A 1.2 THz third-harmonic gyrotron with novel azimuthally segmented electron beam Open Access

Terahertz (THz) waves exhibit distinctive characteristics that have opened new frontiers across various scientific domains, including biomedicine, spectroscopy, and communication.¹ Numerous applications within these fields, such as electron spin resonance (ESR) and dynamic nuclear polarization nuclear magnetic resonance (DNP-NMR), require high-power THz radiation.^2,3 However, the power of conventional slow-wave devices or quantum cascade lasers typically remains below the watt level within the 0.3–3 THz frequency range.^4,5 This limitation arises from the constraints associated with miniaturized slow-wave structures and the reduction in quantum energy levels, resulting in the so-called THz gap. Fortunately, the gyrotron, which operates based on the electron cyclotron maser (ECM) mechanism,^6–8 exploits the interaction between gyrating electrons and high-order modes in a smooth-walled resonator, enabling it to surpass the limitations of slow-wave structures and generate long-pulsed or continuous-wave high-power radiation in the submillimeter and terahertz regime.⁹ Consequently, gyrotrons have emerged as a unique option of THz wave source in DNP-NMR.³ 

However, operating at high frequencies and high harmonics presents significant challenges due to intense mode competition, resulting from a dense mode spectrum and the inherit weak coupling strength of harmonic modes compared to fundamental parasitic modes.^8,13,14 In conventional gyrotrons with off-axis gyrating electron beamlets, also referred to as small-orbit gyrotrons, it is challenging to generate radiation at harmonics higher than the second in the THz regime.⁹ The advent of the large-orbit gyrotron (LOG), which utilizes an axis-encircling electron beam, enables strong mode selection of mode whose azimuthal index coincides with the harmonic number: $m=s.$¹⁵ This has facilitated the demonstration of the remarkable 1 THz/400 W radiation at the third harmonic using a 15 T pulsed magnetic field.¹⁶ However, a key challenge with LOGs is the generation of the large-orbit electron beam (LOB), which requires sophisticated theoretical design and complex engineering for the electron gun.¹⁷ Furthermore, operation at the THz regime with a high magnetic field usually demands high voltage of the LOB gun.¹⁵ For instance, typically 80 kV is required for 1 THz at the third harmonic with a magnetic field of 15 T.^16,18 In contrast, conventional small-orbit gyrotron has a simpler electron-optical system (EOS), which is beneficial for engineering implementation. It can also be generated at lower voltages,^19,20 offering an advantage for increasing the compactness of gyrotron systems while delivering power levels in the tens of watts range, which are sufficient for applications such as DNP-NMR spectroscopy. It should be noted that second-harmonic conventional gyrotrons with frequencies from 0.3 to 1 THz (up to 20 T SCM) and voltages from 3.5 to 30 kV have been successfully demonstrated^{3,9,12,20–22} and some of them have been also implemented in a commercial DNP-NMR spectrometer.²² However, as the frequency continues to increase—such as 1.18 THz CW radiation for a 1.8 GHz DNP-NMR spectrometer—the CW operation with the existing SCM necessitates the use of third harmonic operation. It is essential to address the challenges posed by severe mode competition in high-frequency high-harmonic conventional small-orbit gyrotron.

The azimuthally segmented electron beam facilitates mode-selective enhancement of beam–wave coupling strength for the operating mode, providing a promising strategy to reduce mode competition in high-frequency high-harmonic gyrotrons.²³ This method is based on the induced degeneracy and cooperation of two oppositely rotating waves of a transverse cavity mode, as illustrated in Fig. 1. By introducing an azimuthally segmented electron beam that aligns with the azimuthal periodicity of the desired operating mode, we can break the orthogonality of the two oppositely rotating waves and selectively induce their degeneracy. This cooperation enhances the coupling strength of the desired mode to the electron beam. In this paper, the proposed beam–wave interaction system is studied more thoroughly for a 1.2 THz third-harmonic conventional gyrotron, where the competition among modes is exceptionally fierce. This paper is organized as follows. Section II presents the theoretical basis for the analysis of interaction between electromagnetic (EM) waves and an electron with azimuthal periodicity. In Sec. III, the interaction circuit is design and studied systematically. Section IV is the conclusion and discussion.

