Analysis of multipactor in a rectangular waveguide using Spark3D software Free

Multipactor discharge is a resonant nonlinear electron multiplication that may occur in high power microwave devices at very low pressures, such as those operating in particle accelerators and satellite systems. This phenomenon has been known to cause vacuum window failure,¹ generation of excessive noise in communication satellites, detuning resonant cavities, and an increase in surface outgassing that could lead to a more destructive plasma discharge. Thus, multipactor research has been a topic of continued interest, for instance, see Ref. 2 and references therein.

Multipactor is caused by multiple secondary electron emissions due to repeated (multiple) electron impact onto material surfaces. The number of emitted electrons depends on the electron impact energy and is best captured by the Secondary Emission Yield (SEY) curve that provides the ratio of secondary to impacting (primary) electrons as a function of impact energy. It is noted that the SEY may take on fractional values since it considers the statistical average of many primary impacts. Obviously, an SEY value > 1 is necessary to have a multiplication at all. Additionally, for two-surface multipactor, resonance of the microwave field and electron oscillations is also needed for multipactor to occur. That is, freshly emitted secondary electrons should be accelerated in phase with the microwave field to then themselves become primaries and impact the surface with an energy that carries a high SEY. One notes that in single surface multipactor,³ e.g., for microwave-driven surface flashover across a dielectric window, the period of the electron motion may be much shorter than the microwave period, at least for microwave frequencies below 20 GHz.⁴ 

Limiting the discussion to two-surface multipactor, classical multipactor theory with the electron trajectories limited to one dimension has been developed early in Refs. 5–8. These provide impact energy and the SEY at a certain rf voltage, frequency, gap of the parallel plate, initial velocity of emitted electrons, rf phase starting electron emission, and multipactor order in one dimension, assuming resonance of the electron and rf field. Since then, it has been worked out in reasonable detail that actual secondary emission is caused with various emission velocities and angles, resulting in variations of electron phase and electron passing time. Thus, statistical theories considering distribution of velocity and angle of secondary electron emission were developed.^9–17 Vdovicheva et al.¹¹ and Anza et al.¹³ developed a statistical theory using Maxwellian distribution for the normalized initial velocity of secondary electron u, that is,

where d, m_e, ω, and $υt$ are the gap distance of the parallel plate, electron mass, angular frequency, and the velocity thermal spread, respectively. Likewise, Anza et al. also developed a multipactor theory in multicarrier systems and a numerical simulation including space charge effect, assuming the secondary release velocity of Maxwellian distribution and the release angle following the cosine law distribution.^14–16 

One finds a large body of work investigating multipactor based on the theory and techniques outlined in the references mentioned above, including many that analyze multipactor in rectangular waveguides. They elucidate multipactor threshold power of rectangular waveguides^18–21 and identify the conditions susceptible to multipactor depending on the frequency and gap size product.^{6–8,13,15,22,23} However, these studies provide little to no information on the impact of a spatially varying field distribution on the impact energy, SEY, order, and position for multipactor in a waveguide structure.

This study includes the spatially varying electric field distribution transverse and parallel to the microwave propagation direction in a TE₁₀ waveguide with varying heights. The presented relies on the extensive usage of a commercial multipactor analysis software “Spark3D” developed by Anza et al., which enables simulating multipactor in complicated 3D structures.¹⁷ In this paper, we introduce the simulation method utilized, provide an in-depth analysis of results with values obtained from the Spark3D simulations, and compare with available experimental data.

Building upon experimentally tested geometries, the height of two kinds of WR284 rectangular impedance transformer waveguide structures changes exponentially over its length, reaching a minimum waveguide height before exponentially transitioning back to the standard WR284. Note that TE₁₀ mode at 2.85 GHz is unaffected by the narrowing of the waveguide height (waveguide losses and propagation speed change, but the mode is conserved). Two values for the minimum height were chosen, 2.0 and 5.5 mm, respectively, primarily for consistency with the experimental setup geometry.²⁴ 

The electric field distribution without multipactor in the structure was carefully simulated employing a high frequency solver, Ansys HFSS, for 1 W microwave power, and then imported into Spark3D, cf. Fig. 1. To obtain a range of power/field levels, the electric field for the multipactor simulation was simply scaled with the square root of the input power.

The SEY values for electron impact onto the bounding surfaces, see Fig. 2, are based on the parametric model of Rodney and Vaughan,²⁵ using the maximum SEY and its impact energy of technical copper surface, obtained from the experiment by Baglin et al.;²⁶ then, an SEY curve of 0° electron impact was imported to Spark3D, which automatically adds the angle dependence of each electron impact.

