On the limits of multipactor in rectangular waveguides Free

Multipactor is a widely discussed topic in specialized experiments and practical applications such as satellite communication systems.^1–7 The term multipactor, a portmanteau of the words “multiple” and “impacts,” describes an electron resonance effect in which a population of electrons within a high vacuum space (mean free path much larger than the container) moves in sync with an applied RF electric field. Due to the large mean free path, these electrons are allowed to accelerate to energies ranging from tens to hundreds of eV, and then collide with the container wall. These walls, normally made of metal, then have a probability of ejecting a secondary electron, which then adds to the electron population. This electron multiplication leads to detuning in high Q systems and even to a gaseous discharge owing to possible outgassing from wall adsorbents.^8,9 While a high secondary electron yield is useful for devices such as multiplier tubes, it is a hindrance to RF systems in its repercussions.

Previous experimental work has demonstrated that direct detection techniques are superior in their ability to pinpoint electron growth within a system. Care must be taken in implementing these detection methods as some may be intrusive and perturb the RF fields of the system, while others may only offer limited time resolution (on the order of μs).¹⁰ An example of such direct detection methods comes in the form of a charged copper wire. The wire itself is biased such that electrons emitted from the multipactor region are attracted to the probe so that direct current measurements are made. As stated previously, care must be taken as to not allow secondary electrons to be emitted from the probe itself. Previous work on this detection method overcame this issue by the coating of the electron probe in multiple layers of carbon.⁴ These direct detection methods coupled with global detection methods are providing more insight into multipactor by quantifying formative and statistical delays. A wide range of geometries have been studied, but planar geometries are of particular interest herein.^11–14 

The planar geometries within are akin to WR-284 waveguide dimensions operating at S-band frequencies. Multipactor thresholds for test gaps are reported, but specific interest is taken in the upper and lower limits, as well as the dynamics of this phenomena.

Much of the early theory regarding multipactor makes assumptions that are not reflective of a true waveguide environment.^8,9 While this work is paramount in establishing an understanding of the multipactor phenomena, over the years, it has been found that the processes involved are more complex and chaotic in nature.^15–18 In light of this, a simplistic model was used in the design of this experimental setup to quickly determine appropriate test cell gap sizes, as well as the required power ranges of the RF sources.

An acute understanding of secondary electron emission is necessary for the development of a multipactor test bed. The secondary electron yield of various materials has been measured and scrutinized over the years with many results agreeing with Vaughan's empirical formula for high energy electrons (hundreds of eV). However, recent studies have shed light on secondary emission yield (SEY) curves measured at low electron energies (tens of eV) in which the formula provided by Vaughan begins to falter due to the state of the test material when an SEY measurement is carried out (surface treatments, measuring as received, etc.). For this experimental setup, yield curves in both tens and hundreds of eV were most relevant in determining multipactor thresholds in a waveguide test cell.^1,19–21

For the ease of a quick estimate, one generally assumes an infinite, parallel-plate geometry with an applied sinusoidal electric field of specific frequency f and plate distance d. The acceleration of a single electron, initially at rest near one of the plates in a perfect vacuum, is then considered, sans collisions with a background gas. This implies that one of the most important criteria of multipactor is found in the mean free path of the experimental setup being much larger than the gap distance of the test cell (typically at pressures below $10−5$ Torr). This ensures that the primary electron has sufficient energy upon impact to cause (on average) a secondary electron to be emitted from a material. Electron impact energies at which the SEY is unity are referred to as the materials crossover points; this manuscript primarily being concerned with the first crossover point where, starting from low primary energies, the SEY crosses over from below to above unity. The material used throughout the experiment described within being copper.

Since the involved electron impact energies are well below the electron rest-mass energy, the non-relativistic kinematic equations are invoked for calculating the necessary electric field amplitudes that cause secondary electron yield. One may limit the Lorentz force equation to the electric force (due to the electrons being slow and thus neglecting the magnetic field component of the propagating wave). The parameter N, also known as the multipactor order, is introduced, denoting the number of odd half-cycles required for the electron to cross the test cell gap. Further reading on the specifics of multipactor order is found in Refs. 22 and 23,

where E_RMS is the RMS electric field, d is the test cell gap size, f is the operating frequency of the applied field, m_e is the mass of an electron, and q_e is the magnitude of the electron charge. Determining the impact energy is found by manipulating the kinetic energy formula, yielding

where E_KE is the kinetic energy of the incident electron at the peak electric field.^9,21,24 While simplistic in nature, it is commonly accepted that Eqs. (1) and (2) describe the basic dynamics of multipactor and are consistent with much of the experimentally observed multipactor behavior reported in this manuscript. Focusing specifically on the multipactor threshold governed by the first crossover point in the SEY yield and zero phase difference between the microwave and multipactor, Fig. 1 is generated that reveals the lower boundary susceptibilities up to the nineteenth order (N = 19). As elucidated in Sec. IV, there is no upper threshold to multipactor within a rectangular waveguide structure operating in the dominant propagating mode. It is noted that additional susceptibility diagrams, which include the emission phase and the second crossover point, may be found elsewhere.²² 