In gyrotrons, the hollow electron beam can simultaneously interact with multiple TE modes within an oversized cylindrical cavity. These modes interact and compete with each other through their interaction with the same electron beam.^8,24 Generally, gyrotrons operate stably in a single rotating TE mode, also known as a vortex wave,²⁵ which is characterized by a variation in the azimuthal phase while maintaining a constant amplitude: $e±jmφ$⁠. This behavior is associated with the conventional use of an azimuthally uniform electron beam. The direction of rotation [co-rotating (+) or counter-rotating (−)] is determined by the respective coupling strength to the beam. Typically, the two oppositely rotating modes are orthogonal and compete with each other, with the one exhibiting a higher coupling strength being excited, thereby suppressing the other mode.

However, when the azimuthal distribution of the electron beam becomes non-uniform, the interaction behavior between modes can change significantly. Notably, for an azimuthally segmented electron beam, when the azimuthal beam period aligns with the mode's azimuthal phase beating period, as shown in Fig. 1, it creates conditions conducive to the coexistence and cooperation of two oppositely rotating modes. Intuitively, this occurs because the coexistence of two modes forms a standing wave pattern, allowing the antinodes to match exactly with each beamlet for maximal interaction.²³ 

For suppressing the intense mode competition, the interaction circuit incorporates specific characteristics, including unique mode selection, a relatively long resonant cavity, an electron beam with high velocity ratio, moderately large velocity spread, and azimuthal periodicity for inducing phase locking of two oppositely rotating wave of the desired TH operating mode. The overall design parameters are summarized in Table I. Detailed design considerations and simulation results will be discussed in Secs. III A–III G.

The third-harmonic (TH) mode faces severe mode competition from the lower-harmonic parasitic mode. The use of a small cavity radius and low-order operating mode with a rarer mode density can considerably reduce the risk of mode competition. However, for operation at a frequency as high as 1.2 THz, a major challenge lies in the drastic down-scaling of the cavity dimension with the shortening of the wavelength, which lead to small power capacities and fabrication difficulties. This, on the other hand, makes necessary the use of high-order operating mode and an electric-large cavity. Balancing mode competition and fabrication feasibility of state-of-the-art milling machines, a cavity radius of R_c = 1.4∼1.55 mm was chosen, corresponding to an eigenvalue approximately 35–39 for the modes with a frequency of 1.2 THz, as shown in Fig. 2(a). To avoid mode competition, it is necessary to locate the TH operating mode away from the fundamental-harmonic (FH) and second-harmonic (SH) parasitic modes in the frequency (eigenvalue) distribution. Notably, a gap in the FH modes can be observed between 35.4 and 38. For the SH modes, TE_8,5(2) mode is well separated from other SH modes and it is also a classical mode for SH gyrotrons.^21,27,28 By this, a gap exists between SH TE_8,5(2) and TE_14,3(2) modes, which is favorable for third harmonic operation. It is worth mentioning that the backward wave (gyro-BWO) operating regime of a lower-harmonic parasitic mode can extend aggressively into the high-frequency side, posing a huge risk of mode competition for stable excitation of a higher-harmonic mode.^21,29,30 Therefore, priority was given to estimating the frequency separation in the lower-frequency side of the TH modes with respect to the FH and SH modes. Additionally, to realize a maximal effect of coupling strength amplification when inducing the degenerate of two oppositely rotating waves, it is favorable that both rotating waves of the desired operating mode can achieve strong coupling to the electron beam simultaneously.²³ 

