About consideration of electron transverse motions in resistive wall amplifier

Resistive wall amplifier is a vacuum microwave device, which has not been spread, but attracts attention again. Main features – no need to slow-wave structure and almost complete lack of feedback between output and input. The amplification occurs due to the phase shift between the electron beam and field variables that arise due to the presence of absorbing walls.
The classic version of this device has high values of gain, however, if you replace the usual coating of a dielectric for metamaterial, gain increases substantially.
The linear theory of resistive device with an infinitely wide electron beam and a magnetic field, that coaxially to the beam, are considered. The transverse motion of the electrons are considered. The cases, when the beam moves through the environment with $\varepsilon>0$ and $\varepsilon<0$ are considered.
This paper is original, because the first analytical account of the transverse motion of the electrons for a resistive amplifier, beam modulation and the removal of the signal using segments of transmission lines are produced.
It is shown that the influence of transverse motion of electrons and co-current magnetic field leads to the dispersion equation of the sixth order and the emergence of cyclotron waves. It is considered an approximation that reduces the order of the dispersion equation to the fourth and showes a good agreement with the general case. The gain calculations demonstrated growing second wave (cyclotron wave). That becomes comparable with the growing wave of space charge. On large lengths it leads to higher gain.
In the case of $\varepsilon<0$ the imaginary component of the dispersion equation roots increases and suffers gap. The calculation shows higher gain on the smaller length of the resistive section than in the case of $\varepsilon>0$.