Equations (1) and (2), enable pinpointing of the expected multipactor 'hot spots' within a waveguide structure given an applied electric field specific to input power and microwave mode; here, the dominant $TE10$ in the WR-284-like test cell. Thus, the electric field between the two plates exhibits a half-sine distribution.

As an example, multipactor conditions one may find for two selected waveguide heights, 5.5 mm and 2.1 mm are outlined (waveguide broadside dimension being that of standard WR-284 waveguide). Just above the multipactor power threshold, the multipactor hot spot is expected to form at the center of the waveguide geometry with high values of N, always associated with a low power, see Fig. 2. As the power is increased, multipactor will stay mostly in the center while reducing its order, N. For very high powers, the resonant conditions are no longer met in the center, and the multipactor will be located more toward the sidewalls of the waveguide, see Fig. 3. For the presented calculations, values of 30 and 2000 eV were adopted for the first and second crossover points of the SEY curve, respectively. Lower and upper impact energies reflect those crossover points found in the literature (cf. Refs. 1, 21, and 25–27) which reports a lower first crossover in comparison to Vaughan's earlier experiment.⁸ 

Note that the blue-shaded regions in the figures represent the area in which a primary electron with normal incidence will strike the opposing wall with its kinetic energy positioned between the two SEY crossover points. As was found from the experiment, N = 19 was observed for mechanically polished copper (not vacuum baked or conditioned) with a 5.5 mm gap (cf. Sec. IV A). The calculated hot spot for Fig. 3 is for input powers exceeding the conditions viable for N = 1 in the center of a 2.1 mm gap (225 kW calculated, while only 100 kW are needed to have N = 1 in the center of the gap).

By incorporating microwave sources covering a broad power range, it is revealed in Sec. IV B that the field distribution of a waveguide operating in the dominant mode has a significant effect on the upper threshold of multipactor, while surface chemistry and adsorbates have a profound impact on the lower threshold of multipactor.^21,25,28

A planar test cell with the broadside dimensions of a WR284 waveguide and a varying height was implemented. Operating in the dominant TE₁₀ mode allows for the manipulation of the waveguide height without affecting the cutoff frequency in the form of a stepped impedance transformer. This test cell allows a sufficient electric field for multipactor to be localized in the vicinity of multiple diagnostic devices (see Ref. 24 for more details).

That is, the treatment process for the material under test being mechanically polished with a high grit sandpaper, followed by CeO microparticle buffing for high gloss shine, followed by ultrasonic cleaning in a commercial acid solution (Citranox), and finally rinsing with DI water. A process similar to that in Ref. 24.

Two sources were used in this series of experiments. An S-Band coaxial magnetron with a nominal output of 4 MW and a 4 μs pulse width was implemented as a high power microwave source. An adjustable tuning stub was inserted in line with the magnetron output in order to achieve output powers as low as 2 kW within the test cell. A high power circulator attached to a matched load was placed between the inserted stub and magnetron in order to better protect the source from large reflections. To extend the pulse duration beyond 4 $μs$⁠, a frequency agile, solid–state RF source was also implemented with a nominal power output of 3 kW with an adjustable pulse width of up to 100 $μs$⁠. These two sources allow for the experiments to cover a large power range (2 kW to MW) at a fixed S-band frequency, and a large portion of the S-band (2.5 to 2.9 GHz) with variable power (3 kW to 10 kW). For the purposes of the experiments described throughout, a constant operating frequency of 2.85 GHz was used (a frequency within the band commonly used for broadcasting-satellite service).²⁹ Future experiments may incorporate frequency sweeps of similar test cells. Uncertainties of the experimental setup are 0.5% and 3% for the power measurements and test gaps, respectively. Thus, the error in the electric field values is smaller than 2%. The errors in the EMT and PMT signal amplitudes are of a similar magnitude.