Based on the aforementioned consideration, the TE_18,5 mode was chosen. It is located far away from the SH TE_8,5 mode and the FH TE_6,2 mode on its lower frequency side. The beam–wave coupling strength as functions of the beam radius is shown in Fig. 2(b), which is defined as the square of the coupling factor $G2=Jm∓s2K⊥,kRe$⁠. In order to fully leverage the benefits of cooperation between the co- and counter-rotating waves of the TE_18,5 mode by employing an azimuthally segmented electron beam, the beam guiding radius is chosen as R_b = 0.907 mm, where the two waves can achieve strong coupling to the beam. The coupling strength and frequency distribution of the potential competing modes are shown in Fig. 2(c). Among these modes, the FH TE_+6,2(1), TE_+4,3(1) and SH TE_+8,5(2), TE_−11,4(2) can be identified as the most critical within their respective harmonic categories, due to their high coupling strength and close frequencies from the TH operating mode. This has been verified in the subsequent simulation, and the SH TE_+8,5(2) mode turns out to be the most critical one among all. Most notably, by positioning the beam at R_b = 0.907 mm, the competition from TE_8,5 mode is greatly suppressed compared to the beam position at the first radial peak of the TE_+18,5(3) mode, because the strong beam–wave coupling associated with the TE_+8,5(2) and TE_−8,5(2) mode is effectively mitigated.

Due to the weak beam–wave coupling strength of the third harmonic mode, a long cavity is usually employed to increase both the interaction time and the quality factor, thereby reducing the starting current. This, however, incurs drastically increased Ohmic losses. It is crucial to strike a balance between Ohmic losses and mode excitation, and more importantly, to consider the competition among modes. In terms of mode competition, following the scaling relation between the diffractive quality factor and wavelength Q_d ∼ (L/λ)² (Ref. 29) the harmonic mode's shorter wavelength amplifies the sensitivity of its quality factor to the cavity length variations ΔL compared to lower-harmonic parasitic modes. In addition, a longer cavity reduces the continuous frequency tuning range limited by beam velocity spread,^31,32 which tends to suppress the lower-harmonic parasitic modes that are usually excited in the gyro-BWO regime.²¹ The increase in the cavity length also narrows the cyclotron resonance band defined as $Δω≈1/L/vz$ (⁠ $vz$ is the electron axial/transit velocity).²⁴ Therefore, extending the cavity length in a specific manner can also help to bolster the competitive advantage of the harmonic mode over the lower-harmonic parasitic mode, but as mentioned, it also drastically increases the Ohmic loss, limiting the output power. Considering the enhanced coupling strength enabled by the proposed azimuthally segmented electron beam and leveraging the interplay of mode competition, starting current, and the Ohmic loss, we choose a cavity length of 13 mm. This uniform section is connected to the output waveguide by two slightly taper sections with a length/angle of 2 mm/1° and 5 mm/3.6°, respectively. The structure and cold field profiles are shown in Fig. 3. The diffractive quality factor Q_d of the TE_18,5 mode in a cold cavity is about 163 500.

The operating parameters of the gyrotron are shown in Table I. The electron voltage is 30 kV. The velocity ratio is 1.5 with a velocity spread of 30% to suppress parasitic modes.^31,33 Considering the frequency difference and the beam–wave coupling strength, the most dangerous competing modes for the TH operating mode are the FH TE_+6,2(1), TE_+4,3(1) and SH TE_+8,5(2), TE_−11,4(2) located on its lower- and higher-frequency side, respectively. The other competing modes either exhibit a weaker coupling to the beam or are located farther away from the desired TH operating mode in the frequency distribution. Since parasitic modes with the same harmonic number share similar operational characteristics with their counterparts, it is expedient to focus the mode excitation analysis solely on their most dangerous representatives using the single-mode frequency-domain approach. The dynamic interaction process that considers a more comprehensive range of parasitic modes will be investigated later using advanced multimode time-domain simulations.

The starting current of the modes is calculated using a single-mode stationary self-consistent model,³¹ and the results are shown in Fig. 4. It is worth noting that for a coaxial electron beam with azimuthally uniform beam–wave coupling strength, the single-mode starting currents are independent of the electron beam's azimuthal nonuniformity, meaning the starting currents are identical whether the beam is uniform or nonuniform (noted that I_b is the total beam current integrated over the beam's transverse area). Here, the conductivity of cavity wall is taken as 5.8 × 10⁷ S/m. The dash lines and solid lines correspond to the cases of zero velocity spread and 30% velocity spread, respectively. For an ideal electron beam with no velocity spread, the FH TE_+6,2(1) and SH TE_+8,5(2) parasitic modes can be excited at a broad range of magnetic field (gyro-BWO regime) and both have a lower starting current than that of the TH TE_−18,5(3) mode. This is a major obstacle impeding the excitation of the TH operating mode. By introducing 30% velocity spread in the electron beam, the starting currents of the FH and SH parasitic modes in their gyro-BWO regime are drastically increased, which opens a narrow zone free of competition for the excitation of the TE_−18,5(3) mode.^31,33

