig. 7(a), the 3 mm width is predicted to lead to higher mitigation as compared to the wider (and more porous) 4 mm groove, especially at longer times. As for the spacing, both the 4 mm and 6 mm separations for the W = 3 mm case in Fig. 7(b) show promise. Based on a simple porosity argument alone, one might expect the S = 4 mm case to have a stronger population suppression compared to S = 6 mm. Although the 6 mm spacing is seen to have a slight advantage in controlling population growth at shorter times, the two are predicted to be roughly equal at longer times. This outcome results from the local electric field distributions. The results of Figs. 3(a) and 3(c) reveal the peak field to be slightly higher for S = 6 mm but with a much lower spread. The larger spread for the S = 4 mm arises from the stronger coupling. Hence, not as many electrons are affected by the field for the S = 6 mm case. However, the situation is dynamic, and as secondaries form with random emission angles, the distinction begins to blur at later times.

After obtaining the electron swarm response without any external magnetic field, simulations were performed by adding a constant longitudinal magnetic field. Magnetic field effects have recently been reported in the context of single-surface multipactor.⁶⁰ Other reports⁶¹ have probed the effects of external magnetic fields in coaxial waveguides and found changes in the RF power thresholds. However, to the best of our knowledge, there have not been any studies on electron population dynamics relating to multipactor in rectangular waveguides with the inclusion of both different groove structures and an axial magnetic field on an equal footing.

The superposition of a constant magnetic field could have two, somewhat conflicting effects. One outcome would be the nudging of electrons in the swarm due to the added Lorentz force. This could move the swarm away from the high field regions, thus affecting the energy gain and SEY operating point. Second, due to trajectory curving, the electrons could strike the electrode surface at different angles and either hit a vertical groove surface or the horizontal part between grooves. As is well known, shallower angles lead to higher SEY.³⁴ As a result, the overall population of secondary electrons emitted from waveguide surfaces can be expected to depend on the axial magnetic field strength, waveguide dimensions, groove spacings, input power, operating frequency, etc., while possibly leading to anomalous population increases if the shallow angle scenarios were to dominate.

Simulation results obtained for the time-dependent electron population in a rectangular waveguide, with and without a static magnetic field, are shown in Fig. 8. Different scenarios with and without surface grooves were simulated at an electric field of 9.5 × 10⁴ V/m. A magnetic field of 10⁻⁴ T was assumed, which is roughly 2.7 times the Earth’s magnetic strength. For simulations involving grooves, a pair of six grooves at each of the top and bottom waveguide surfaces were used with W = 3 mm and S = 4 mm. In one case, the pair of six grooves lay directly one below the other, while in the other scenario, an offset of W/2 between the two six-groove patterns at the top and bottom was assumed. Results for the time-dependent electron population in Fig. 8 show that, as expected, the highest increase occurs for a waveguide without any grooves, regardless of the presence or absence of a magnetic field. However, with the 10⁻⁴ T field applied, the population growth is predicted to be somewhat smaller, but exhibits an increase from an initial value of 100 electrons to about 500 electrons over 35 ns. With six matching grooves on the top and bottom surfaces, electron populations at the final times of 35 ns, with and without a magnetic field, are about even. The total population after 35 ns seems to remain at about the starting level, though the magnetic field does produce a slight advantage. The best result is predicted for six grooves at the top and bottom with an offset between the two patterns. Both with and without the low 10⁻⁴ T magnetic field, the electronic population is predicted to decrease over time in Fig. 8. After about 35 ns, less than 30 electrons remain in the system, starting from an initial population of 100. Thus, the composite effect of multiple grooves on parallel surfaces in concert with a modest magnetic field is seen to produce multipactor suppression.

However, it may not be necessary to have two sets of six grooves for multipactor suppression. This had already been seen in the results of Fig. 8 without a magnetic field. Simulations for a waveguide with only one set of six grooves at the bottom surface yielded the results shown in Fig. 9 for a 9.5 × 10⁴ V/m electric field. For the grooves, the width W and spacing S were kept at 3 mm and 4 mm, respectively. With neither the grooves nor an external magnetic field applied, the electron population is predicted to grow quickly. Without any grooves but the 10⁻⁴ T field, smaller population growth is predicted. With just six grooves, the electron population is seen to reduce over time, both with and without an external magnetic field present. The best result is predicted in the presence of the external magnetic field, with the population decreasing from the initial 100 to about 5 electrons. Hence, for the cases of successful multipactor mitigation, the results for the pair of groove sets do not show a clear benefit of adding a static axial magnetic field after 35 ns like the case of a single set of grooves does.

The 10⁻⁴ T value is slightly less than the rough threshold predicted by Geng et al.²⁴ for a smooth waveguide without grooves. At an electric field F₀ of 9.5 × 10⁴ V/m, an associated power P of 619 W, and waveguide dimensions of 72.1 mm (=L_x) and 2.1 mm (= L_y), a value of 2.1 × 10⁻³ T is yielded for the requisite magnetic field as given in Eq. (1) of Geng et al.²⁴ In the present case, with grooves on the waveguide surfaces, the multipactor suppression is expected to be stronger or would not require as high a magnetic field. Hence, the suppression in Figs. 8 and 9 at the 10⁻⁴ T level seems promising and reasonably in line with expectations.

As a final comment, it may be mentioned that there have been previous reports on SEY reductions through surface treatments such as pre-baking, polishing, and exposure to plasmas for surface cleaning.^62,63 This arises because the treatments can change the effective mass, surface roughness, permittivity (which affects the energy loss function of incident primaries³⁴), local workfunction, and electron density of states that dictates the “supply” of surface electrons. The multipactor response of low energy electrons can be altered by such processing as electron entry into the electrode material under these conditions is relatively shallow. The response in high energy/power systems, however, is usually “hard” and persists even after extended processing due to the deeper penetration of energetic incident electrons. The behavior, trajectories, and inelastic energy loss are then dictated by the bulk characteristics rather than the surface properties or conditions. Hence, for such applications, the dual scheme of patterned grooves and an axial magnetic field studied here would be useful.

Multipactor can lead to a nonlinear breakdown, adversely impact communication systems and degrade the reliability of satellites. Although the physics is well known, the present contribution has focused on model evaluations for a mitigation strategy based on the dual use of surface modifications and a DC applied magnetic field. The analysis comprehensively included Monte Carlo based swarm dynamics, with electric field perturbations arising from the grooved geometries, details of the energy- and angle-dependent secondary electron emission from the emitting surfaces, and a static, axial magnetic field. Changes in the numbers of grooves, their spacings, and placement for a given waveguide structure were included in the analysis.

The results demonstrate that combinations of grooves and a DC magnetic field can effectively mitigate multipactor growth. For instance, a set of grooves with a 1 mm depth on only one side of the waveguide has been shown to suppress multipactor at a ∼10⁵ V/m electric field. Although this demonstrates evidence in support of such strategies, optimal combinations for an arbitrary field and operating frequency would be needed for more realistic and practical applications. Finally, higher electric fields deserve attention and will be reported elsewhere.

This research was supported, in part, by Department of Defense MURI Grant No. FA9550-18-1-0062 on “Multipactor and Breakdown Susceptibility and Mitigation in Space-Based RF Systems,” through a subaward from Michigan State University to Texas Tech University.

The data that support the findings of this study are available from the corresponding author upon reasonable request.