While direct observation techniques are used in this experiment, global detection techniques were also used. A traveling-wave resonator ring is implemented with the test cell due to its sensitivity to small phase shifts which subsequently detune the high Q structure. The ring also provides gain to the input power due to the high Q. This detuning and gain are better seen in Sec. IV.^30–32 

Diagnostics included an electron multiplier tube (EMT), photomultiplier tube (PMT), and directional couplers situated on the traveling wave resonator in order to measure the forward and reverse power. The EMT serves as a local detection technique in which electron population growth due to multipactor may be observed in real-time. Note that the electron numbers may be estimated from the EMT signal, see Ref. 24. The PMT signal simply serves as a timing marker for the UV LED.

Due to the short pulse width of the magnetron, UV seeding (utilizing a 265 nm LED) was implemented onto the traveling wave resonator via a UV transparent window. The UV source is implemented to supply seed electrons for multipactor to occur. Two windows sit on boresight with the test cell in order for UV to be transmitted at one end and detected by the PMT at the other, cf. Fig. 4.

As with any electron amplification process, the initial number of available electrons plays a crucial role in the overall dynamics and is estimated as follows. A pulsed UV LED (Mightex LCS-0265–02-23) was used throughout the experiment, with a typical optical output power of 5 mW with an initial 23 mm beam waist diameter and a half diverging angle of 1 deg. This yields 50 nJ within the 10 $μs$ seed pulse, which translates to 0.8 m distance between the LED and test gap to an estimated fluence of 2.87 $nJ/cm2$⁠. Further taking into account the surface area of interest being the calculated hot spot region with a length of $∼$5 mm (width of the multipactor hot spot from Fig. 2), and applying Lambert's cosine law, one finds the effective illuminated area to be $∼$1 mm², resulting in an estimated incident optical energy of 0.02 nJ.³³ This corresponds to roughly 10⁷ photons striking the copper surface. From Ref. 34, a photo-emissive yield of $8×10−3$ electrons per incident photon is seen for copper with incident photon energies of 4.6 eV, with Ref. 35 corroborating such low photoelectric yields for pulsed UV LEDs. It is then approximated that the number of electrons within the gap will reach a maximum of $∼104$ accumulated during the 10 $μs$ UV pulse.

By utilizing the solid–state source, measurements were carried out with a 5.5 mm test gap. Multipactor threshold was measured and then subsequent measurements were taken to over a kilowatt past the detected threshold power. The full pulse width of the solid–state microwave source was used (100 $μs$ pulse width) with the 265 nm UV LED source. Throughout the experiment, the UV pulse width was set to 10 $μs$⁠, which was found to be close to the threshold where the multipactor appeared inconsistently (∼5 $μs$ and below). Further increasing the UV pulse width had an undesired impact on the formative time conceivably by flooding the gap with electrons. This approach is in line with the testing procedures outlined elsewhere.³⁶ Special interest was taken in what will henceforth be referred to as the formative delay time, or the time from the noticeable beginning of the multipactor population growth to the significant detuning of the high Q test setup. Unless otherwise mentioned, waiting periods of several minutes were added in-between measurements, thus avoiding the multipactor threshold being affected by prior multipactor events.

One easily calculates that the temporal growth of the multipactor from initial n₀ electrons is governed by the SEY yield and the electron transit time from wall to wall for a one-dimensional case.^9,37 That is,

Considering Sec. II, an increase in power with a constant frequency and gap will reduce the multipactor order and thus the electron transit time of $N×T2$⁠, while the SEY yield increases initially with power. That is, not too far from the threshold, both transit time and SEY contribute to a faster increase in the electron number with increasing power. In the presented experiment, the multipactor electron density will grow to a point where the phase shift introduced by the electrons will detune the resonant ring. Thus, Eq. (3) is used to define a formative delay determined by the delay between the onset time of the multipactor and detuning of the resonant ring.

In this context, it is important to assess how much phase shift is required to detune the ring, tuning properties at a low power with initially atmospheric pressure air in the test structure were examined. Noticeable detuning was observed with a pressure increase of ∼100 Torr. This corresponds to a detuning phase shift of $10−3$ rad, which is obtained from the detuning pressure by factoring in a ring circumference of 3.7 m, a cutoff frequency of 2 GHz (the cutoff for WR-284 waveguide), and a refractive index of atmospheric pressure air (1.000272).

