Characterization of the energy of runaway electron bunches by analyzing emitted microwave radiation Open Access

One of the most important characteristics of a bunch of runaway electrons (RAEs)^1,2 formed in a gas discharge gap with a nonuniform electric field^3–6 is the kinetic energy (W_k) that the electrons gain on arrival at the anode. Unlike the case of an electron beam (e.g., an explosive emission one⁷) accelerated in vacuum,⁸ the cathode voltage amplitude $|Uc|max$ is unusable for estimating the RAE maximum energy from the potential difference across the distance traveled by the electrons. Actually, a picosecond duration flow of RAEs^9–13 is emitted from the boundary of the gas plasma produced at a pointed cathode when a critical electric field is reached at this place early in the voltage rise time.^14,15 Then, if the RAEs are accelerated within a time comparable with the voltage rise time, their energy will be lower than $e|Uc|max$ (e is the elementary charge). In addition, the RAEs, while traveling toward the anode, lose energy in inelastic collisions with molecules.^16–18 However, in this case, an increase in W_k is possible due to the presence of plasma at the front of the fast ionization wave^19–21 following the RAEs. This plasma expels the electric field toward the anode, thereby increasing its intensity and the energy of accelerated electrons.^22–24 

To estimate W_k of RAEs downstream of the anode, magnetic spectrometry is used.^14,25 However, this is often done by recording the current of RAEs (using a collector) after their passage through different cutoff foils,²⁶ which act as low-energy filters.²⁷ In the first case, for better energy resolution, a small electron fraction is isolated from the RAE flow diverging from the cathode.^11,28 This, however, noticeably reduces the charge of the RAEs, thus worsening the sensitivity of the technique. For recording the signal of RAEs downstream of the filters,^9,10 collimation is also necessary, since to detect a diverging electron flow, a collector of large diameter is required, which lengthens the signal and reduces its amplitude.²⁹ Thus, the data required for restoring the energy spectrum are garbled.²⁶ Two above-mentioned methods and time-of-flight technique²⁴ for the estimation of the energy of runaway electron bunches require multiple shots to produce the measurements and/or the use of multichannel registration system.

A number of previous studies considered the possibility of diagnostics of RAE flows by analyzing the optical radiation induced by RAEs in condensed media (see, e.g., Refs. 30 and 31, and citation therein). However, unlike the case of ultrarelativistic electrons, where Cherenkov radiation (CR) detectors are widely used, for the W_k hundreds-of-keV range of our interest, the threshold for the occurrence of the Cherenkov effect is critical. This threshold implies that the electron velocity (v) exceeds the phase velocity of the electromagnetic (EM) wave propagating in the medium (v_ph). It is also necessary to tune out competing radiation mechanisms such as scintillation, which may occur in the radiator under the influence of the accompanying bremsstrahlung, or pulsed cathodoluminescence.^32–34 However, electrons may emit microwave CR when they pass near the surface of a dielectric^35,36 or in dielectric-loaded or corrugated waveguides where slow EM waves are excited.³⁷ In waveguides of this type (as well as in hollow ones, where the waves are not slowed down), a competitive cyclotron mechanism of radiation generation by electrons is possible.³⁸ In this article, we compare the two mechanisms and show the possibility of estimating energies of hundreds of keV in an RAE bunch for a single shot. To do this, we use the spectrum of the microwave CR that occurs when the bunch passes through a slow-wave structure (SWS) designed as a circular waveguide partially loaded with a dielectric.