In conventional gyrotron with an annular uniform electron beam, the oppositely rotating modes are orthogonal and compete each other due to the azimuthal averaging effect of the mode phase difference over the electron beam guiding phase.⁸ It has been found that the azimuthal periodicity of electron beam can lead to the phase lock and the cooperative coexistence of the two oppositely rotating modes.²³ For achieving such mode degeneracy with mode selection, it is important to have electron period Δφ_e,beam = 2π/N_φ matches with the beat period of the oppositely rotating waves Δφ_e,beat = 2π/2m of the desired mode exclusively Δφ_e,beam = k⋅Δφ_e,beat, i.e., m = k N_φ (k = 1, 2, …).

Considering the azimuthal index of the operating TE_18,5 mode and those of the most dangerous parasitic modes (e.g., FH TE_6,2(1), TE_4,3(1) and SH TE_8,5(2), TE_11,4(2)), we choose an electron beam with an azimuthal period of N_φ = 18, i.e., the nonuniform electron beam consists of 18 beamlets evenly distributed in the guiding phase φ_e,segbeam= φ_e0 + i_b ⋅ π/9 (i_b = 1, 2,…, 18, φ_e0 is an arbitrary initial phase). This serves to selectively induce the degeneracy of two oppositely rotating waves only in the desired TH TE_18,5 operating mode, as will be elucidated in the subsequent demonstration.

Here, we first consider the case of two-mode interaction dynamics as an illustration of the cooperative excitation of the two oppositely rotating modes. We consider two operating points denoted “A” and “B” in Fig. 4. The results of multimode simulation considering are shown in Fig. 5, including the time-dependent variation of the power and frequency of the two modes, as well as their temporal phase difference: $Δψt=ω−−ω+t$⁠. Noted that the carrier frequency of modes is adaptively updated according to the actual wave frequency to more accurately describe the frequency pulling effect and reduce the computational errors.^34,35

In the case of a uniform electron beam, for operation at point “A” located within the single-mode self-excitation regime of TE_−18,5(3), this mode is excited, while its counterpart TE_+18,5(3) is suppressed due to a higher starting current, as shown in Fig. 5(a). This is the typical scenario of mode suppressing between two orthogonal modes. By contrast, when the nonuniform electron beam is implemented, the two modes are excited simultaneously and reach at a stable coexistence of two modes, as shown in Fig. 5(b). The frequencies of the two modes are first pulled from their initial cold cavity value and then gradually converge to the same value, with the phase difference arriving at a constant value for the stable coexistences of the two modes. These phenomena indicate that the two oppositely rotating modes are phase locked or degenerate, and the two oppositely rotating modes can cooperate with each other and “share” the same electron beam by introducing azimuthal periodicities of the electron beam. The coexistence of the two modes will form a standing-wave like pattern. Naturally, the fixed azimuthal anode should be located at the guiding phase of the beamlet, which gives the maximal coupling and interaction strength between the beam and the wave, as shown in Fig. 6. The reason for such phase-lock, can also be interpreted from the perspective that the phase difference of the two modes is fixed for each of the 18 beamlets $Δψ=2mφe,segbeam=36φe0+ib ·2π$⁠.

For the operation at point B located beyond the single-mode operating regime of TE_−18,5(3), either the single TE_−18,5(3) or TE_+18,5(3) mode cannot be excited due to the operating current being lower than their starting current, as shown in Fig. 5(c). However, when the nonuniform beam is implemented, the amplitudes of the degenerate TE_−18,5(3) mode and TE_+18,5(3) modes start to grow simultaneously and arrive at stable values, as shown in Fig. 5(d). The frequencies of the two modes arrive at the same value with a constant phase difference. Therefore, the cooperation of the two modes allows for lowering starting current. By performing two-mode simulation, we can determine the starting current for this phase-locked degenerate mode TE_(±)18,5(3) as the red line (dash and solid lines) in Fig. 4. Correspondingly, the competition-free operating zone of the third harmonic mode is broadened into lower beam current as denoted in Fig. 4.

