**Investigation of electromagnetic processes in periodic waveguide structures**

The theoretical investigation of electromagnetic processes in periodic waveguide structures is of interest from the viewpoint of application of rippled waveguides and high-power gyrotrons in the accelerator and microwave technique and, also, for solving the problems of controlled thermonuclear fusion [8–13,34–40]. In 2006 the main achievements in this field are related with

**the international activity of SPE RESST**[8,34] within the framework of the STCU project (Project manager Zaginaylov G.I., Doctor of Science (Physics and Mathematics)). The project goal was to develop the methods for mathematical simulation of coaxial gyrotron cavities with a rippled conductor (see the Table).

Name of the foreign organization, country | Instrument for cooperation | Results |

Technical University Hamburg-Harburg Germany Research Center Karlsruhe, Germany Technical University Helsinki, Finland. |
STCU project #3227 "Optimization of coaxial gyrotrons cavities for use in the controlled thermonuclear fusion devices" (2005…2007). Project manager — Zaginaylov G.I., Leading Researcher of SPE RESST at NSC KIPT, Doctor of Science (Physics and Mathematics). |
Multiwave analysis was performed on a coaxial gyrotron cavity with rippled internal conductor. Computer programs were developed for resonance frequency calculations, Q factors, field distribution in the cavity and for calculations of the attenuation in the internal conductor of a coaxial. |

In particular, the theory of background plasma influence on the electromagnetic properties of the high-power gyrotron cavity was developed [8]. It has been established that the background plasma decreases the resonance frequency of the cavity, increases its Q-quality at the operating mode and Q-factors of parasitic modes. The calculated correction to the cavity frequency is in good agreement with measurements at 140 GHz 1 MW gyrotron developed in Karlsruhe for plasma heating in the stellarator WX-7 in the long pulse mode. Besides, the dispersion properties of rippled plasma waveguides were investigated analytically [9,35]. The theoretical explanation of experiments on the electron confinement by injection of a velocity-smeared tubular electron beam into the Penning trap is suggested [10,11,36,37]. The self-simulated solutions of the Schrodinger nonlinear multidimensional equation were obtained [38] by the correct choice of self-simulated variables providing the integral conservation. The processes of electron beam reflection from the semi-bounded plasma are studied [12,39,40].