The experiments were performed on the setup shown schematically in Fig. 1(a). A setup of the same design, but without a dielectric insert, was previously used to estimate the energy of RAEs using the time-of-flight method.²⁴ The latest measurements and those presented below took into account the influence of an external magnetic field on the formation of fast electron flows in gases in the pre-breakdown stage of a gas discharge (see Ref. 39 and citation therein). They became feasible due to the recently demonstrated possibility of transporting RAE bunches [Fig. 1(b)].^40,41 The applied magnetic field allowed the electrons not lost in scattering and transverse diffusion in the gas to be collected on the collector. At a small transverse-to-longitudinal velocity ratio (i.e., at a small pitch factor), the helical trajectory of the electrons only slightly lengthened their paths.⁴² Therefore, the energy lost by the RAEs in the transportation section was relatively small, making ≤10% of their initial energy that ranged between 200 and 350 keV [Fig. 1(c)].²⁴ 

In the experiment described below, a subnanosecond voltage pulse U_in was applied to the cathode via a homogeneous coaxial line. The pulse amplitude was −165 kV [Fig. 1(d)]. The amplitude of the pulse reflected from the steel cathode U_ref1 was −95 kV [Fig. 1(d)], i.e., it was smaller (in absolute value) than the amplitude U_in. Note that before the emission starts, the idle mode is realized on the cathode, so the voltage on it doubles compared to U_ref1. The reflection amplitude for the graphite cathode U_ref2 [Fig. 1(d)] was even smaller than that for the steel one. This indicates earlier emission of RAEs and, accordingly, their lower energy on the anode.

The jitter of the tubular RAE bunch (with a pulse width of 30 ps) relative to the U_in pulse leading edge was ≈10 ps [Fig. 1(e)]. Its current amplitude (10 A on average) was unstable to within approximately 30%.

Figure 1(f) shows schematically the measuring system used to record radio frequency (rf) pulses. The output radiation was diffracted by a horn antenna. The sensor, made as a miniature pin antenna (E-dot sensor), was placed at the end of a high-frequency cable connected directly to the oscilloscope. The horn-to-sensor distance was chosen so that the amplitude of the rf signal to be recorded could be reduced to a level at which no additional attenuation was required.

When the transverse velocity of an electron increases,³⁸ it emits radiation under cyclotron resonance condition while moving in a magnetic field H along a spiral trajectory with gyrofrequency ω _H = eHc/W (here, W = mc²γ, c is the speed of light, m is the electron rest mass, and γ is the Lorenz factor). In an ensemble of such electrons, orbital phase bunching is possible in the field of the excited waveguide mode.⁴³ As a result, for short beams, coherent radiation can be excited due to the so-called super-radiance.⁴⁴ 

In the experiment where the dielectric insert shown in Fig. 1(a) was not used, only cyclotron radiation was observed. Let us consider its features. Figure 2(a) demonstrates a typical rf pulse from the E-dot sensor [position A in Fig. 1(f)]. The spectra of rf pulses obtained at longitudinal magnetic fields H = 21.5 and 10.7 kOe [Figs. 2(b) and 2(c), respectively], have maxima at frequencies f = 29.2 and f = 22.7 GHz, respectively. In Fig. 2(d), these frequencies correspond to the intersections of the dispersion characteristics of the waveguide modes, $f=[(hc/2π)2 +fc2]1/2$⁠, and electrons, f − hv/2π = ω_H/2π, that were obtained for a given H by properly choosing electron energy W_k. Here, h is the longitudinal wave number, f is the radiation frequency, and f_c is the critical frequency of the waveguide mode.

As can be seen from Fig. 2(d), precise cyclotron resonance with a frequency of f = 29.2 GHz occurred for the TM₁₁ mode at H = 21.5 kOe and W_k= 220 keV [solid violet line in Fig. 2(d)] corresponding to the inverse branch of the f (h) curve. It should, however, be noted that this mode was not identified in a single shot, and at the same magnetic field, resonances with lower modes (up to the TE₁₁ one) were possible at W_k ≈150 keV [(dashed green line, Fig. 1(d))]. Taking this into account, to determine the excited mode (and, accordingly, the electron energy W_k), an alternative method has to be used. For instance, the energy spectrum of a single thin bunch of RAEs (the tubular bunches shown in Fig. 1(b) consist of just such “jets”) emitted and transported in a field H = 21.5 kOe was reconstructed from the spread (in amplitude and duration) of a picosecond current pulse of RAEs, which traveled a distance of 6 cm in air.⁴⁰ The electron energy W_k at the distribution maximum was also ≈220 keV.

