We theoretically study specific non-stationary dynamics of gyrotrons generation in conditions of significant excess of the operating current over the starting value. In this case, gyrotrons with continuous electron beams unpredictably emit “giant” ultrashort radiation spikes, which can be interpreted as microwave rogue waves. Within the frame of the average approach and direct 3D particle-in-cell simulations, we demonstrate that the occurrence of gyrotron rogue waves can be regularized by periodic injection of an electron beam. More regular formation of rogue waves would enhance interest in the implementation of such systems for possible applications.

Currently, the most powerful radiation sources in the millimeter and sub-millimeter ranges are gyrotrons,1–4 whose operation is based on stimulated excitation of a near-cutoff waveguide mode by a beam of electrons gyrating in a homogeneous magnetic field. Typically, gyrotrons operate in a steady-state regime. However, since gyrotrons are distributed nonlinear systems, these devices can exhibit complex dynamic behavior, including self-modulation and chaotic generation.5–9 As shown in Refs. 10 and 11, when the operating current significantly exceeds the starting value, gyrotrons can sporadically emit “giant” radiation spikes with intensities a factor of 100–150 greater than the average power and a factor of 6–9 exceeding the power of the driving stationary electron beams. Together with the statistical features such as a long-tail probability distribution, this allows the interpretation of generated spikes as microwave rogue waves. The mechanism of rogue waves formation in gyrotrons is related to the simultaneous cyclotron interaction of a rotating electron beam with forward and backward waves near the waveguide cutoff frequency as well as with the longitudinal deceleration of electrons.

Note that the generation of short pulses with high peak power in the millimeter-wave range is of interest for a number of applications. In this regard, certain efforts are being made to develop typically pulse-periodic radiation sources based on electron–wave interaction. For example, periodic generation of ultrashort pulses can be obtained based on self-mode-locking in electron generators with delayed external reflections,12,13 or based on the effect of passive mode-locking when using a saturable cyclotron resonant absorber installed in a feedback circuit.14 

Compared to the described methods, gyrotron rogue waves may have a higher peak power especially at high frequencies. For example, according to the simulations,15 in the range of 0.5 THz, based on this effect, it is possible to obtain radiation pulses with a duration of several tens of picoseconds and a peak power of up to 5 MW. However, the obvious disadvantage of rogue waves for applications is their extremely rare and unpredictable occurrence. This paper proposes a method for regularizing formation of rogue waves in gyrotrons due to periodic injection of a driving electron beam. The duration of a single current pulse is chosen so that it initiates only one powerful rogue wave. At the same time, the injection period of such pulses must be large enough so that the radiation generated in the system has time to leave the region of electron–wave interaction and does not affect the formation of the next powerful rogue wave. Theoretical analysis is carried out within the framework of the average approach as well as based on direct 3D PIC (particle-in-cell) simulations.

Let us consider the gyrotron model in which an interaction space in the form of a quasi-regular cylindrical waveguide with radius R is powered by a tubular axisymmetric rotating electron beam [Fig. 1(a)]. The interaction space with the length of the regular part (interaction length) l is bounded by a cutoff neck on the cathode edge z = 0 and has a smooth widening on the collector edge l < z < zout.

FIG. 1.

The gyrotron model under consideration.

FIG. 1.

The gyrotron model under consideration.

