**Influence of the inner conductor roughness on power and efficiency limitation for coaxial gyrotron**

Coaxial cavity gyrotrons are the promising sources of microwave radiation for fusion devices. At present, theoretical investigations are in progress with the aim to develop gyrotrons of new generation.

An urgent purpose is to investigate the effect of wall roughness on the scattering and the absorption of microwave radiation in coaxial gyrotron cavity. For this purpose, the method of integral equations of macroscopic electrodynamics is used.

The power required for plasma heating in electron-cyclotron heating systems of fusion machines with magnetic confinement must exceed 2 MW with the pulse duration of about 10 min. However, in spite of considerable efforts, the last experiments for coaxial gyrotrons are characterized by much shorter pulses (∼ 10

^{–3}s). One of the reasons is the cavity wall heating to several hundred degrees, which occurs for a time of about 10^{–3}s and leads to oxidation and deformation of the wall surface. As a result, the wall conductivity decreases, and the cavity surface becomes rough. This can enhance wave absorption and scattering induced by the cavity walls, and thereby initiates failure of oscillations.
The integral equations of macroscopic electrodynamics have been applied to the problem of wave scattering by a rough dielectric surface.

The problem of electromagnetic and acoustic waves scattering by a rough surface is classical. To date, more than twenty approaches to this problem are available, and further research in this field is actively pursued. This is because the exact solving of the general three-dimensional problem of wave scattering by a rough surface is associated with almost insuperable computational obstacles, despite the high level of the present-day computer technologies.

Most of the approximate methods are based on the assumption that both the roughness height and the slope of tangent to rough surface are small. While the first condition is satisfied in many cases of practical interest, the second condition is often violated. In particular, such is the case for microwave cavities.

To analyze the processes of wave scattering and absorption by a rough dielectric surface, it is convenient to use the integral equation method introduced by M.A. Khizhnyak. It does not contain integrals over the rough surface. This makes it possible to construct an effective iterative solving scheme with much weaker suppositions than those discussed above.

For small irregularities, an iterative scheme has been realized.

At each iterative step, the problem has been reduced to a system of Wiener-Hopf equations. These equations are related to each other by free terms and can be studied analytically.

The correction term, which relates to the scattered radiation induced by the rough surface, has been obtained in analytical form and analyzed qualitatively.

Equations remain valid in the case of no incident wave. In this case they describe the well-known surface waves for a half-space with a negative permittivity.

The results obtained have been analyzed qualitatively

For a rough surface, the propagation of surface waves can be investigated regardless of the surface profile. Contrary to this, the results of other approaches are only available for surfaces having smooth profile.

It has been determined that the wave scattering by a rough dielectric surface depends strongly on the typical scale of the roughness. If this scale is about or above the wavelength, the energy of the incident wave is scattered mainly in the ambient space. For small wavelength, the energy is transformed into that associated with the surface-like fields and thus can lead to substantial wave absorption. The above-described effects grow in importance with increasing impedance Z of the surface.

The theory has been applied for calculating the gyrotron efficiency

The theory has been applied for calculating the gyrotron efficiency ÒÅ

mode (Fig. 1).Fig. 1 – Start-up scenario for the coaxial cavity gyrotron with the operating ÒÅ

_{34,19}mode

As can be seen from the curve 3 of Fig. 1, the roughness of the cavity surface initiates the premature failure of oscillations.

The results obtained can be used for more realistic estimates of the scattered power and ohmic losses in coaxial gyrotron cavity, and their effect on the gyrotron operation.