Simulation was carried out in two regions for each waveguide structure. One is the whole region of the impedance transformer waveguide structure. The other is the local region located at center of the structure, where the highest field exists, cf. Fig. 1. The field of the local region is almost constant spatially; therefore, multipactor in this region is akin to that in simple parallel plate geometry, which has been analyzed by many researchers. The temporal electric fields of the local region with 2.0 and 5.5 mm, which are normalized by the square root of the input power, are depicted in Fig. 3. In this simulation, electrons are limited to each region, and absorption and emission of electrons are determined by the SEY of the region boundary. Thus, the local region simulation does not include electron diffusion outside the local region, which has a minor impact on the multiplication of electrons. However, it is expected that it hardly affects impact energy and SEY in the local region.

Figure 4 depicts the variation of the number of electrons in the whole of the waveguide model with a 2.0 mm gap, obtained from the Spark3D simulation with the initial electron number set to 300. It is expected that multipactor will occur in the structure if the number of electrons increases with time from the simulation. At which level multipactor will saturate is not topic of this investigation as the impact of the multipactor on the microwave propagation/mode and the space charge effect are neglected here.

Additionally, Spark3D can make advanced statistics in the Paraview package that provides the average SEY, average impact energy, impact density, and emission density for each surface mesh in a model. As an example, the average SEY for the 2.0 mm gap impedance transformer for 10 kW input power clearly shows a high SEY in the center region, see Fig. 5. Using those values at each mesh, impact energies and SEYs for all electron impacts in a simulation region were averaged. The equations for that are

where E_Ave is the average impact energy of the electron in a simulated region, σ_Ave is the average SEY for electron impacts in a simulated region, N is the total number of mesh cells, ρ_k is the impact density of the electron at mesh k, $Ek$ is the average impact energy of the electron at mesh k, and σ_k is the average SEY at mesh k. In the case of 10 kW, E_Ave and σ_Ave calculated using Eqs. (3) and (4) are 80 eV and 1.28, respectively. Likewise, E_Ave and σ_Ave for various input power values were obtained to investigate their power dependence.

Also, Spark3D computes an average multipactor order, and the multipactor position is deduced from the emission density drawing in Paraview such as in Fig. 5, thus enabling investigating variations in the multipactor order and position.

To build confidence into the reliability of the Spark3D simulations, the following was considered:

First, it was confirmed using the approximation found elsewhere¹³ that E_Ave and σ_Ave calculated using Eqs. (3) and (4) and the computed order are consistent with the computed electron amplification,

where N(t) is the number of electrons at time t, f is the input rf frequency, and n_Ave is the average of orders of electrons running in the simulated region. Figure 4 also shows dotted lines obtained by Eq. (5), which are very consistent with the solid line computed by Spark3D. Thus, average values calculated by Eqs. (3) and (4) and the computed average order are accepted with reasonable confidence.

Second, consistency of Spark3D with Anza's theory¹³ was confirmed, through the following step. Spark3D simulation was carried out in the local region in Fig. 1, where the SEY of silver which Anza used for his theory analysis¹³ is imported; then, the average SEY for varying power was obtained by Eq. (4), and peak voltage and power of input microwave at $σAve=1$ were found. Values of the obtained peak voltage were superimposed on the susceptibility curve drawn from Anza's theory,¹³ cf. Fig. 6. It is found that the Spark3D simulation is consistent with Anza's theory, which is also consistent with some experimental results reported elsewhere.²⁷ Having established confidence in the simulations, we proceed to probe the multipactor dynamics in rectangular waveguides under TE₁₀ mode.

The average electron impact energies and average SEYs at each input power, obtained by Spark3D simulation in the regions in Fig. 1, are shown in Fig. 7. Interestingly, the average impact energy not only increases with an increase in input power but also sometimes decreases, even if the simulation is done in the local region where the rf field is almost constant spatially. As to the reason why average impact energy in the local region decreases, it is expected that the ratio of electrons decelerated by the rf field increases with an increase in power due to the growing non-resonance of the microwave field and electron oscillations. Since it results in slower electron velocity or traveling back to an emitted plate, the electron impact becomes weak. Consistent with the up and down of the impact energy, the SEY in the local region also exhibits an initial increase with the increase in power, until ∼70 kW in the 2.0 mm gap and ∼200 kW in the 5.5 mm gap. After exceeding these power levels, the SEY declines, while the electron impact energy increases, due to exceeding the energy for the maximum SEY. The lowest and the highest multipactor thresholds where SEY = 1 in the local region are about 6 and 150 kW in the case of 2.0 mm gap and about 22 and 900 kW in the case of 5.5 mm gap, respectively.