Close modal
Assuming that the electron beam excites only one TEmn near-cutoff mode at the fundamental cyclotron resonance, the electron–wave interaction can be described by the following system of equations, which includes the parabolic equation for the dimensionless wave amplitude a, the averaged weakly relativistic motion equations for the transverse p x , y = m e V x , y γ, and longitudinal p | | = m e V z γ electron momenta,10,
i 2 a Z 2 + a τ = i f ( τ ) I 0 2 π 0 2 π p ̂ + p ̂ | | d θ 0 , [ Z + g 0 2 4 τ ] p ̂ + + i p ̂ + p | | ( Δ 1 + | p ̂ + | 2 + p ̂ | | 2 1 g 0 2 ) = i a p ̂ | | + β 0 2 2 a Z , [ Z + g 0 2 4 τ ] p ̂ | | = g 0 2 β 0 2 2 Re ( a Z p ̂ + * p | | ) .
(1)
Here, we use the following variables and parameters (index “0” refers to the initial values):
τ = ω c β 0 4 t 8 β | | 0 2 , Z = β 0 2 ω c z 2 β | | 0 c , a = e A J m 1 ( ν m n R 0 / R ) m e c ω c β 0 3 γ 0 ,      p ̂ + = ( p x + i p y ) e i ω c t + i ( m 1 ) φ m e V 0 γ 0 ,
p ̂ | | = p | | / m e V z 0 γ 0, g 0 = β 0 / β | | 0 is the pitch factor, β 0 = V 0 / c, β | | 0 = V | | 0 / c, and I 0 = 16 ( e I b / m e c 3 ) β 0 6 β | | 0 G is the normalized excitation factor, I b is the electron current, G = J m 1 2 ( ν m n R 0 / R ) ( ν m n 2 m 2 ) 1 J m 2 ( ν m n ) is the form-factor, R 0 is the beam injection radius, ν m n is the nth root of the equation J m ( ν ) = 0, Δ = 2 ( ω c ω H 0 ) / ω c β 0 2 is the initial mismatch between the operating mode cutoff frequency ω c and the unperturbed electron gyrofrequency ω H 0 = e H / m c γ 0, and γ 0 = ( 1 β 0 2 β | | 0 2 ) 1 / 2 is the relativistic mass-factor. Function f ( τ ) characterizes the electron emission, which in the model under consideration (unlike10) can be non-stationary.

The boundary conditions for the motion equations in (1) are set in the input cross section Z = 0 and correspond to the uniform distribution of electrons over the phases of cyclotron rotation and the absence of an initial spread in velocities and energies: p ̂ | | ( Z = 0 ) = 1; p ̂ + ( Z = 0 ) = e i θ 0 , θ 0 [ 0 , 2 π ). The boundary condition for the wave in the input cutoff narrowing is a ( Z = 0 ) = 0. At the system output Z = Z out, we apply the well-known reflectionless boundary condition,16 which corresponds to perfect matching with the output waveguide.

Further, we will consider the dynamics of a gyrotron with the normalized interaction length L = β 0 2 ω c l / 2 β | | 0 c = 15, the electron energy of 20 keV, and the pitch factor g0 = 1.3. We assume that a stationary injection of the electron beam takes place, i.e., f ( τ ) = 1. For chosen configuration, the starting value of the excitation factor is I0 ≈ 0.004, the transition to periodic modulation occurs at I0 ≈ 0.016, and stochastic regimes of generation are realized at I0 ≥ 0.1. When the excitation factor increases to I0 = 3.0, isolated radiation spikes (rogue waves) with a peak power that significantly exceeds the average radiation power unpredictably appear in the output signal. It is convenient to characterize the radiated spikes by the conversion coefficient K, which is the ratio of the microwave radiation power P to the electron beam power Pbeam,
K = P P beam = g 0 2 1 + g 0 2 2 I 0 Im ( a a * Z ) ,
(2)
where “*” denotes the complex conjugation. Note that the negative sign of K corresponds to electromagnetic radiation propagating in the -z direction. For the case I0 = 3.0, Δ = −0.7 shown in Fig. 2, and the maximum value of the conversion coefficient at the gyrotron output reaches K ∼ 6.
FIG. 2.

Simulations of rogue wave generation for the case of stationary beam injection f ( τ ) = 1. Time dependence of the conversion coefficient K at the output (Z = 15) and at the up-stream (Z = 2) cross sections of the interaction space (I0  = 3.0, Δ = −0.7).

FIG. 2.

Simulations of rogue wave generation for the case of stationary beam injection f ( τ ) = 1. Time dependence of the conversion coefficient K at the output (Z = 15) and at the up-stream (Z = 2) cross sections of the interaction space (I0  = 3.0, Δ = −0.7).