Noteworthy is the specificity of the rf pulse shown in Fig. 2(a) that was observed for the case of synchronism at f = 29.2 GHz in which the TM₁₁ wave moved toward the electrons and had a negative group velocity v_g ≈ 0.62c. A numerical simulation carried out using the КARAT code⁴⁵ has revealed a reflection of the electrons from the diaphragm [Fig. 1(a)] placed in the inlet of the waveguide. Due to this reflection, ≈80% of the radiation power could be extracted lengthwise with the beam, which was observed in the experiment. The bunch emits during the entire time of its passage through the transport channel. So, since the EM wave propagated first opposed the electron flow and then moved in line with it over a time of ≈1 ns, the rf signal was quite long. The spectra shown in Figs. 2(b) and 2(c) were calculated for an interval of 1 ns [dashed window in Fig. 2(a)]. One of the reasons for the appearance of a later fraction of the rf signal might be a mismatch between the waveguide duct and the horn antenna.

Note that the spectrum shown in Fig. 2(b) contains a set of closely located peaks at frequencies <29.2 GHz. They might occur in the range between 29 and 25 GHz due to the resonances highlighted by black boxes on the inverse branches of the dispersion characteristics of the modes shown in Fig. 2(d). Even lower frequency modes might be excited, since the magnetic field decreased from 21.5 to ≈5 kOe in the region where the tubular RAE bunch was dumped on the waveguide wall. In this case, at W_k ≈ 220 keV, synchronism might occur at frequencies up to the critical f_c ≈ 11 GHz for the TE₁₁ mode. In addition, modes lower than the TM₁₁ mode might appear if conditions for synchronism matched the forward branches of the dispersion characteristics.

For a low uniform magnetic field, H = 10.7 kOe, the spectral maximum at frequency f = 22.7 GHz [Fig. 2(c)] corresponds to the synchronism occurred at the critical frequency of the TM₁₁ mode [dash-dot orange line in Fig. 3(d)] if we chose W_k = 170 keV. Such a reduction in W_k was in principle possible given the effect mentioned in Ref. 22. This can be explained as follows. When the magnetic field was reduced from 21.5 to 10.7 kOe, the compression of the electric field became less effective, since the cross section of the RAE flow increased^24,40,41 and the plasma density at the front of the ionization wave decreased. However, the choice of W_k = 320 keV for the same H = 10.7 kOe increased the synchronization frequency only by 1 GHz [Fig. 2(d)], which was half the width of the observed spectral peak at half maximum [Fig. 2(c)].

For f ≈ 20 GHz, the condition for Cherenkov synchronism was satisfied for electrons with W_k ≈ 280 keV. This is a realistic value, since the measurements were performed at an increased magnetic field strength. In this case, the current density of the tubular bunch of RAEs (and the plasma density in the ionization wave) increased, enhancing the effect of electric field compression.²² When a graphite cathode was used in the experiment, the maximum voltage on the cathode [≈2U_ref2 in Fig. 1(d)] was reduced. As a result, the RAEs were accelerated to lower energies and the CR spectral peak was shifted to the frequency f ≈ 23.6 GHz. For the experimental conditions, this corresponded to the Cherenkov synchronism for electrons with W_k ≈ 210 keV. Control measurements by the time-of-flight method²⁴ showed similar energy values: ≈200 and ≈285 keV for the current peaks of electron bunches for the cases of the cathodes made of a steel and graphite, respectively.