On the other hand, the average SEY in the whole region maintains a higher value, while the average SEY in the local region decreases with the power, and the average impact energy in the whole region remains below 700 eV when the average SEY in the whole region is more than one. Such behavior is consistent with the variation of the multipactor position moving away from the local region as the power increases to higher levels, see Fig. 8. Based on this observation, the position of the multipactor discharge, defined by the center point of the emission densities, is shown as the distance from the center of the narrowest waveguide gap, cf. Fig. 9. In the case of a 2.0 mm gap, exceeding 80 kW from which the SEY in the local region starts to decrease with the increase in power, the multipactor position stays in the gap but moves sideways toward the wall into the lower field region of the TE₁₀ mode, keeping the SEY below the second crossover point and thus the impact energy in the range from 300 to 700 eV. Exceeding 270 kW, multipactor moves to a region outside the 2.0 mm gap. Likewise, in the case of the 5.5 mm gap, when the SEY in the local center region decreases, multipactor goes away the center region, keeping a high SEY. Thus, the multipactor moves to relatively low electric field regions when the second crossover point energy is exceeded in the regions that initially exhibited the lowest multipactor onset threshold. Note that in Spark3D, electrons are seeded not only in the center region but also in lower field regions with the release angle (polar angle) distribution of emission electron following the cosine law described in Ref. 28. Thus, multipactor may be initiated in the lower field region due to direct multiplication of the initially seeded electrons or diffusion of multipacting electrons from the center region due to the added tangential velocity components in the SEY process.

Variation of average multipactor order computed by Spark3D is also shown in Fig. 9. The orders of the lowest threshold power where SEY = 1 are about three and eight in cases of 2.0 mm and 5.5 mm gaps, respectively, and with an increase in power, those decrease due to an increase in electron velocity and almost saturate to certain values. It is expected as the reason for the saturation that not only accelerated electrons but also decelerated the increase in electrons with the increase in power, due to de-resonance with the rf field and due to the multipactor position moving to the lower field. In the case of the waveguide with a 2.0 mm minimum gap, the average order increases after the saturation since multipactor is caused at positions of a larger gap than 2.0 mm.

The relation between average impact energy and average SEY of the local region is represented in Fig. 10, compared with the imported SEY curve. Arrows on the solid line connecting plots mean the direction of variation with an increase in power so that the solid line shows how average impact energy and SEY vary with power. It is found that a curve showing the relation between average impact energy and average SEY is much different from a normal SEY curve such as the imported SEY curve. Particularly, this curve sometimes goes back to the lower energy side and the SEY also decreases than that at lower power even if the impact energy is the same. It is surmised that those are due to the spread in the energy distribution caused by a growing de-resonance. For instance, if input power increases to more than 200 kW in the case of the 5.5 mm gap, due to de-resonance, electrons might not only have higher impact energies, e.g., around 2000 eV, but also have lower energies less than 100 eV. Since energies around 2000 eV or less than 100 eV correspond to a low SEY close to one, if the ratio of electrons with lower energy less than 100 eV is large, it is possible that both of averages of impact energy and SEY become lower than those of lower power.

From Fig. 7, the susceptibility lines for the local region are determined, see Fig. 11. The horizontal axis is the product of microwave frequency f and gap d between plates. The vertical axis is peak voltage in gap, converted from input power. Each circle and triangle plots obtained from Fig. 7 represent peak voltages by which the average SEY becomes one and maximum, respectively, at each f × d. Solid lines connecting circle plots represent the voltage range of multipactor occurrence for f × d. The dotted line connecting triangle plots represents voltage of the strongest multipactor occurrence for f × d. Between the solid lines, there are some circle plots since the average SEY sometimes drops below one despite the ranges of multipactor occurrence, cf. Fig. 7.

We also diagnosed multipactor experimentally for conditions similar to the above-mentioned Spark3D simulation. As a diagnostic method, a method of direct electron observation using an electron multiplier tube (EMT), which we devised, was adopted, see Refs. 24 and 29 for a detailed description. As a test piece of multipactor observation, a step impedance transformer whose minimum gap is 2.1 mm was utilized. Although this is different from the tapered impedance transformer used in Spark3D simulation, it was found from field simulation that the field distribution in the center region of both cases is identical. This was corroborated in the experiment where the forward voltage gain, S21, of the 2.85 GHz microwave in both of the impedance transformer waveguides was found to be nearly equal to one as expected. Hence, until the MP moves away from the center another gap region at the high powers more than 270 kW, we can compare both the waveguides. At the center in the top, the broadside wall of the 2.1 mm gap part, a 1 mm aperture is located. An EMT is mounted above the aperture so that a portion of electrons at the center of the 2.1 mm gap enters the EMT. The test piece was incorporated into a traveling wave resonator ring constructed from the WR-284 waveguide³⁰ in which a 2.85 GHz signal will operate in the dominant TE₁₀ mode. By the resonance of the signal of the source, much higher power is delivered to the test piece than the source can supply. The experiment was carried out in two conditions, that is, with and without UV irradiation to seed initial electrons via photoemission from the waveguide walls. Note the 5% gap size difference between experiment and simulation, which has, however, a minor effect on the conclusions drawn in this paper. That is, comparing straight waveguide sections whose gaps are 2.0 and 2.1 mm, respectively, one obtains an increase in less than 500 W and 15 kW compared to the 5.5 kW and 150 kW of onset and end point powers for breakdown, respectively.