Close modal

Figure 3 shows in detail the mechanism of gyrotron rogue waves formation for one such spike emitted at time τ ≈ 360. At the first stage ( τ 340 350), the electron beam excites a counter-propagating (backward) wave. After reflection from the cutoff neck, this wave is absorbed by newly arrived electrons, resulting in significant increasing their transverse energy (see axial distribution of the electron transverse momenta in Fig. 2, at τ 356). At the second stage, this electron fraction initiates the excitation of a co-propagating (forward) electromagnetic pulse. Since the pulse has a group velocity exceeding the translational velocity of the beam, it slips relative to the electrons, accumulates their energy, and is effectively amplified (top row at τ 358). This process is accompanied by a reduction in the pulse duration with a simultaneous increase in the steepness of its fronts and the appearance of a strong transverse magnetic field H a / Z. Thus, at the last stage, the energy of the translational motion of electrons is also converted into radiation energy, which is demonstrated by a sharp decrease in their longitudinal momenta (bottom row at τ 358).

FIG. 3.

Axial distribution of the conversion coefficient K (top row), instantaneous (dots), and averaged (solid line) values of the transverse (middle row) and longitudinal (bottom row) momenta of electrons before the emission of a giant radiation spike (rogue wave).

FIG. 3.

Axial distribution of the conversion coefficient K (top row), instantaneous (dots), and averaged (solid line) values of the transverse (middle row) and longitudinal (bottom row) momenta of electrons before the emission of a giant radiation spike (rogue wave).

Close modal
After the rogue wave leaves the interaction space, the excitation of the seed backward wave can again be initiated in the system. This process is quasi-regular with a period determined by the group velocities of the backward and forward waves, which can be estimated based on the analysis of the dispersion diagram of the electron-wave synchronism. Within the framework of the model used, such a dispersion diagram can be constructed in the absence of electron-wave coupling ( I 0 = 0) using the representation a , p ̂ + exp ( i Ω τ i Γ Ζ ), where the normalized shift Ω of the generation frequency ω from the cutoff one and the normalized longitudinal wave number Γ are given by the following relations:
Ω = ω ω c ω c 8 β | | 0 2 β 0 4 , Γ = 2 β | | 0 c β 0 2 ω c h .
Thus, from (1), we obtain the following expressions for the dispersive characteristics of the electromagnetic wave and the electron beam,
Ω = Γ 2 , Ω = 4 g 0 2 ( Γ Δ ) .
(3)
Figure 4 shows the typical relative position of these characteristics under the conditions of the formation of rogue waves, when the double resonance of electrons with the waveguide mode takes place. The points 1 and 2 correspond to synchronisms with backward and forward waves, respectively, and are characterized by the following values of the longitudinal wave numbers:
Γ 1 , 2 = 2 g 0 2 2 g 0 2 1 g 0 2 Δ .
(4)
FIG. 4.

Dispersion diagram of the waveguide mode (solid line) and the electron beam (dots) in the regime of rogue wave generation. The points 1 and 2 correspond to synchronisms with backward and forward waves.

FIG. 4.

Dispersion diagram of the waveguide mode (solid line) and the electron beam (dots) in the regime of rogue wave generation. The points 1 and 2 correspond to synchronisms with backward and forward waves.

Close modal
Taking into account that the normalized group velocity can be found as V ̂ g r = d Ω / d Γ = 2 Γ, the period T of radiation travel along the interaction space with length L can be estimated as
T = L V ̂ g r ( 1 ) + L V ̂ g r ( 2 ) = L 2 Δ 1 g 0 2 Δ .
(5)
Substituting the values L = 15, g 0 = 1.3, and Δ = −0.7 gives the period T ≈ 16, which, however, is several times less than the period of formation of seed pulses on backward waves T ∼ 30–40 [see Fig. 2(b)]. A significant increase (1.5 times) in the transverse moment of electrons upon absorption of the backward wave can be seen in Fig. 3 (τ = 343 ns). With further interaction, the magnitude of the transverse moment additionally increases (τ = 356 ns) due to the absorption of the wave emitted by electrons previously injected into the interaction space. Thus, the resulting pitch factor of the electrons can be estimated as ∼3 to 4, which should be used in (5) instead of the initial value g0 = 1.3.

Using an increased pitch factor value in (5) gives a T value close to the simulation results.