Interestingly, the starting current is reduced by a constant factor of $Ist,−/Ist,degen±=1.71$ throughout different values of magnetic field. This coincides with the ratios of the sum of the coupling strength of the two modes to that of the counter-rotating mode: $G(+)2+G(−)2/G(−)2=0.088 24/0.0517=1.71$ in Fig. 2(b). Additionally, the ratios between the power of the two modes in Figs. 5(b) and 5(d) are the same (⁠ $P−/P+=1.41$⁠), and they also coincide with ratios between the coupling strength of the two modes $G(−)2/G(+)2=0.0517/0.0366=1.41$⁠. Moreover, the normalized axial field profiles of the two modes at their coexistence equilibrium are nearly identical. All these linear and nonlinear phenomena observed in self-consistent simulation indicate an increase in the beam–wave coupling strength of the degenerate TE_{(−/+)18,5(3)} mode with respect to a single rotating mode: (⁠ $G(±)2=G(+)2+G(−)2$⁠).

The results in Fig. 5 showed that the implementation of the nonuniform electron beam with azimuthal periodicity not only lowers the starting current of the TH operating mode, but also increases the total output power of co- and counter-rotating waves of the TH operating mode. For operation at point A, the total output power is increased from 20 W [Fig. 5(a)] to 30 W [Fig. 5(b)] by a factor of about 1.5. For a better comparison between the uniform beam (case 1) and nonuniform beam (case 2) cases, the total output power and frequencies as functions of the magnetic field for different values of beam current are shown in Figs. 7(a) and 7(b). For a beam current of I_b = 0.6 A, the maximal output power rises from 27 W (case 1) to 90 W (case 2), showing an increase of about three times under the same beam current. However, due to the competition from the SH TE_+8,5(2) mode (as shown in Fig. 4), the maximal available power of case 2 might be lower to about 50 W, which is still twice that of the case of uniform electron beam. For a beam current of I_b = 0.4 A, the output power can reach up to 20 W for a nonuniform electron beam, whereas a conventional uniform beam exhibits negligible oscillations under similar conditions.

Additionally, the total output power as functions of the beam current is shown in Fig. 7(c) for the two cases. In general, the power increases with an increase in the beam current. The implementation of the nonuniform electron beam lowers the operating beam current, thereby increasing the efficiency and power in the low-current regime compared to the case of conventional uniform electron beam. However, as the beam current increases in the nonlinear regime, the increase in the power becomes slower, indicating a decrease in the output efficiency. Notably, the excess of total power for a nonuniform beam over that of a uniform beam gradually diminishes, and the power become comparable at large currents.

These results highlight the substantial differences in quality factors between cold and hot cavities in such high-frequency gyrotrons, which will lead to completely different results of mode excitation and overall output efficiencies. This outlines the importance of using self-consistent formulas with inclusion of cavity wall loss for accurate simulation of the beam–wave interaction dynamics of high-frequency gyrotrons. In addition, the results demonstrate an interplay between the Ohmic and electron efficiencies: it is beneficial for the high-frequency gyrotrons to operate near the linear regime with a low Ohmic loss, but the electron efficiency would be low too.

It is beyond doubt that selective enhancement in the beam–wave coupling strength is beneficial, as it can lower the starting current and increase the power several times. Seeking a maximal power can yield high output near the boundaries of competition-free zones in Fig. 4; however, these areas might be susceptible to significant mode competition due to practical variations in beam or cavity parameters. We would like to emphasize that the implementation of a nonuniform beam uniquely reduces the beam current, expanding the operating zones toward competition-free regimes, while still achieving comparable output power in the tens of watts range. Here, we have made a detailed nonlinear self-consistent analysis to provide a comprehensive picture of the effect of coupling strength enhancement on the output power in THz gyrotrons, where the high-frequency Ohmic loss plays an important role. Notably, utilizing a nonuniform beam with increased coupling strength also enables a reduction in cavity length, further minimizing Ohmic loss, as discussed in Sec. II B on the cavity design.