Increasing the magnetic field strength (to 33 kOe) compared to those used in the experiment (see Fig. 2) was justified by the fact that a strong field ensures a better passage of a RAE bunch through the channel of the slow-wave structure. In addition, the frequency of the cyclotron radiation (parasitic in this case) shifted upward on the inverse branches of the f (h) curves for the TM₀₁ and HE₁₁ modes, as shown in Fig. 3(a) by the blue dashed line. The resonance, which occurred at a frequency of ≈28 GHz, manifested itself as a local spectral peak [see Fig. 3(d)]. Unavoidable cyclotron radiation could also occur at the short section of magnetic field drop after the SWS end. This explains the appearance of side peaks at f ≈ 23 GHz [Figs. 3(b) and 3(c)] in the case of synchronism of the 280-keV electrons with the TE₁₁ mode, corresponding to the f (h) inverse branch, or with the TE₂₁ mode, corresponding to the f (h) forward branch, when H was reduced to 22 and 7 kOe, respectively.

Let us consider the features of the CR emitted by a bunch of RAEs traveling through the dielectric-loaded SWS. In our case, unlike that of Cherenkov masers with similar SWSs and high-current beams,^47–51 the microwave generation mechanism was based on the summation of the CRs of individual electrons. The total CR is coherent³⁵ for a bunch, the length of which is small compared with the radiated wavelength: L_e ≪ λ (λ is the CR wavelength). The radiation of this type is not accompanied by the development of convective instability in the electron flow⁵² and is not an amplification⁵³ of EM seed arising from shot noise or representing coherent spontaneous radiation of high-frequency current at the steep beam rise time.^54–58 To clarify this, assume that our electron bunch behaves in time like the δ function. The velocity of an electron bunch and the group velocity of a CR wave are noticeably different (0.763c and 0.51c, respectively). Therefore, as follows from Fig. 3(a), the electrons when having traveled a distance of ≈4.5 cm in ≈200 ps, will overtake the wave by a distance of ≈1.5 cm (equal to λ). The real RAE bunch of duration τ_e ≈ 30 ps also overtook the CR wave, but the wave was not amplified since there were no electrons in the region of retarded high-frequency fields.

The described scenario of CR excitation is illustrated by the results of a numerical experiment (Fig. 4), where the particle-in-cell method (r-z version of the KARAT code⁴⁵) was used to simulate the CR of an electron bunch (W_k = 280 keV) traveling through the vacuum channel of the dielectric-loaded SWS [see Fig. 1(a)]. Figure 4(a) demonstrates quadratic rise in the CR peak power with the bunch current, i.e., the number of radiating electrons (for the same pulse width). Comparison of Figs. 4(a) and 4(b) shows that the CR peak power is determined also by the bunch duration. It increases with decreasing bunch length L_e due to better coherence of the overall radiation, that is, a less phase shift between the emissions of different bunch fractions. In the case illustrated by Fig. 4(c), where the SWS length was doubled compared to that used in the experiment [Fig. 1(a)], while the bunch current was the same, the duration of the delayed CR pulse increased proportionally to the time for which the bunch traveled in the SWS. The calculated spectral maxima of the CR pulses shown in Fig. 4 occurred at a frequency of ≈20 GHz, as in the experiment (see Fig. 3).

Noteworthy is the small width of the spectral peaks in Figs. 3(b)–3(d), which is typical for coherent radiation. This, however, requires confirmation. Indeed, in the experiment we used a tubular flow of RAEs, which, as can be seen in Fig. 1(b), unlike the simulation model presented in Fig. 4, consisted of the order of ten discrete bunches (jets). As already noted,³⁵ the CR emitted by all bunch electrons is coherent for a bunch of length L_e ≪ λ. This is partly the case for the CR of any discrete bunch of RAEs with a duration of τ_e ≈ 30 ps,⁴⁰ since it has a length of L_e ≈ 0.7 cm while CR wavelength of λ = 1.5 cm. The problem is that individual bunches are emitted with a jitter of δt ≈ 10 ps relative to the leading edge of the cathode voltage.²⁴ Therefore, for the given set of bunches with the ratio of δt/τ_e = 1/3, the CR phases are spread in the interval of {0, 2π/3}. Let us consider the quadratic summation of the field intensities for ten identical radiation pulses of unit amplitude with a filling frequency of 20 GHz and an envelope defined as sin²(t). The duration of the envelope is not important in this case. In coherent summation, such that $ϕi≡0$ for all phases, we obtained the overall field intensity $IΣmax =100$ (Fig. 5). When a set of phases was taken randomly from the interval {0, 2π/3}, I_Σ ranged between 50 and 90. That is, the total radiation retained the features of coherence.