Thus, one can expect that an electron bunch (current pulse) with a duration of more than T can form a powerful output spike—a rogue wave. However, it should be taken into account that the electron beam needs time to fill the interaction space. This time T f can be estimated using the value of the normalized translational velocity of electrons V ̂ | | 0 = 4 / g 0 2,
T f = L V ̂ | | 0 = L g 0 2 4 6 .
(6)
Note that the value of T f may be larger taking into account the above-mentioned increase in the pitch factor. As a result, we can conclude that the normalized duration of the current pulses should not be less than T e = T + T f 50.
Thus, it can be assumed that when feeding the gyrotron with a periodic sequence of such current pulses, one can obtain periodic or at least quasiperiodic generation of rogue waves. The repetition period of current pulses can be made taking into account the fact that the electromagnetic fields excited by each current pulse should not affect the formation of the subsequent rogue wave. To do this, it is necessary that during the period of repetition of current pulses, the electromagnetic fields generated in the system have time to leave the interaction space. The characteristic decay time t0 of oscillations in the cavity can be estimated as
t 0 Q ω ,
(7)
where ω = 2πf and f is the operating frequency. The minimum value of cavity quality factor Q is given by following relation:17,18
Q min 4 π ( l λ ) 2 ,
(8)
where λ is the radiation wavelength. Accordingly, the minimum normalized value of the decay time T 0 can be estimated as
T 0 4 L 2 π 300 .
(9)

Let us further simulate the formation of rogue waves in the considered gyrotron for the case of periodic injection of the electron beam, defining the function f(τ) in such a way so that trapezoidal current pulses with normalized duration of the leading and trailing fronts τe = 5 and the duration of the regular part τ1 are injected with a period τ2. The duration of the pulse fronts was chosen to be approximately equal to the transit time of electrons through the interaction space Tf. Figure 5 shows the dynamics of the gyrotron depending on the current pulse duration τ1 for a given injection period τ2 = 1000, when the condition τ2T0 is fulfilled.

FIG. 5.

Simulations of rogue wave generation for the case of periodical beam injection. Time dependence of the conversion coefficient K at the gyrotron output for different current pulse durations τ1 and fixed injection period τ2 = 1000.

FIG. 5.

Simulations of rogue wave generation for the case of periodical beam injection. Time dependence of the conversion coefficient K at the gyrotron output for different current pulse durations τ1 and fixed injection period τ2 = 1000.

Close modal

It is seen that with a duration of τ1 = 50, radiation spikes of very low power are generated at the gyrotron output. As the duration increases to τ1 = 70, the peak power of the spikes also increases to values K ∼ 6, but they appear quite rarely. At τ1 = 100, the frequency of occurrence of powerful radiation pulses increases significantly. However, with a further increase in duration to τ1 = 150, a situation arises when two rogue waves are formed during one current pulse. With a further increase in τ1, the generation of double radiation spikes becomes more frequent. Thus, we can conclude that the optimal duration of the current pulse is τ1 = 100 ( 2 T e).

Figure 6 shows formation of the gyrotron rogue waves depending on the injection period of current pulses τ2 at the optimal value of the pulse duration τ1 = 100. It is seen that the result depends on the ratio of the injection period τ2 and the decay time T0 is given by the relation (9). At τ2 =100 < T0, the dynamics of the system are practically no different from the case of continuous beam injection. At the same time, as the injection period increases to τ2 = 300 ∼ T0, the dynamics of the system becomes more ordered. With a further increase in the injection period, a transition occurs to a situation where almost every current pulse generates a high-power rogue wave.

FIG. 6.

Simulations of rogue wave generation for the case of periodical beam injection. Time dependence of the conversion coefficient K at the gyrotron output for different injection periods τ2 and fixed current pulse durations τ1 = 100.

FIG. 6.

Simulations of rogue wave generation for the case of periodical beam injection. Time dependence of the conversion coefficient K at the gyrotron output for different injection periods τ2 and fixed current pulse durations τ1 = 100.

Close modal

The results obtained were confirmed based on 3D PIC (particle-in-cell) simulations using the KARAT code,19 which allows taking into account a number of real physical factors. We consider a Ka-band gyrotron with TE11 operating mode excited by a 20 keV/2.1 A rotating electron beam with a pitch factor of 1.3. The cavity profile and the instantaneous position of the electrons are shown in Fig. 7. The length of the regular section of the cavity is 15 cm. The simulations take into account the initial spread of electron transverse velocities at the level of 20%, typical for gyrotrons. The electron beam in the cavity is guided by a uniform magnetic field, and after interaction, the electrons are deposited on the cavity wall due to a decrease in the magnetic field magnitude. To simulate matching with the output waveguide, an absorbing layer with variable conductivity is placed at the collector end of the interaction space, the reflection coefficient of which does not exceed 1% of the incident radiation power. The number of grid nodes is about 2 × 105, and the number of macroparticles in the interaction space is approximately 2 × 104.