The single-mode starting current of each orthogonal mode serves as a pivotal indicator of the competitive relationship among modes. However, the interaction among these modes through the electron beam necessitates a deeper investigation into the mode interaction dynamics using the more advanced multimode time-domain simulations. In the multimode simulation, we consider 36 modes listed in Fig. 2(c). The results of mode competition are shown in Fig. 9 for a magnetic field of 14.88 T and a beam current of 0.5 A. Such an operating point locates beyond the operating regime of the single rotating TE_−18,5(3) mode but falls into the operating regime of the degenerate TE_±18,5(3) mode. It is assumed that the nonuniform electron beam consists of 18 beamlets uniformly distributed in the azimuthal coordinates as illustrated in Fig. 6. The TE_−18,5(3) and TE_+18,5(3) modes are excited simultaneously and become phase-locked. Meanwhile, the rotating waves of parasitic modes maintain their orthogonality, with the excitation of the desired TH mode effectively suppressing other parasitic modes. This verifies the stable operating regime of the TH mode as defined in the single-mode analysis.

In terms of mode interaction, the nonsynchronous interaction among orthogonal modes with close frequencies tends to favor the early excited mode, which in turn suppresses the excitation of other modes.²⁴ This behavior is generally consistent with the mode competition results anticipated by single-mode simulation analysis. Close attention should be paid to the synchronous interaction among modes, as it can lead to the parametric instability or the coexistence of multiple modes.^8,24,39 It is worth noting that even though TE_18,5(3) and TE_6,2(1) modes satisfy the spatial synchronous condition m₁s₂ = s₁m₂, their frequencies are well separated (ω₁s₂ ≠ s₁ω₂) so that the excitation of TH TE_18,5 mode remains suppressing the FH TE_6,2(1) parasitic mode.

In practice, the nonuniform electron is not an ideally point-like structure, but exhibits guiding radius spread. This might affect the mode degeneracy of the oppositely rotating waves. The radial spread will result in slight variation of the beam–wave coupling strength, an effect that has already been studied in numerous works.^31,40,41 Here, we will focus on the azimuthal spread d of the nonuniform electron bunches and study its impact on the interaction of two oppositely rotating waves. When the azimuthal periodicity of the electron beam matches with that of the phase beating of the oppositely rotating modes, i.e., Δφ_e,beam = k ⋅ Δφ_e,beat (k = 1, 2,…), the effect of the azimuthal spread of the beamlets also exhibits periodicity. Then, it can be characterized by the azimuthal spread radian d normalized to the azimuthal wavelength of a nonsymmetric TE_m,n mode (with m azimuthal periods) at the beam radius, which can defined as $λm=2πRe/m$⁠. The output power as a function of different azimuthal spread value is calculated using the multimode simulation, and the results are shown in Fig. 10 for B = 14.88 T and I_b = 0.5 A. It can be found that as the azimuthal spread d starts to increase from zero, the two oppositely rotating mode can still remain phase-locked, yet their output power slightly decreases. When the spread value is smaller than 0.2λ_m, the power degradation is small. As d continuous to increase, the power of the weakly coupled rotating mode (TE_+18,5(3)) experiences a more pronounced decline. Upon reaching a spread value that is half of the azimuthal wavelength 0.5λ_m, the two counter-rotating waves disappear, indicating that the two modes are orthogonal and compete with each other. This observation is consistent with the higher starting current required for a single rotating mode compared to the operating beam current. As the spread value further increases, the degeneracy of the two modes reemerges. At a spread value of d = λ_m, the scenario resembles the case of a uniform electron beam, and the two rotating modes are orthogonal again.

