# Gyroton with the Corrugated Resonator

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## Abstract

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_{11}wave, then it gives up longitudinal energy to the same wave with more than 78% efficiency, and an output power up to 30 MW. The developed mathematical model of the interaction of the relativistic electron beam with an irregular circular waveguide and resonator fields presented in this article can be used to calculate and optimize the processes occurring in various microwave electronic devices, such as gyrotrons, gyrotons, TWT, Gyro-TWT, and BWT.

## 1. Introduction

_{11}wave which is strongly coupled with the TE

_{11}wave in the corrugated waveguide. The waveguide corrugation leads to the slowdown of the phase velocity along the waveguide. This makes it possible to substantially decrease the required magnetostatic field value to achieve the synchronism between the electron beam and the traveling slow wave.

_{11}wave phase velocity is less than the longitudinal velocity of electrons, and the following equation is satisfied

_{11}wave phase velocity, ${\mathsf{\Omega}}_{\gamma}$—relativistic cyclotron frequency. Electrodynamic calculation of phase velocity for one corrugation crest for TM

_{11}wave yields the following result in the considered case ${\beta}_{w}=\frac{{v}_{w}}{c}=\frac{\mathsf{\Delta}\overline{z}}{\mathsf{\Delta}{\varphi}_{w},}=0.718$ and the longitudinal electrons velocity is ${\beta}_{z}=\frac{{v}_{z}}{c}=0.82$. The traveling wave gyroton is one of the possible realizations of the maser creation idea based on the anomalous Doppler effect with revolving fields.

_{11}wave phase velocity (${\beta}_{w}=\frac{{v}_{w}}{c}=1.7$) is greater than the longitudinal electrons velocity. There is a coupled resonator-chain in such gyroton, where the TM

_{11}wave resonates, and the coupling is achieved through the TE

_{11}wave. Therefore, in this case we do not deal with the anomalous Doppler effect. Synchronous regime here is determined by the following condition:

_{11}waves are present, but the wave traveling together with the beam performs the dominant role. In the output waveguide only one TE

_{11}wave propagates and other waves are evanescent. However, the TM

_{11}wave has a primary role in the resonator.

_{11}rotating with the frequency ω in the presence of longitudinal magnetostatic field H

_{z0}. Rectilinear electron beam in the waveguide center moves from the z-axis under the action of the TM

_{11}wave magnetic field H

_{t}and it gains transverse velocity V

_{t}. Then electrons start to revolve along the Larmor orbit and fall into the TM

_{11}wave phase. As a result, E

_{z}field of the wave extracts energy from electrons.

## 2. Mathematical Model

- -
- the electromagnetic waves excitation equations in the axisymmetric longitudinally irregular waveguide by a relativistic electron beam;
- -
- the electron beam motion equations in the excited electromagnetic fields.

_{ni}are the Bessel function roots (J

_{n}(ν

_{ni}) = 0), and μ

_{ni}are the Bessel function derivative roots (J’

_{n}(μ

_{ni}) = 0).

**E**,

**H**,

**J**are defined via the calculated ones as

**E**and

**H**wave modes with the identical azimuthal indices n are coupled that is caused by the azimuthal waveguide symmetry.

_{vc}—normalized radius of the electrons Larmor orbit center, $q=\frac{{v}_{\perp}}{{v}_{\parallel}}$—ratio of the electrons transverse velocity to the longitudinal velocity.

## 3. Algorithm and Calculations Results

_{0}= 381 kV, beam current I

_{0}= 8.7 A, resonator length l = 2πL/λ

_{0}= 25, corrugation internal radius g

_{1}= 2πr

_{1}/λ

_{0}= 2.902, corrugation amplitude Δg = 2πΔr/λ

_{0}= 2.05, the output ridge radius g

_{3}= 1.902, the corrugation edges number n = 14, the magnetostatic field value F = H/H

_{s}= 1.113.

_{11}wave propagates in the output waveguide. The corrugation cavities form five coupled double-humped resonators operating mostly on TM

_{11}wave. In total, 16 wave modes were considered in the calculations: eight waves TM

_{11}–TM

_{18}and eight waves TE

_{11}–TE

_{18}. The obtained results were verified by increasing the calculations accuracy: the waveguide base functions number was varied from 16 to 64 and the integration steps number—from 2000 to 8000. The accuracy change did not exceed 1% in this case.

_{11}wave. The electronic efficiency of this gyroton reaches 78%. The resonator profile is corrugated, and it is connected at the left side with the drift tube that is below-cutoff for all the wave modes (at the operating frequency); the rectilinear electron beam is injected through this tube center. To the right, the corrugation ends with the internal contraction, which increases the quality of this resonator.

_{11}falling to the right is presented in Figure 4. Figure 4 shows that the resonator cold Q-factor is Q

_{n}= 5250. Therefore, the field in this construction of gyroton can be considered as fixed with a high degree of accuracy.

_{11}wave high-frequency magnetic field. The electrons gyration radius and the rotation leading center radius increase simultaneously in this case (curves 4 and 6), this means that the electrons orbits only touch upon the resonator axis. Consequently, beginning with z > 3, the electron beam begins to give up its longitudinal energy to the longitudinal electrical component of the TM

_{11}wave (curve 2). The efficiency increases almost monotonically with the growth of z (curve 1), and the electron’s rotation center radial coordinate gradually moves away from the device axis (curve 4). So, device high efficiency is achieved only due to use of the electrons longitudinal energy.

## 4. Discussion and Conclusions

_{11}wave, then it is possible to substantially lower the resonator quality and it is, respectively, essential for the increasing of the gyrotons output power. The mathematical model of the relativistic electron beam interaction processes with the irregular circular waveguides and resonators fields presented in this paper can be used for the calculation and optimization of the processes occurring in various microwave electronic devices, such as gyrotrons, gyrotons, BWOs, and TWTs.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Schematic illustration of the gyroton waveguide, 1—the electron trajectory; 2—the waveguide.

**Figure 3.**Fields distributions of the modes in the resonator. Curve 1 corresponds to the wave amplitude for TM

_{11}mode; 2—the wave amplitude for TE

_{11}mode; 3—the dependence of efficiency on longitudinal coordinate z; 4—the profile of the resonator b(z)/λ

_{0}.

**Figure 5.**Integral characteristics of the gyroton: curve 1 corresponds to the dependence of efficiency on longitudinal coordinate z; 2—the longitudinal electrons velocity ${\beta}_{z}={v}_{z}/c$; 3—the resonator profile $g=b/{\lambda}_{0}$; 4—the electrons rotation center radius; 5—the electrons transverse speed ${\beta}_{t}={v}_{t}/c$; 6—the electrons gyration radius ${r}_{e}/{\lambda}_{0}$.

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**MDPI and ACS Style**

Kolosov, S.; Kurayev, A.; Rak, A.; Kurkin, S.; Badarin, A.; Hramov, A.
Gyroton with the Corrugated Resonator. *Plasma* **2019**, *2*, 1-13.
https://doi.org/10.3390/plasma2010001

**AMA Style**

Kolosov S, Kurayev A, Rak A, Kurkin S, Badarin A, Hramov A.
Gyroton with the Corrugated Resonator. *Plasma*. 2019; 2(1):1-13.
https://doi.org/10.3390/plasma2010001

**Chicago/Turabian Style**

Kolosov, Stanislav, Alexander Kurayev, Alexey Rak, Semen Kurkin, Artem Badarin, and Alexander Hramov.
2019. "Gyroton with the Corrugated Resonator" *Plasma* 2, no. 1: 1-13.
https://doi.org/10.3390/plasma2010001