Generation of Rogue Waves in Gyrotrons Operating in the Regime of Developed Turbulence

Within the framework of the average approach and direct 3D PIC (particle-in-cell) simulations, we demonstrate that the gyrotrons operating in the regime of developed turbulence can sporadically emit “giant” spikes with intensities a factor of $100–150$ greater than the average radiation power and a factor of $6–9$ exceeding the power of the driving electron beams. Together with the statistical features such as a long-tail probability distribution, this allows the interpretation of generated spikes as microwave rogue waves. The mechanism of spikes formation is related to the simultaneous cyclotron interaction of a gyrating electron beam with forward and backward waves near the waveguide cutoff frequency as well as with the longitudinal deceleration of electrons.

Figure 1

Dispersion diagram of the waveguide mode $\mathrm{\Omega }={\mathrm{\Gamma }}^2$ and the electron beam line $\mathrm{\Omega }=4g_0^{-2}(\mathrm{\Gamma }-\mathrm{\Delta })$ in the regime of rogue wave generation in a gyrotron (center). Radiation spectra at the entrance $Z=0.2$ (left) and output $Z=15$ (right) of an interaction space. ($L=15$, $g_0=1.3$, ${\beta }_{\perp 0}=0.2$, and $\mathrm{\Delta }=-0.7$).

Figure 2

The output power time traces and histograms of the pulse-height distributions for different values of the normalized current parameter $I_0$ corresponding to the zone of chaotic generation. $N_p$ is the number of spikes in the intensity bins with a given ratio $P/⟨P⟩$. The distribution shown by a gray color in Fig. 2 is obtained when the change of a longitudinal momentum is neglected.

Figure 3

Evolution of the main physical variables inside the gyrotron interaction space in the process of the rogue wave formation: (a) normalized electromagnetic energy flux $\widehat{P}=\mathrm{Im}(a\partial a^{*}/\partial Z)$, (b) the amplitude of the RF current $|J|$, (c) and (d) the amplitudes of the electric $|a|$ and magnetic $|\partial a/\partial Z|$ fields, (e) and (f) the normalized transverse ${⟨|{\widehat{p}}_{\perp }|⟩}_{{\theta }_0}$ and longitudinal ${⟨{\widehat{p}}_{\parallel }⟩}_{{\theta }_0}$ momenta of particles averaged on initial cyclotron phases.

Figure 4

(a) Geometry of a 35-GHz gyrotron used in the 3D PIC simulations and instantaneous positions of macroparticles ($U=20\mathrm{kV}$, $I=2\mathrm{A}$, $g_0=1.3$, $H_0=13\mathrm{kOe}$). (b) Increase of transverse momenta of electrons in the field of the backward wave. (c) Typical spatial profile of the forward propagating giant spike and longitudinal deceleration of electrons in the interaction space.

Figure 5

Results of the PIC simulations: (a) the time dependence of the output power and typical temporal profile of a generated spike, (b) the corresponding long-tail pulse-height distribution, and (c) the broadband radiation spectrum.