Intrinsic resonance in gyrotron using non-resonant RF structure Open Access

Gyrotrons, utilizing the electron cyclotron maser interaction to generate coherent radiation,^1–3 have been experimentally demonstrated as high-power (up to sub-gigawatt level) and high-frequency (up to the terahertz range) sources.^4–8 These powerful radiation sources have found numerous applications in plasma heating for fusion, communication, sensing, security, and biomedical diagnostics.^9–12 Based on the feedback mechanism, gyrotron oscillators are traditionally categorized into two types the gyrotron backward-wave oscillator (gyro-BWO) and the gyromonotron. In gyro-BWO, oscillation¹³ arises as the forward-moving electron beam undergoes continuous modulation through the backward radiation emitted by preceding electrons. The modulation induces a stronger backward radiation tendency in the beam, leading to the gradual formation of an internal feedback loop and absolute instability.^14,15 This instability allows the gyro-BWO mode to grow and the field profile contracts to the beam entrance end.¹⁶ As the magnetic field (B₀) or the beam voltage (V_b) is tuned, the electron transit angle of gyro-BWO modes varies simultaneously to satisfy the beam–wave resonance condition, thereby enabling frequency tuning.¹⁴ On the other hand, the gyromonotron operates in a resonant structure.^17–19 Its oscillation is believed to be formed primarily by reflective feedback due to structural nonuniformities.^1,6,20–23 Previous studies have demonstrated that in a gyromonotron device, the electron beam tends to resonantly interact with the fundamental cavity axial mode,¹⁸ built up by the interference of forward and backward waves. The resonant conditions of the interaction cavity therefore restrict the frequency tunability of each gyromonotron mode as B₀ or V_b is adjusted.²⁴ However, the resonant enhancement renders the gyromonotron a high-efficiency microwave source.^25,26

Almost half a century ago, Bratman et al. conducted a study on the non-fixed fields of transverse electric (TE) modes within a semi-infinite low-Q regular tube, featuring a structure cutoff at the upstream end and an abrupt drop of static magnetic field (magnetic-field cutoff) right before the downstream end²⁷ to physically obtain a finite interaction region. The outgoing-wave (no-reflection) boundary condition was adopted in the model to analyze gyrotron devices in a self-consistent manner. They observed an exceptionally high transverse peak interaction efficiency of 0.75, at which the field profile closely resembles that of a resonator mode with weak phase variation across the entire interaction length. The feedback pathway was attributed to the downstream reflected wave generated at the cross section where resonant interaction of electrons with the radio frequency (RF) field completely ceases. As this backward reflection propagates to the upstream, it is turned into the forward wave due to the presence of cutoff, thereby forming the feedback loop. A subsequent study using a similar three-section low-Q cavity with an oversized end pointed out the involvement of both the forward and backward waves in shaping the high-Q resonant field profile.²⁸ After these works, an increasing number of research efforts^{18,19,29–34} have been devoted to self-consistent investigation of low-Q gyromonotron systems. Fliflet et al.¹⁹ further showcased the persistence of strong beam–wave interaction of the TE₀₁ mode in a low-Q RF system considering a uniform magnetic field, and the calculated results were compared with the experimental verification.

It is important to notice that those low-Q gyromonotron systems usually feature an upstream structural irregularity (e.g., cutoff) and a downstream beam irregularity (by either closing the magnetic guiding field or applying simulation boundaries to terminate the beam–wave resonant interaction). However, the extent to which these irregularities induce reflection and whether such reflection is adequate to produce a strong feedback for generating high-efficiency forward output has not been thoroughly examined. That raises a fundamental question regarding the underlying mechanism of the observed oscillations. Do the oscillations necessarily rely on the recurrent conversion between forward and backward waves formed by either tube irregularities or inhomogeneities of the electron-stream conductivity caused by the variation of the static magnetic field²⁸ or the presence of the imposed boundary conditions?