In conclusion, the above-mentioned analysis of experiments and calculations performed for bunches of RAEs traveling through an empty waveguide has shown that the kinetic energy of the bunch electrons cannot be estimated from the spectrum of the radiation emitted by the electrons when they are in cyclotron resonance with the waveguide modes at frequencies matching the inverse branches of the dispersion characteristics of the modes or at their near-critical frequencies. To do this estimation would require identifying the spectral peak modes in a single shot.

Such identification is not required in cases where intense microwave CR is emitted by the electrons synchronous (v ≈ v_ph) with the slow modes of an SWS based on a partially dielectric-loaded circular waveguide. The dispersion characteristics of the two lowest modes of such an SWS, TM₀₁ and HE₁₁, are degenerate starting from a certain boundary frequency, which sets the upper limit for estimating the bunch energy (in our case, it was ≈400 keV). The CR peak at a frequency of 20 GHz, corresponding to the synchronism of a 30-ps bunch having typical energy of ≈280 keV with any of the mentioned modes, was observed for the case of a stainless steel cathode regardless of where the registration point was located: in the center or on the periphery of the radiation pattern. When using a graphite cathode, earlier RAE emission was accompanied by a decrease in the amplitude of the accelerating voltage. This determined an increase in the frequency of the excited CR peak to 23.6 GHz, which is possible with a decrease in the energy of RAE bunch to ≈210 keV. The bunch energies evaluated from CR spectral peaks in both cases were confirmed in time-of-flight measurements. It should be noted that, when recording the CR peak excited in the dielectric-loaded SWS, it was possible to tune out the parasitic cyclotron radiation by applying a stronger magnetic field. This shifted cyclotron resonances to the high-frequency region.

Observed narrow spectral peaks of CR are characteristic of coherent emission. The coherence has been confirmed by a numerical simulation of the excitation of CR by a short electron bunch traveling through a vacuum channel of the dielectric-loaded SWS. It has also been shown that the overall radiation of a set of discrete 30-ps bunches forming a tubular flow of RAEs should also be coherent, even if they are emitted with a jitter of ≈10 ps.

The obtained results indicate that the proposed method for single-shot determination of the characteristic kinetic energy of a picosecond bunch of runaway electrons traveling through a gas-filled SWS using spectral analysis of the accompanying microwave Cherenkov radiation is quite effective. This method is applicable for electron bunches of a certain energy range formed in vacuum.

This work was financed by the A. V. Gaponov-Grekhov Institute of Applied Physics, Institute of Electrophysics of UB, Institute of High-Current Electronics of SB, and P. N. Lebedev Physical Institute. The experiments were carried out on the equipment provided by the Collective Use Center of the IEP UB RAS. Also, it was helpful that measurements were performed using the Tektronix DPO73304D oscilloscope provided by the Ural Federal University.

The authors have no conflicts to disclose.

N. S. Ginzburg: Methodology (lead); Writing – review & editing (equal). L. N. Lobanov: Investigation (equal); Writing – review & editing (equal). V. V. Rostov: Formal analysis (equal); Writing – original draft (equal). K. A. Sharypov: Data curation (lead); Investigation (equal). V. G. Shpak: Conceptualization (equal); Supervision (lead). S. A. Shunailov: Data curation (equal); Investigation (lead). A. A. Vikharev: Software (lead); Writing – review & editing (equal). M. I. Yalandin: Conceptualization (lead); Investigation (equal); Writing – original draft (lead). I. V. Zotova: Formal analysis (lead); Writing – original draft (equal). N. M. Zubarev: Conceptualization (equal); Writing – review & editing (lead).

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