With UV seeding, the EMT signal was observed at input power levels exceeding 2.25 kW, such as shown in Fig. 12. However, Spark3D shows 5.5 kW of threshold. Two reasons are considered for this threshold difference. One is the difference of SEY on the low energy side. While it is noted that the imported SEY almost matches with the SEY obtained experimentally from technical copper by Baglin,²⁶ there are some example of copper SEY curves whose energy of initial crossover with one is much lower.^15,31 Actually, we have confirmed that Spark3D shows 2.25 kW threshold when a SEY, which has its first crossover point at 9 eV, which is consistent with SEYs in the references given for the lower energy side, is imported. Another reason may be found in the difference of the initial emission velocity distribution of secondary electrons. Since the Spark3D simulation is consistent with Anza's theory, it is expected that the Maxwellian distribution of Eq. (1), which is affected by thermal spread $υt$⁠, is used in Spark3D as the distribution of secondary electron emission velocity. Therefore, if thermal spread is different, the distribution of the emission velocity changes and, consequently, the spatial electron distribution between plates for the rf phase and impact velocity distribution also change, resulting in different multipactor thresholds.

In the case of without UV seeding, such as in Fig. 13, the multipactor onset power is much higher than the case of with UV and changes randomly since it is affected by differences of dispersion of the spatial electron position for the rf phase due to sporadically appearing and fewer initial electrons. In cases of peak of 35 kW and 100 kW, the EMT signal rose rapidly immediately after power achieves onset voltage and then power falls down a little. This means that multipactor originated in the center part of the 2.1 mm gap where the electron detection aperture is located. The observed peak power is primarily determined by the multipactor de-tuning of the traveling resonator, thus preventing further resonant growth. This is consistent with the Spark3D result since Spark3D shows an SEY larger than unity in the local center region, cf. Fig. 7(a), and the multipactor position almost in the center position, cf. Fig. 9(a), in the range of 25 kW–150 kW. However, in the case of higher power levels, e.g., bottom Fig. 13, the electron signal is still low when the microwave power peaks. Consistent with this observation, it is confirmed from Spark3D simulations that in this high power case, the initial multipactor is formed outside the local center region, followed by a migration of the multipactor toward the center region when the microwave power drops from 190 to 150 kW, cf. Fig. 9(a). That is, the initial off-center multipactor detunes the resonant ring such that the power peaks at 190 kW (without the detuning, the power would climb to a few MW). The power level continues to drop owing to the multipactor further detuning the structure. With the power level dropping to about 150 kW, conditions favorable for multipactor in the center region are attained, thus enabling the delayed detection of the electron signal. The observed time delay between detuning and the EMT signal, see Fig. 13 bottom, is clear evidence and does not exist for powers below ∼150 kW, see Ref. 29 for other experimental results.

Multipactor in a rectangular copper waveguide with the TE₁₀ mode field was analyzed using average impact energy, average SEY, average order, and multipactor position, obtained from Spark3D software which considers continuous distribution of emission velocity and the angle of secondary electron emission.

It was confirmed that the Spark3D simulation results are consistent with the general equations of electron multiplication by multipactor and Anza's theory. We found no upper power limit in the rectangular waveguide structure investigated. The multipactor location simply moves to a lower field region that keeps the electron energy locally below the second crossover point.

Through analysis of the copper waveguide, we revealed that there are cases where the average impact electron energy decreases despite an increase in power, resulting in a decrease in average SEY. As a consequence, a graph showing the relation between the average impact energy and the average SEY draws a curve with a part going back to lower energy and SEY side, and with not only two threshold points where SEY = 1, but also several more. It is surmised that those are due to the growing non-resonance of the microwave field and electron oscillations. For a conclusive proof, the detailed electron energy distribution would have to be determined in future work.

Additionally, it is uncovered that when the average SEY in the highest field region is close to or less than one, multipactor may be caused in a lower field region where the SEY is effectively higher than one.

The numerical results are compared with data from the experiment. While there is some deviation between the thresholds obtained from Spark3D and the experiment, the results at higher power levels are consistent with the experiment in the view of the SEY for each power level.

This work was supported by the AFOSR under Grant No. FA9550–18-1–0062.

The data that support the findings of this study are available within the article.