For the sake of simplification, recent studies have intended to investigate the gyrotron oscillator in its relatively simplified configuration, i.e., a uniform tube under a constant DC magnetic bias.^{14,16,35–37} In such a setup, the outgoing-wave boundary conditions are adopted at both ends, as Bratman et al. did in their model.^27,28,30 Among these works, Hung et al.³⁵ have successfully attained a low starting current and achieved a high backward output efficiency in generating the TE₅₂ mode. This work also demonstrated an oscillation frequency very close to the cutoff and a resonance mode profile, all achieved without requiring any external feedback. On the other hand, in the case of the transverse magnetic (TM) mode gyrotron implementing TM₁₁-mode, a nonlinear and self-consistent simulation demonstrated a peak interaction efficiency of 32%, primarily contributed by the generation of the forward wave.³⁷ 

These observations suggest that a system without any external feedback can still effectively function as a gyromonotron-like oscillator, delivering a strong output toward the collector end, regardless of the operating mode being TE or TM. Several thought-provoking questions then arise: What is the underlying feedback mechanism that governs the gyrotron in the near-cutoff region, resulting in the generation of strong forward waves? Does this interaction encompass an absolute instability characterized by the internal feedback, reminiscent of the gyro-BWO,^1,14–16 or does it resemble a convective instability as seen in gyro-traveling-wave tube (TWT)?^1,13,38,39 Moreover, what are the corresponding inherent bunching mechanisms for TE and TM modes at the near-cutoff gyromonotron regime?

In this paper, we present comprehensive analyses of the TM₁₁-mode and the TE₀₁-mode gyrotrons, elucidating the physics contributing to their high output efficiencies (>30%) when implemented with a uniform tube biased by constant static magnetic field. Our investigation begins with a nonlinear and self-consistent particle-tracing simulation of the TM₁₁-mode gyrotron (Sec. II). By examining the behavior of the forward and backward components under magnetic field tuning, we provide a qualitative explanation based on their detuning from the beam–wave resonance conditions. To further clarify the underlying feedback mechanism, the impact of introducing heavy Ohmic wall loss to both the upstream and downstream sections of the system is explored (Sec. III). In Sec. IV, we delve into the linear stage of the system and discuss the linear behavior (start-oscillation currents and field profiles) of the TM₁₁ mode. By examining the electron transit-angle phases, we unveil that the high efficiency observed in a uniform tube results from the concurrent presence of a forward convective energy exchange process and a backward internal feedback mechanism. Furthermore, we study the TM-mode bunching mechanisms in Sec. V, revealing that the co-generated forward and backward waves can together enhance electron bunching near the cutoff region of the TM mode. In Sec. VI, the analyses in Secs. II–V are applied to the TE₀₁-mode gyrotron. An analogous conclusion is drawn, demonstrating that the proposed fundamental mechanisms hold true under near-cutoff operation regardless of the mode choices. The identification of the dominated instability and several comparisons and the remarks of the present findings are provided in Sec. VII. Finally, in Sec. VIII, we present our conclusions, summarize the key findings and implications of our study, and provide a concluding remark.

For the demonstration purpose, the TM₁₁ mode is selected as the first testing mode in a uniform waveguide with a radius (r_w) of 0.2 cm and a constant static magnetic field B₀ across the entire tube. An axis-encircling beam with an electron's guiding center radius (r_c) of 0 cm is chosen for the optimum beam–wave coupling strength.^40,41 Throughout this work, the beam voltage V_b and the pitch factor α (electron transverse-to-axial velocity ratio) are set at 70 kV and 1.0, respectively. These parameters are consistent with those employed in the recent publication.³⁷  Fig. 1(a) shows the TM₁₁ dispersion (⁠ $Re[kz]$⁠) and the beam–wave resonant lines (⁠ $ω−Re[kz]vz0−Ωc0=0$⁠) at multiple magnetic fields B₀, where $ωc$ is the cutoff frequency of the mode, c is the speed of light in vacuum, v_z0 is the electron's initial axial velocity, and $Ωc0=eB0/γ0me c$ is the initial relativistic gyrating frequency of the electron. e and m_e are, respectively, the charge and the rest mass of electron, and $γ0=(1−v02/c2)−1/2$ is the Lorentz relativistic factor, in which $v0=(v⊥02+vz02)1/2$ and $v⊥0$ are, respectively, the electron's initial velocity and transverse velocity. Only the fundamental harmonic is considered to make the discussion concise. The exact formula of waveguide mode's propagation constant $kz$ can be found in the  Appendix [Eq. (A2)]. In the present case, the corresponding grazing magnetic field $Bg=(γ0me c/e)⋅(ωc/γz0)$⁠, where $γz0=(1−vz02/c2)−1/2$ is $34.96 kG$⁠, at which the beam–wave resonance line is tangent to the mode dispersion curve. Beyond B_g, the intersection between the beam mode and the waveguide mode signifies the most preferred emission direction of oscillation. In the following, the terms “gyromonotron region” and “gyro-BWO region” refer to the conditions where the beam line intersects, respectively, with the forward branch and the backward branch of the waveguide mode.