Further considering the vacuum operational mode of the ring, the same detuning phase shift is obtained for an electron density of $108electrons/cm3$ assuming a test section volume of 13 mm wide, 5.5 mm tall, and 51 mm long. For details on how phase shift and electron density are related in the WR-284 waveguide, see Ref. 38. Thus, a numerical value of n_e on the order of $108$ at the point when significant detuning occurs is inferred.

At the lowest possible power that led to multipactor, see Fig. 5, the formative delay is measured to be approximately 800 ns, which shrinks to roughly 160 ns when the power is increased to 6 kW, see Fig. 6. Note that no multipactor was observed below the power level of ∼5.15 kW, regardless of the LED being pulsed or not. One also notes that the shorter delay time is accompanied by a stronger detuning (drop of forward power in the ring). Furthermore, while fed with a square electrical pulse, the drop in the strength of the PMT signal is due to the pulsed characteristics of the UV LED.

In general, as the forward power is increased, the formative delay time decreases, see Fig. 7, where three measurements were taken for each power level. This is believed to be due to an increase in forward power causing a slightly wider portion of the waveguide field distribution to become conducive to causing multipactor. Thus, the slight decrease in delay time until about 5.65 kW. As will be discussed below, the order of multipactor then jumps from N = 19 to N = 17, resulting in the downward step in formative delay visible in Fig. 7.

Following Hatch and Williams²² calculations of multipactor order, we find that the experimental results are most consistent with the ratio of primary impact to secondary emission velocity, k = 50, first crossover point of 25 eV, zero phase difference between multipactor and microwave field, and a multipactor order of N = 19 and 17 for an RMS voltage of 627 and 700 V for N = 17, cf. Fig. 7. Other combinations of k, crossover point, multipactor order, and phase difference provide an inferior match with the experimental data. Since k is large and the phase difference is zero, simply applying Eqs. (1) and (2) yields an identical result.

With the calculated multipactor orders, Eqs. (2) and (3) are used to determine an effective SEY by substituting the average measured formative delay times and the electron transit time in the multipactor gap. The initial electron number, n₀, may be estimated from the maximum number of electrons released during the 10 $μs$ UV pulse (⁠$∼104$⁠), cf. Sec. III. For the data depicted in Fig. 8, three values for the initial electron population, one, one hundred, and ten thousand electrons were adopted to elucidate the error associated with the initial electron number assumption. The corresponding ratio of $nen0$ of 10⁸, 10⁶, and 10⁴ serves as a parameter in Fig. 8. Note that the next higher order of multipactor, N = 21 with an associated impact energy of 25 eV was not observed. One thus concludes that the SEY first crossover point falls between 25 and 30 eV for mechanically polished copper cleaned as described in Sec. III. This may be compared with the reported values in the literature that list the first crossover point for 'as received' copper to be above 25 eV.^25–27 Note that this first crossover point SEY was estimated for an actual multipactor developing in the test section rather than the usual electron gun–target setup.^19,28

One concludes that two conditions must be met for the multipactor to occur during a microwave pulse. First, the power/electric field must be sufficiently large in order to accelerate electrons to cause impact energies beyond the first crossover point of copper.^9,21 Second, the appearance of initiating electrons must occur at some time during the pulse. Thus for the demonstrated measurements utilizing the 5.5 mm gap, a UV pulse was routinely applied to initiate multipactor, yielding a threshold of 5.15 kW (100 $μs$ pulse). To be precise, without the UV pulse, the multipactor would not appear for powers up to the tested 6.5 kW, 100 $μs$ duration, if the system had not been fired for an extended period of time (multiple minutes).

However, very repeatably, applying a few successive microwave pulses after up to about 30 s (as long as 1 min), multipactor events were still observed, without the UV pulse applied. This is, the system exhibited a roughly 30 s memory that produced initiating electrons within the gap. It is unlikely that the electrons themselves have a lifetime of tens of seconds in the test section with no electric field applied. It is also unlikely that they would simply sit loosely on the surface of the copper. However, electron emission from nitrogen metastable interactions has been previously reported as affecting breakdown delays due to their long life (reported at tens of seconds and higher in low-pressure gases of $75 μ$ Torr).^39–41 The exact de-excitation mechanism which produces these electrons has yet to be conclusively quantified and its determination is beyond the scope of the manuscript. It is noted, however, that the initiation of multipactor via long life metastables is consistent with those reported in the literature.⁴² Figure 9 demonstrates low power multipactor without UV seeding. Similarly, multipactor thresholds below those reported (∼5 kW) were observed when an ion gauge was utilized to measure the vacuum pressure. Even though the gauge was roughly 1 m away from the test section, multipactor could still be initiated below the threshold when the ion gauge was turned on. Again, one may argue that the transport of excited neutrals or ions to the test section produced initiating electrons via wall collisions. This general behavior was also observed elsewhere.⁴³ 