FIG. 7.

Axial (a) and transverse (b) structure of a gyrotron interaction space used in PIC simulations (1—cutoff neck, 2—regular section of the cavity, 3—electron beam, 4—output waveguide, and 5—absorber).

FIG. 7.

Axial (a) and transverse (b) structure of a gyrotron interaction space used in PIC simulations (1—cutoff neck, 2—regular section of the cavity, 3—electron beam, 4—output waveguide, and 5—absorber).

Close modal

The selected physical parameters correspond to the normalized length L ≈ 15 and the excitation factor I0 ≈ 3.0 in the averaged model (1). A high value of the excitation factor at an acceptable operating current is achieved due to interaction at the lowest TE waveguide mode, which has a strong coupling with the electron beam. Similar to Sec. II, in the case of continuous injection of the electron beam, the output radiation at the output cross section z = 20 cm is a random sequence of electromagnetic spikes with a maximum conversion coefficient K ∼ 3 (Fig. 8), which is somewhat reduced due to the influence of velocity spread and the space charge. Note that, as in Sec. II, the conversion coefficient here is defined as the ratio of the power of the electromagnetic pulse to the average power of the electron beam. The radiation power is calculated as an integral of the Poynting vector over the cross section; thus, the sign of K is determined by the sign of this vector, which is positive for a pulse propagating in the +z direction and negative when propagating in the −z direction toward the electron beam.

FIG. 8.

Results of 3D PIC simulations for the case of stationary beam injection. Time dependence of the conversion coefficient at the output cross section z = 20 cm and at the up-stream cross sections z = 2 cm of the interaction space.

FIG. 8.

Results of 3D PIC simulations for the case of stationary beam injection. Time dependence of the conversion coefficient at the output cross section z = 20 cm and at the up-stream cross sections z = 2 cm of the interaction space.

Close modal

At the up-stream cross sections z = 2 cm of the interaction space, a quasi-regular generation of a sequence of pulses on the backward wave is observed, the repetition period of which is about 10 ns. Thus, the optimal duration of the current pulse for the formation of a single rogue wave cannot be lower than the specified value, and in fact must exceed it several times.

The characteristic decay time of the electromagnetic field by a factor of e, calculated based on the minimum diffraction quality factor Q ≈ 4000, is t0 ≈ 20 ns. This gives a lower estimate for the repetition period of current pulses in the case of periodical beam injection. However, due to significant reflections of the quasi-cutoff wave from the output cone of the cavity, the radiation can be retained longer in the interaction space. Therefore, in order to eliminate the mutual influence of radiation fields on the formation of rogue waves by successive current pulses, the injection period should be significantly increased.

Figure 9 shows the dynamics of the considered gyrotron for the case of periodical beam injection with fixed period of 200 ns and different pulse duration t1. The current pulses have trapezoidal shapes with leading and trailing fronts of 2 ns. Note that, in general, the results obtained are very close to the results of simulations of averaged equations (1). Namely, there is an increase in peak power and an increase in the regularity of the formation of rogue waves with increasing duration of current pulses t1. At t1 = 20 ns, output radiation pulses of very low power are formed. As the current pulse duration increases to t1 = 25 ns, radiation spikes with the conversion coefficient K ∼ 1 appear. The maximum conversion coefficient K ∼ 3 is reached at t1 = 30 ns, and in this case, the corresponding rogue waves occur quite often. At t1 = 40 ns, a situation arises when two high-power electromagnetic spikes are generated by one current pulse. With a further increase in t1, the generation of double spikes becomes more frequent. Thus, we can conclude that the optimal duration of the current pulse is t1 ∼ 30 ns. In normalized variables, this corresponds to values of τ1 ∼ 60, which is slightly less than the optimal value obtained based on model (1).

FIG. 9.