Figure 1(b) shows $εfwd$ and $εbwd$ with superscripts “NL,” meaning they are calculated based on the solved oscillation frequency in the nonlinear region (ω₀). Note that subscripts “0” of v_z0 and Ω_c0 represent their initial values at the entrance of the beam. Although Eq. (1) might introduce errors due to the omission of the beam spread caused by nonlinear beam–wave interaction, these quantities still serve as useful approaches for qualitative elucidation. When B₀ is adjusted, $εfwdNL$ displays a valley pattern as indicated by the L.H.S. arrow in Fig. 1(b), while $εbwdNL$ remains relatively unchanged. A smaller detuning factor implies better synchronization and hence higher interaction efficiency in the soft-excitation regime. $εfwdNL$ and $εbwdNL$ therefore exhibit inverse responses to the output efficiencies η_fwd and η_bwd, respectively. Since $εfwdNL$ is smaller than $εbwdNL$ at the near-cutoff region (roughly below 37.3 kG), the forward wave appears to dominate the beam–wave interaction. At B₀ ∼ 37.3 kG [marked by the R.H.S. arrow in Fig. 1(b)], $εfwdNL$ and $εbwdNL$ intersect with each other, indicating the crossover behavior of η_fwd and η_bwd, as can be seen by comparing Figs. 1(d) and 1(e). This manifests the gradual transition from gyromonotron to gyro-BWO, where the dominant beam–wave interaction shifts from the forward wave to the backward wave. Note that in cases of hard excitation, even higher efficiency may be achieved by large detunings.⁷ 

Once L exceeds 3.0 cm, the fluctuation in η_beam becomes apparent across the entire B₀-tuning region. A similar response is observed for η_fwd. Regardless of the increase in L, η_fwd remains primarily within the upper bound observed as L ∼ 3.0 cm. The fluctuating η_fwd spectrum arises from the iterative beam-energy deposition and absorption. To illustrate this concept, the forward-wave power profile P_fwd at B₀ = 36 kG is plotted in Fig. 2(a). For L = 1.7 to 3.0 cm, the longer tube allows more space for the beam–wave energy exchange, leading to the gradual increase in the peak forward-wave power. As L exceeds a certain threshold, the electron beam starts to reabsorb the wave energy gained at the earlier stage (upstream), leading to a decrease in the forward output power at the downstream end. The cases with L = 3.0 to 5.0 cm presented in Fig. 2(a) evidence this statement. Accordingly, the fluctuating η_fwd spectra observed at different L is attributed to the truncation of the beam–wave energy deposition and re-absorption process at the downstream port. On the contrary, η_bwd in Fig. 1(e) gradually saturates and remains flat even when L approaches 5.0 cm. At the gyro-BWO region (e.g., B₀ = 38 kG), when L exceeds a relaxation length,¹⁵ the backward output efficiency would saturate due to the nonlinear field contraction. Any further increase in length beyond the relaxation length only contributes a marginal tailing field structure to the overall field distribution.¹⁶ Interestingly, a similar phenomenon is also observed in the gyromonotron case of B₀ = 36 kG, as evident from the saturated and contracted backward-wave power profiles P_bwd in Fig. 2(b).

For gyro-BWOs, the backward wave plays an essential role in energy extraction, which is confirmed by negligible η_fwd at B₀ = 38 kG. As for the case in the gyromonotron region (B₀ = 36 kG), while the beam is coupled strongly with the forward wave, the backward wave always survives. The origin of the backward wave is understood by the intrinsic internal feedback, whereas that of the forward wave remains elusive, especially for operating in a uniform tube without any external feedback, such as the absence of structural or magnetic-field inhomogeneities. Although Fig. 2(a) appears to demonstrate the convective growth of the forward wave, the coexistence of forward and backward components operating near cutoff seems to imply the simultaneous presence of the forward-manner convective interaction and the backward-manner internal feedback. To further clarify the underlying mechanism, lossy-section tests are employed in Sec. III to assess the response of forward and backward waves.

Applying distributed wall loss to the interaction circuit is an effective way to eliminate the unwanted oscillations.^38,42–44 This method is adopted here to test the response of output signals. Based on the nonlinear power profiles displayed in Fig. 2, a tube with a moderate length of L = 3.5 cm is selected. This length ensures adequate spacing between the major-field distributions of the forward and backward components, enabling their individual manipulation. Two lossy-section tests are applied, during which a lossy section of 1 cm is present at the downstream [Figs. 3(a) and 3(b)] and the upstream [Figs. 3(c) and 3(d)].