In this paper, we present the development of a 1.2 THz third-harmonic gyrotron using a novel azimuthally segmented electron beam that facilitates mode-selective enhancement in the beam–wave coupling strength for the desired mode. The interaction circuit is systematically studied. First, the third-harmonic operating mode TE_18,5(3) is carefully selected, considering the unique requirement of the novel beam configuration. Next, the cavity is designed to balance factors such as starting current, Ohmic loss, and mode competition. Then, the single-mode starting current of the modes is analyzed using a self-consistent stationary model. Following this, we introduce a nonuniform electron beam consisting of 18 beamlets, evenly distributed in the electron guiding phase, to selectively induce the degeneracy of the oppositely rotating waves of the TE_18,5(3) mode. The two-mode dynamic simulation (including the co-rotating TE_+18,5(3) and counter-rotating TE_−18,5(3) mode) shows that the coexistence of the two modes can lower the starting current of the degenerate TE_18,5(3) mode. The scaling relation indicates that this improvement is attributed to the enhanced total coupling strength resulting from the cooperation of the two oppositely rotating modes. Furthermore, a detailed nonlinear self-consistent analysis reveals that the enhancement of coupling strength increases the output power by three times under the same beam current. The results also highlight the significance of the self-consistent effect and Ohmic loss in the analysis of high-frequency gyrotrons. Thus, the implementation of the azimuthally segmented electron beam serves to broaden the competition-free operation regime of the third-harmonic TE_18,5(3) mode and significantly increase the output power. Subsequently, we conduct a multimode simulation to investigate the nonlinear interaction of different modes. The simulation results confirm the effectiveness of the design, as the excitation of the degenerate mode does not induce nonlinear excitation of parasitic modes. Finally, the effect of the electron guiding phase spread is studied. The results indicate that the degradation of output power is relatively small when the guiding phase spread is within 0.3λ_m.

The required electron beam can be generated by modulating the emitter ring into periodic segments in a conventional magnetron injection gun.²³ However, in our present case, it is desirable to control the guiding phase spread of the electron beamlet to be considerably small preferable within 0.3λ_m, so as to maximize the effect of the azimuthally segmented electron beam. This can be realized by simply covering some parts in an emitter ring, but the emitter filling ratio will be lower. It is necessary to reach a balance among velocity spread, guiding phase spread. The use of elliptical emitters can help to increase the filling factor.⁴² It is worth noting the azimuthal phase drift of the beamlet can be neglected, because the resonator is axisymmetric and the antinode of the standing waves can always adjust themselves to matches the beam location. This greatly simplifies the requirement of the magnetron injection gun (MIG) compared to the multibarrel gyrotrons.^42,43 In addition, the mechanism of the proposed azimuthally segmented electron differs from previously reported methods that achieve the coexistence of the oppositely rotating modes through resonant cavities with azimuthal asymmetry.^44,45 Instead, it relies on the intrinsic modal phase relationship of multimode beam–wave interactions, enabling efficient, cooperative, and spontaneous coexistence of two oppositely rotating waves. The cavity-induced mode degeneracy can achieve a stronger mode selection, but it may suffer from beam–wave coupling degradation.⁴⁶ This degradation can also be interpreted as the coupling loss resulting from the electrons located at the node (weak EM field) of the azimuthal standing wave pattern formed by the coexistence of the two oppositely rotating waves. Furthermore, for applications that do not require a Gaussian beam, the coexistence of the two modes presents no problems. However, most applications necessitate a Gaussian beam, making the conversion of the two modes into one or two Gaussian beams highly beneficial. This can be accomplished using a quasi-optical launcher equipped with a specific mode-converting wall, as successfully demonstrated in Ref. 47.

This work was supported by the National Natural Science Foundation of China (Nos. U21A20458 and 52407007), the National Key Research and Development Program of China (No. 2021YFA1600303), and the China Postdoctoral Science Foundation (Nos. 2024M751004 and 2024T170294).

The authors have no conflicts to disclose.

Xianfei Chen: Conceptualization (equal); Methodology (equal); Writing – original draft (equal). Houxiu Xiao: Funding acquisition (equal); Supervision (equal); Writing – review & editing (equal). Weijian Liu: Formal analysis (equal); Software (equal). Runfeng Tang: Investigation (equal); Software (equal). Shaozhe Zhang: Formal analysis (equal); Investigation (equal). Donghui Xia: Validation (equal); Writing – review & editing (equal). Xiaotao Han: Funding acquisition (equal); Supervision (equal); Writing – review & editing (equal).

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