Results of 3D PIC simulations for the case of periodical beam injection. Time dependence of the conversion coefficient K at the gyrotron output for different current pulse durations t1 and fixed injection period 200 ns.

FIG. 9.

Results of 3D PIC simulations for the case of periodical beam injection. Time dependence of the conversion coefficient K at the gyrotron output for different current pulse durations t1 and fixed injection period 200 ns.

Close modal

It is quite interesting to compare the spectra of the output radiation of the gyrotron in the case of stationary and periodic beam injection. In the first case, we have an almost continuous spectrum of noise-like radiation in the range of 35–37 GHz [Fig. 10(a)]. At the same time, in the second case, the output spectrum is a set of lines with a noise pedestal [Fig. 10(b)].

FIG. 10.

Results of 3D PIC simulations. Output radiation spectra for stationary injection (a) and for periodic injection of a beam with a period of 200 ns and a pulse duration of 30 ns.

FIG. 10.

Results of 3D PIC simulations. Output radiation spectra for stationary injection (a) and for periodic injection of a beam with a period of 200 ns and a pulse duration of 30 ns.

Close modal

Table I allows everyone to compare the results obtained within the two approaches used, namely, the numerical solution of the averaged equations and direct 3D PIC simulations. The normalized value of the current pulses' duration is converted into a dimensional value taking into account the physical parameters used. It can be seen that in PIC simulations, in comparison with the simplified averaged model, lower conversion coefficients are obtained and, in addition, the optimal duration of electron pulses is somewhat shorter. Note that, in addition to the influence of the velocity spread, one of the factors reducing the conversion coefficient in the simulation may be the space charge, which increases in the region of rogue wave formation due to the strong deceleration of electrons. The formation of regions with high space charge is observed in numerical simulations.

TABLE I.

Comparison of the main results obtained by solving the averaged equations and based on 3D PIC simulations.

Averaged approach PIC simulations
Maximum conversion factor in the case of stationary beam injection  ∼6  ∼3 
Optimal duration of electron pulses  ∼50 ns  ∼30 ns 
Maximum conversion factor the case of periodical beam injection  ∼5  ∼3 
Averaged approach PIC simulations
Maximum conversion factor in the case of stationary beam injection  ∼6  ∼3 
Optimal duration of electron pulses  ∼50 ns  ∼30 ns 
Maximum conversion factor the case of periodical beam injection  ∼5  ∼3 

Thus, within the framework of two independent approaches, we demonstrated the possibility of quasi-regular formation of rogue waves in gyrotrons due to periodic injection of electron beams. We believe that the developed approach opens up new possibilities for creating powerful sources of pulsed millimeter radiation. A possible problem in the experimental implementation of the considered regime of quasi-regular generation of rogue-waves may be the formation of electron current pulses with the required duration and repetition rate, since it is known that the typical duration of the accelerating voltage pulse rise in gyrotrons is several microseconds.20,21 At the same time, we believe that this problem is purely technical. In particular, in Ref. 22, a generator of rectangular voltage pulses with a rise time of 4 ns, a total pulse duration of 100 ns or more, an amplitude of up to 9 kV, operating with a repetition rate of 10 Hz, is described. Recent work23 presents a voltage pulse generator with an amplitude of up to 20 kV, a rise time of 8 ns, and a total pulse duration adjustable from 25 ns to microseconds, operating at a repetition rate of 10 kHz. On this basis, sources of current pulses can be implemented.

In particular, as a further development, it is of interest to study rogue in gyrotrons with high-current (kiloampere) electron beams and sub-gigawatt output power.24 Note that in the latter case, it is easier to ensure the required excess of the operating current over the starting value for the transition to the regime of rogue wave formation with significantly high radiation peak power.

The work was performed at the Scientific and Educational Mathematical Center (Agreement with the Ministry of Science and Higher Education of the Russian Federation No. 075-02-2020-1632).

The authors have no conflicts to disclose.

R. M. Rozental: Investigation (lead); Methodology (equal); Writing – original draft (lead). N. S. Ginzburg: Supervision (lead). S. R. Rozental: Data curation (equal). A. S. Sergeev: Software (lead). I. V. Zotova: Methodology (equal); Writing – review & editing (lead).

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

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