Figure 3(a) demonstrates that the forward output efficiency is significantly suppressed to below 5% and becomes flat as the downstream Ohmic loss increases from 10¹ρ_Cu to 10⁴ρ_Cu. This outcome is not surprising since the location of the lossy section coincides with the major field of the forward wave. The energy emitted by the electron beam therefore gets dissipated by the surface current on the waveguide wall directly. This test also illustrates an important fact that the generated forward wave must correspond to the cold-tube mode rather than the hot-tube mode.¹³ In other words, the highest achieved forward efficiency (η_fwd ∼ 23%) does not rely on the field energy residing within the gyrating electrons themselves. Instead, it relies on the energy being carried by the surface current and the guided mode, which can be subsequently extracted using appropriate couplers. Moreover, due to the nonlinear field contraction, the increased downstream Ohmic loss has a negligible impact on the backward output in the gyro-BWO region. η_bwd can even be slightly enhanced in the region of B₀ = 35.5 ∼ 37.5 kG. This slight enhancement might result from the increased reflection at the interface between the two regions with high-contrast in their wall losses, reflecting the generated forward wave back to the upstream direction.

In the other case with the upstream lossy section, it is expected that the increased wall loss should effectively eliminate the backward wave, while having a lesser effect on the generation of the forward wave. Figure 3(d) confirms the first point, showing that η_bwd is reduced by half at 38 kG (gyro-BWO region) and is completely suppressed at 36 kG (gyromonotron region) when the wall resistivity reaches 10⁴ρ_Cu. The distributed upstream loss thus serves as a key to suppress spurious oscillations from the absolute instability, which was utilized to achieve a gyrotron traveling-wave tube (gyro-TWT) with a ultrahigh gain of ∼70 dB.¹³ However, the result in Fig. 3(c) contradicts the aforementioned expectation on the forward wave generated at the gyromonotron region. It is discovered that η_fwd also decreases significantly as the wall loss increases, synchronizing with the reduction in η_bwd. This observation provides valuable insight into the origin of forward oscillation, suggesting a possibility that a similar internal feedback mechanism, originally associated with the backward wave, may also play a crucial role under the near-cutoff operation adopted by most gyromonotron devices. As a result, when this internal feedback loop is disrupted by the heavy Ohmic wall loss at upstream, all subsequent nonlinear beam–wave interactions are suppressed, causing the complete absence of output. To simplify the analysis of the interplay between forward and backward waves and illustrate their origins, Sec. IV will focus on the physics during the linear growth stage, avoiding the complicated nonlinear effect.

Figure 4(a) plots the starting current I_st and the oscillation frequency ω_st of the TM₁₁ mode in a 3.5 cm uniform tube and a constant B₀. These values are calculated by solving the relativistic Vlasov equation and Maxwell's equations together. Following linearization using the small-signal approximation, either the Laplace transformation method or the integral method can be employed for the solution.³⁶ To manifest the strong beam–wave interaction at the gyromonotron region, the inverse of I_st is also displayed, which mirrors the trend observed in the nonlinear forward-wave efficiency in Fig. 1(d). Despite the nonlinear effects such as the truncation of beam-energy deposition [Fig. 2(a)] and the nonlinear field contraction [Fig. 2(b)], it becomes apparent that the lower the starting current, the higher the output efficiency. This finding emphasizes that the feedback mechanism at near-cutoff operation continues to play a significant role, even in the linear regime. The presence of such a feedback loop in a finite-length system could lead to the onset of self-oscillation during the linear stage, introducing a new energy-balance condition.¹⁶ 

Figures 4(b) and 4(c) present the amplitude and the phase of the start-oscillation field f(z) in the cases of 36.3 (gyromonotron region) and 38 kG (gyro-BWO region), respectively. In the former case, the primary field profile extends toward the downstream. The same decomposition method outlined in the  Appendix [Eq. (A10)] can be applied to obtain the forward and backward field coefficients [f_fwd(z) and f_bwd(z)] at the moment when the oscillation begins. The decomposed two field amplitudes (|f_fwd(z)| and |f_bwd(z)|) with similar maximum amplitudes reveal that both of the forward and backward waves exhibit fairly good beam–wave synchronization. The forward wave then grows convectively toward the downstream, while the backward wave is generated as it counter-propagates with the electron beam. In addition, the phase of the overall field [ $∠$f(z)] remains nearly constant along the axial direction, suggesting the presence of an intrinsic “standing-wave pattern.” Combing the distributions of |f_fwd(z)| and |f_bwd(z)|, we can conclude that the observed standing-wave pattern is constructed by forward and backward waves with similar amplitudes and nearly equal phases. Under this circumstance, as shown in Fig. 4(a), the oscillation frequency is almost fixed at a “resonant frequency,” however, within a uniform structure without external feedback. In the case of gyro-BWO [Fig. 4(c)], it is observed that the backward wave experiences superior beam–wave synchronization and thus f_bwd(z) dominates the oscillation. Unlike the former case, the phase [ $∠$f(z)] now evolves toward the upstream during the generation and the propagation of the backward wave, indicating the traveling-wave nature of gyro-BWO. These observations align with the previous research findings.^14,28