Utilizing the HPM magnetron setup (cf. Sec. III), the upper thresholds of multipactor were tested. Figure 10 demonstrates input powers below 20 kW will not produce multipactor within a 2.1 mm gap. This threshold measurement does not incorporate UV seeding. Thus, the statistical delay is expected that adds to the formative time delay of the previous section. After increasing the power in the gap to ∼22 kW, multipactor onset was seen within the test gap, as demonstrated in Fig. 11. It is important to reiterate that UV seeding was turned off for any waveforms where the power within the gap is above 20 kW as it was not needed to initiate multipactor consistently at these power levels. This is believed to be due to the sufficiently large electric field amplitudes causing electron emission from the field stressed surfaces. One notes, however, that the associated macroscopic field (5 $kVcmRMS$⁠, ∼7 $kVcmpeak$⁠) is low for simple metallic cold field emission.

Applying Eqs. (1) and (2) yields that the order of multipactor in Fig. 11 is consistent with order N = 3, while input powers demonstrated in Fig. 12 and beyond are consistent with N = 1, with a normal incident energy of $∼$ 280 eV. This incident energy exceeds the first crossover point of pure copper reported in the literature, meaning that a majority of electrons are in viable conditions to produce secondaries.⁹ 

The lower threshold within a 2.1 mm gap without UV seeding is demonstrated in Fig. 11 which is of N = 3 based on Eqs. (1) and (2). When power is further increased as in Fig. 12 to 100 kW input power (required for N = 1), severe detuning is observed in the forward power (drop of 25 kW) after the initial electron population growth. Based on Sec. II, a further increase in power would disrupt the resonance condition between the electron transit time and the operating frequency half-period. At this point, the multipactor would no longer be supported in the center of the test section and one may indeed argue that an upper multipactor threshold has been reached. Obviously, the often found assumption of a homogeneous RF-field between the two surfaces falls in the presented practical waveguide geometry.^3,11,18,44 Thus, owing to the lateral field distribution in the TE₁₀ mode, the multipactor simply moves from the center of the broadside dimension toward the waveguide walls, cf. Fig. 3.

One finds evidence for the off-center multipactor formation from the experiment with the highest tested power level, see Fig. 13. As the power is increased past 150 kW in the test gap (past the calculated requirements for N = 1 in the waveguide center), a delay between the detuning of the forward power in the ring and the peak of the electron population growth was observed. It is fair to argue that the observed delay is due to the multipactor initially forming off-center, in which case it would not be detected by the EMT that collects electrons through the 1 mm diameter orifice in the center of the waveguide broadside dimension. While unnoticed by the EMT, the resonant ring is detuned by the multipactor and the power level drops to the point where resonance conditions support multipactor resonance in the waveguide center.

Thus, it is of the authors' opinion that an upper threshold to the multipactor phenomena does not exist in a rectangular waveguide as theorized.⁴⁵ Mainly attributing this to an inhomogeneous electric field, it would stand to reason that the only realistic upper threshold would be in the case of extremely high electric fields which would, however, cause the multipactor to transition into vacuum breakdown.

Experiments were carried out to observe the two extremes of multipactor thresholds within WR-284-like structures utilizing a 2.1 mm and 5.5 mm test gap. At the low power end, utilizing electron seeding with a UV source, high orders of multipactor within a 5.5 mm gap (N = 17 to 19) were observed and formative delay times reported. An effective SEY for copper at the multipactor threshold was estimated utilizing the measured formative delay times, cf. Fig. 8. Consistent with previous breakdown reports in the literature, excited metastables are the likely cause that facilitates multipactor initiation via electron producing wall collisions long after a previous multipactor pulse is initiated with a UV source (up to several tens of seconds).

At the high power end, reaching input powers of over 200 kW passing through a 2.1 mm height waveguide section, it was revealed that due to the field distribution within the waveguide structure (TE₁₀ mode), there is no practical upper threshold to multipactor formation as one could argue for the homogeneous field case.

This research was supported by the Air Force Office of Scientific Research under Contract No. FA9550–18-1–0062.

Special thanks to Dr. Nicholas Jordan at the University of Michigan and Dr. John Verboncoeur at the Michigan State University for the fruitful discussions.

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