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Within the framework of the average approach and direct 3D PIC (particle-in-cell) Many physical systems exhibit behavior associated with the emergence of the so-called rogue waves which represent dramatically high-amplitude events that occur with low probability but much more frequently than expected in ordinary (e.g., Gaussian or Rayleigh) wave statistics. can be suggested that, similar to lasers [5][6][7], the rogue waves can also occur in microwave oscillators representing active systems which are based on the interaction of electromagnetic waves with non-equilibrium electron beams.One of the widely studied microwave oscillators is a gyrotron in which a helical electron beam excites a waveguide mode near its cutoff frequency [10]. In this paper, we demonstrate that rogue waves can appear in gyrotrons operating in the regime of developed turbulence. It is well known that in electronic oscillators, including gyrotrons, an increase of the excess over the starting conditions leads to a complication of the radiation spectrum caused by the appearance of periodic self-modulation and then chaotic turbulent generation regimes [11][12][13][14][15][16][17]. Typically, realization of self-modulation regimes is a result of combination of a nonlinearity and delay effects [12].

A violation of the local relation between the visible continuum (VC) emissivity and the millimeter wave (MMW) intensity was observed in the experiments on imaging of MMW beams with subcentimeter width using the VC from a slab of the positive column (PC) of a medium-pressure cesium-xenon (Cs-Xe) dc discharge. The spatial distribution of the microwave-induced variation in the VC emissivity of the plasma slab was modeled. For this, perturbations of the electron temperature and density in a slab of the PC of a Cs-Xe discharge affected by an MMW beam were calculated. The relation between the spatial distributions of the VC brightness perturbation and the MMW intensity has been obtained. Good agreement between the results of the experiments and the modeling is demonstrated. The line spread function of a two-dimensional (2D) plasma sensor for MMWs is found, and its width is shown to be equal to 2 mm. We proved that the nonlocality of microwave-induced variations in the intensity of the VC from the PC, as well as the spatial resolution of the plasma technique of MMW imaging, are primarily determined by the influence of the electron heat conduction. Along with the electron heat conduction, the nonlocality of the electron temperature variation is caused by the influence of the hydrodynamic flux of electron enthalpy, diffusion and thermodiffusion fluxes of electrons, and also spatial inhomogeneity of the electron heating by a dc electric field, which results from a microwave-induced variation in plasma conductivity. These factors are responsible for the axial asymmetry of the images of the axisymmetric MMW beams.