To analyze how the forward and backward waves are co-generated and interfere with each other, Fig. 5 illustrates their electron transit-angle phases, i.e., $Θfwd=εfwd×L/vz0$ and $Θbwd=εbwd×L/vz0$⁠, where $εfwd/bwd$ is the detuning factor defined in Eq. (1) and $L/vz0$ can be regarded as the electron transit time across the waveguide. At the gyro-BWO region (37 ∼ 38 kG), $Θbwd$ is almost locked at an optimum value (∼1.05π) due to the automatic adjustment of the oscillation frequency, as shown in Fig. 5. This serves as the key mechanism for frequency-tunable gyro-BWOs.^14,45 At the gyromonotron region (35.5 ∼ 36.5 kG, specifically), both $Θfwd$ and $Θbwd$ vary with B₀. It is important noting that $Θfwd$ and $Θbwd$ are closely aligned with each other (⁠ $Θfwd−Θbwd≈0$⁠), both slightly deviating from $Θ0$ (⁠ $≡(ω−Ωc0)×L/vz0$⁠). This is associated with the fact of negligible Re[k_z] near the cutoff. That is to say, the difference between forward wave and backward wave becomes insignificant from the viewpoint of the electron. Due to this very near-cutoff operation, the group velocity of the wave (v_g) is largely reduced, indicating that the electrons propagate at a speed nearly two orders of magnitudes faster than the transportation of forward wave's energy (indicated by the black curve in Fig. 5). In such scenario, the newly injected electrons are afforded the opportunity to catch up the forward wave generated by the preceding electrons. The as-generated forward wave therefore can modulate the fresh electrons, gradually building up the feedback loop. This analysis confirms that the forward and backward waves generated near the cutoff exhibit very similar internal feedback mechanisms. In other words, the convectively growing forward wave now plays the same role as the backward wave, triggered by the absolute instability.

$Θfwd−Θbwd≈0$ further implies that the forward and backward waves are almost in-phase, corresponding to the constructive interference of the bi-directional propagating signals. The coherent superposition of these waves results in the formation of a standing-wave field pattern, as illustrated in Fig. 4(b). This phenomenon enables the uniform tube to function as a resonator, even without any external feedback. Consequently, this inherent resonant characteristic guarantees high interaction efficiency as observed. The described phase relationships become clearer as the tube's wall loss is reduced to an extremely low level (10⁻⁶ρ_Cu). As demonstrated by the dotted lines in Fig. 5, $Θfwd−Θbwd$ is perfectly locked at zero within the range of 34.5–36.5 kG, corresponding to a perfect on-resonance state. Despite the in-phase relationship and the mutual constructive interreference between the forward and backward waves, it is essential to address whether their corresponding electron bunching effects can effectively cooperate or compete. This issue will be examined in Sec. V.

The beam–wave energy transfer can be further examined by the change in Lorentz factors $Δγ$ of all electron ensembles. Figure 6(a) demonstrates $Δγ$ of 21 representative electrons calculated in particle-tracing simulation at near-cutoff operation (B₀ = 36 kG), while the similar plot at a relative far from cutoff region (B₀ = 38 kG) is shown in Fig. 6(b). In the former case [Fig. 6(a)], it is evident that the electron bunching modulated by the forward and backward waves cooperate with each other, resulting in the enhanced energy extraction from electrons to wave (most $Δγ$ becomes negative when z = 2 ∼ 3.5 cm). As for the far-cutoff gyro-BWOs [Fig. 6(b)], since only the backward wave is present, less energy change in electrons is observed.

The preceding discussions primarily revolve around TM-mode gyrotron oscillators. However, a question still lingers: Does the proposed intrinsic resonance apply to all types of operating modes? To explore this, Sec. VI shifts the focus toward TE-mode gyrotrons.

In order to compare distinctions and similarities between TE-mode and TM-mode gyrotrons, the TE₀₁ mode is chosen due to its degeneracy with the previously discussed TM₁₁ mode.³⁶ The same uniform waveguide with r_w = 0.2 cm and L = 3.5 cm with a constant B₀ throughout the tube are chosen. The optimum electron guiding center radius r_c = 0.48r_w is selected for efficiently interacting with the TE₀₁ mode.^40,44 The beam current used in the TE₀₁ nonlinear particle-tracing simulation¹³ is 1 A, while the other beam parameters hold the same (V_b = 70 kV and α = 1). The obtained nonlinear efficiencies are demonstrated in Fig. 7(a). Similar to the TM₁₁ case, a high beam efficiency of 30% is achieved by a 3.5 cm uniform tube without external feedback. Note that similar hill-shaped efficiency spectrum for the TE-mode gyromonotron has also been observed,²⁷ in which a semi-infinite regular tube with a upstream structure cutoff and a downstream magnetic-field cutoff was employed. This high similarity suggests that the two cutoffs adopted²⁷ may not be the primary reasons for feedback.

Based on the spectra of $ηfwd$ and $ηbwd$ in Fig. 7(a), we can deduce that the interaction in the TE-mode gyrotron under near-cutoff operation involves the simultaneous generation of both forward and backward waves. This statement aligns with the conclusion drawn in Sec. II regarding the TM₁₁-mode gyrotron. On the other hand, similar to the approach in Sec. III, two lossy-section tests are conducted on the TE₀₁-mode gyrotron. The impact of introducing a 1 cm upstream lossy section is illustrated in Fig. 7(b). At relatively low-B₀ region, the heavy wall loss at upstream terminate the forward and backward oscillations at the same time. This indicates the interconnected nature of these two waves, and their reliance on the internal feedback loop closely resemble the trend observed in Figs. 3(c) and 3(d). The results of the downstream lossy section are also quite similar to those in Fig. 3(a) and thus are not displayed here for conciseness.

Figure 7(c) shows the starting current I_st and oscillation frequency ω_st of the TE₀₁ mode, calculated by the method proposed in the literature.^47, I_st of the TE₀₁ mode is three times smaller than that of the TM₁₁ mode [Fig. 4(a)] owing to the larger optimum beam–wave coupling strength.^36,40 Notice that 1/I_st exhibits a striking resemblance to the trend of $ηfwd$ shown in Fig. 7(a), corresponding to the strong beam–wave interaction in the gyromonotron regime. A similar valley-shaped I_st spectrum has also been demonstrated before,^19,30 however, within the structures of nonuniformities. We can therefore conclude that the observed strong beam–wave interaction under near-cutoff operation of gyromonotrons should originate from the proposed “intrinsic resonance” rather than the traditional cavity resonance induced by external feedbacks. Figure 7(d) demonstrates the electron transit angles of the TE₀₁ mode at the gyromonotron region. Once again, $Θfwd$ and $Θbwd$ align with each other, with a minor deviation from $Θ0$⁠. The nearly in-phase condition promises the constructive interferences of the forward and backward waves, giving rise to the intrinsic resonance nature of the TE₀₁ mode when B₀ ∼ 36 kG.

As for the bunching mechanism at the near-cutoff, again, due to negligible k_z, the azimuthal bunching induced by $Δγ$ should dominate as seen from Eq. (2). In other words, E_⊥ of the TE₀₁ mode governs the energy-exchange process, which shows no difference when the wave's propagation direction is reverse. This implies that the dominated azimuthal bunching effects caused by the forward E_⊥ and the backward E_⊥ always cooperate with each other,¹ responsible for the high interaction efficiency observed at the near-cutoff region for TE modes.

As the last part of the discussion, we attempt to identify the type of instability dominating the forward oscillation in the near cutoff region [Figs. 1(d) and 7(a)]. The general mapping analyses³⁹ on the complex-k_z plane, also known as Briggs–Bers criterion (BBC),^48,49 for both the TM₁₁ and TE₀₁ modes are conducted. The result reveals that the current phenomena, i.e., the obtained strong forward wave, correspond to the convective instability rather than the absolute instability. While this finding provides an explanation for the origin of the “convectively growing” forward wave observed in Figs. 2(a) and 4(b), it appears to be in contradiction with the arguments of the “effective internal feedback loop” proposed in Secs. IV and VI. This apparent contradiction may arise from the differing frames of reference. From the waveguide frame's perspective, the oscillation is indeed dominated by the convective instability, manifesting in the growing forward wave. However, the condition with negligible group velocity at the cutoff promises that the forward wave would approach the electrons backwardly, building up the aforementioned internal feedback loop and the “effective absolute instability” for electrons. In addition, the BBC³⁹ for absolute instability is established based on the assumption of infinitely long interaction length, while the classification of the absolute/convective instabilities for a finite-length system is not well-defined.¹⁶ Rigorous definition of instability criteria is, however, beyond the scope of the present work. These questions should be carefully addressed in the follow-up studies. Noteworthily, the absolute instability near the zero group velocity points, similar to the condition in the present study, has recently been carefully re-examined using the BBC, for the conventional traveling-wave tube cases.^50,51

On the other hand, it is important to clarify similarities and differences between the findings presented in this paper and the predictions of classical theories linked to the backward-wave oscillator (BWO) and the traveling-wave tube (TWT). During the linear and start-oscillation stages, as illustrated in Figs. 4(b) and 4(c), the forward-wave component undergoes exponential growth as it propagates, eventually forming a slowly varying spatial plateaus. When operating within the nonlinear regime, if the tube length exceeds a specific threshold, the RF field may redistribute some of its power back to the beam in the steady state, as evident in Fig. 2(a)—a phenomenon akin to the well-known behavior in TWT.⁵² 

Owing to the imposed outgoing-wave boundary conditions, the backward-wave component, in both gyromonotron and gyro-BWO regimes, maintains a zero value at the downstream end (z = L) and grows backwardly to a substantial value at the upstream end (z = 0). These patterns can be observed in both the linear start-oscillation case [Fig. 4(c)] and the nonlinear steady state [Fig. 2(b)], perfectly aligning with predictions from the classical BWO theory of Johnson.⁵³ Additionally, the dip in the starting-current curve [Fig. 4(a) for the TM₁₁ mode and Fig. 7(c) for the TE₀₁ mode] can also be rationalized through the classical BWO theory. While the BBC theory for an infinitely long tube predicts a starting current of zero at the cutoff point, in a finite-length system, there must be a finite, non-zero starting current beyond which oscillations commence. Hence, a minimum starting current should occur near the cutoff for a finite-length system.

Furthermore, in the realm of a gyrotron traveling-wave tube (gyro-TWT), it typically operates below the grazing condition to achieve a steady output with high gain by inducing convective instability.^{1,13,38,39,42} However, absolute instability may still be triggered as the beam current surpasses a certain threshold.⁴³ Lau et al. conducted a comprehensive study on the oscillation mechanism in a gyro-TWT system.⁵⁴ The overdrive of the beam current is believed to induce self-excitation of the “backward waveguide mode” due to the internal feedback mechanism by extending the unstable spectrum to negative values of k_z. This statement aligns with BBC established in the infinitely long system, suggesting that only the backward wave may result from absolute instability, while the forward wave relies on convective instability consistent with our findings using BBC. However, the discoveries in this work reveal that forward oscillation can also occur based on “effective” absolute instability, manifesting in the intrinsic resonance constructed by strongly coupled forward and backward waves in the finite-length system. This clarifies the question raised by Lau et al. regarding the existence of absolute instability when both beam and waveguide modes propagate in the forward direction. Even when the current is not large enough to strongly couple the beam mode with the backward waveguide mode, a steady and strong forward wave oscillation can still be observed. This must be carefully avoided when optimizing gyro-TWT devices because the self-excited forward and backward oscillations might have the potential to significantly spoil the desired functionality.

In this work, we conduct a comprehensive investigation of the TM₁₁-mode and TE₀₁-mode gyrotrons, with a particular focus on a uniform tube without external structural feedbacks. The nonlinear particle-tracing simulation yields an interesting finding: the uniform tube can function as gyromonotron, in the absence of structural and magnetic-field nonuniformities, with a beam efficiency exceeding 30%. By decomposing the fields, we demonstrate that the forward and backward waves are co-generated at this region. A series of lossy-section tests are further employed to test the statement. The upstream wall loss is found to terminate both the backward and forward oscillation, despite the significant deviation of the heavy-loss location from the main steady-state forward field profile. Additionally, a linear analysis conducted during the start-oscillation stage reveals that the oscillation frequency is closely locked to the cutoff frequency of the mode. Operating near the cutoff opens up a possibility: the forward and backward waves generated by preceding electrons may both modulate the newly injected beam, and the internal feedback loop of the forward wave is thus established. From the electron's perspective, the forward and backward waves are nearly degenerate. These in-phase waves coherently form a standing-wave pattern in the steady state, effectively transforming the uniform tube into a resonant cavity, which explains the high interaction efficiency observed at the near-cutoff region.

The findings presented in this study reshape the previous understanding of the open-cavity-based gyromonotrons. Contrary to previous beliefs, it is now evident that the commonly used upstream cutoff end, taper reflection ends, or downstream magnetic-field cutoff only serves as a secondary enhancement rather than the primary driving force of oscillations. Despite the potential challenges in wave coupling that may arise due to near-cutoff operation, the insights provided here are believed to shed light on the underlying mechanism for existing gyrotron devices. The current trend in gyrotron design is shifting toward the development of reflective-type gyro-backward wave oscillators (R-gyro-BWOs),^20,55–57 which combine the high efficiency of the gyromonotron with the advantage of continuous frequency tunability found in gyro-BWOs. However, the underlying mechanism behind the operation of R-gyro-BWOs remains unclear. The interplay between the forward and backward waves discussed in the present work is believed to offer more clues into its working principle.

Last but not least, it is worth noting that while most gyrotron devices have predominantly utilized the TE mode,^1,2,7,8 there has been limited exploration^36,37,40,41 into the TM-mode gyrotrons due to the significantly higher start-oscillation current required, stemming from their weaker mode-coupling strength in a classical open-cavity-type gyrotron adopting weakly relativistic electron beams.⁵⁸ This might lead some to regard the comparative studies between TE and TM modes in the current study as purely theoretical endeavors. However, a recent study by Yang et al. has demonstrated the feasibility of a TM₁₂-mode gyrotron oscillator operating in the sub-terahertz band (∼220 GHz).⁵⁹ The importance of considering a TM mode has been emphasized, especially in the context of harmonic TE-mode gyrotrons using the axis-encircling electron beam, because the mode-coupling strength of a s + 1-harmonic TE mode and that of a s-harmonic TM mode are comparable. Since the competitiveness of TM modes has been confirmed by the multi-mode simulations using particle-in-cell simulator (PIC) solver of computer simulation technology (CST) Studio, the potential of TM-mode gyrotrons warrants further theoretical exploration. This article should help to partially elucidate the similarities and differences between TM-mode and TE-mode gyrotrons, particularly within the near-cutoff gyromonotron regions.

The authors would like to thank Professor Kwo-Ray Chu for valuable comments and suggestions. This work was supported by the National Science and Technology Council, Taiwan, under Contract Nos. NSTC 111-2112-M-194-009-MY3, NSTC 112-2112-M-194-006-MY3, and NSTC 110-2112-M-007-013-MY3.

The authors have no conflicts to disclose.

Tien-Fu Yang: Conceptualization (equal); Formal analysis (equal); Investigation (lead); Methodology (equal); Software (equal); Validation (supporting); Writing—original draft (lead); Writing—review & editing (equal). Hsin-Yu Yao: Conceptualization (equal); Formal analysis (equal); Methodology (equal); Project administration (equal); Software (equal); Supervision (equal); Validation (lead); Writing—review & editing (equal). Shih-Hung Chen: Conceptualization (supporting); Formal analysis (supporting); Validation (supporting); Writing—review & editing (supporting). Tsun-Hsu Chang: Conceptualization (equal); Funding acquisition (lead); Project administration (equal); Supervision (equal); Writing—review & editing (supporting).

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

Equations (A1)–(A8) can be solved together. Beginning from the beam entrance (⁠ $z=0$⁠), the first outgoing-wave wave boundary condition $f′ (z=0)=−ikzf(z=0)$ is specified for ensuring purely backward wave, where $f(z=0)$ is still an unknown factor. The coupled differential equations are then integrated axially from $z=0$ to $z=L$⁠. At the end of the system (⁠ $z=L$⁠), the second outgoing-wave boundary condition $f′ (z=L)=ikzf(z=L)$ is imposed for ensuring a purely forward wave. This complex equation can be solved by an iterative procedure for obtaining the purely real oscillation frequency $ω$ and initial field amplitude $f(z=0)$⁠. Based on these two parameters, the complex field amplitude $f(z)$ and all other electron's dynamic parameters are obtained. The iterative root searching processes are completed if the satisfaction of boundary conditions achieves the level of 10⁻¹⁰, guaranteeing negligible reflections from the boundaries. It is worth noting that a similar set of equations for TE modes can be found in Ref. 13, which were utilized to compute the results presented in Figs. 7(a) and 7(b).