Nonlinear dynamics of an electron beam with a virtual cathode in a finite drift space

Nonlinear oscillations of an electron beam with a virtual cathode in a finite-length drift tube region are examined by numerical simulation (the PIC method). The change in dynamics of the beam complex with decreasing radius of the drift tube is investigated. The dynamical nature of the stochastic oscillations produced by the virtual cathode is demonstrated by an attractor dimension analysis. Physical processes in the electron beam are studied. It is found that one mechanism responsible for the chaotic behavior of the electron beam is due to the nonlinear interaction between the autostructures organized in it.State University, Saratov. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 38, Nos. 3–4, pp. 268–274, March–April, 1995.     

Nonlinear dynamics of longitudinal structures in the electron cloud of a coaxial electron string ion source

The results of 2.5D particle-in-cell simulation of a coaxial electron trap with an internal anode are reported. It is found that, when the circulating current reaches the value of the ultimate vacuum current, first a virtual cathode arises in the trap and then the beam compresses (distributed virtual cathode). The transient preceding the compressed state exhibits complicated nonlinear dynamics, when compressed regions alternate with regions that are in a two-flow state (phase-space bubbles or phase-space holes). Physically, phase-space holes are similar to the well-known Bernstein-Greene-Kruskal plasma structures. Three types of phasespace holes with different dynamics (oscillating holes, flying holes, and chaotic holes) are revealed. Consideration of phase-space holes as quasi-particles makes it possible to find several channels of their interaction in pair collisions. The feasibility of the coaxial trap as a source of highly charged ions is analyzed. Although the compressed beam mode provides a larger amount of accumulated electrons compared with the conventional two-flow mode, the mean kinetic energy in the presence of a virtual cathode turns out to be much lower. A way of elevating the mean kinetic energy is suggested that consists in increasing the limit vacuum current in the axial configuration with an internal electrode.     

The processes of formation and nonstationary dynamics of a virtual cathode in a nonrelativistic electron beam in the decelerating field (two-dimensional approximation)

We present the results of studies within the framework of a two-dimensional numerical model of nonlinear nonstationary processes in a nonrelativistic electron beam in the mode of formation of a virtual cathode in the decelerating field without focusing of the electrons by a magnetic field (low-voltage vircator). The results of numerical simulation were qualitatively confirmed by the experiment. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 49, No. 10, pp. 843–852, October 2006.     

Numerical study of chaotic oscillations in the electron beam with virtual cathode in the external non-uniform magnetic fields

In this Letter the results of theoretical investigations of the chaotic microwave oscillator based on the electron beam with a virtual cathode are presented. Nonlinear non-stationary processes in these electron systems are studied by means of numerical analysis of 2.5D model. It was discovered that the non-uniform external magnetic field value controls the dynamical regime of oscillations in the virtual cathode oscillator. The processes of the chaotization of output microwave radiation are described and interpreted from the point of view of the formation and interaction of electron structures (bunches) in the electron beams. The numerical results have shown that the investigated electron system with virtual cathode could be considered as a promising controlled source of wideband chaotic oscillations in the microwave range.     

Simulation of the Formation of the Squeezed State of an Electron Beam with a Sublimiting Beam Current in a Closed Drift Tube

Technical Physics - The dynamics of formation of a virtual cathode and the squeezed beam state (SBS) in a setup with a sublimiting relativistic electron beam under repeated input of the electron...     

Experimental and theoretical investigation into the effect of the electron velocity distribution on chaotic oscillations in an electron beam under virtual cathode formation conditions

The effect of the electron transverse and longitudinal velocity spread at the entrance to the interaction space on wide-band chaotic oscillations in intense multiple-velocity beams is studied theoretically and numerically under the conditions of formation of a virtual cathode. It is found that an increase in the electron velocity spread causes chaotization of virtual cathode oscillations. An insight into physical processes taking place in a virtual-cathode multiple-velocity beam is gained by numerical simulation. The chaotization of the oscillations is shown to be associated with additional electron structures, which were separated out by constructing charged particle distribution functions.     

Influence of positive ions on oscillatory processes in an electron beam with virtual cathode

We numerically simulate the influence of positive ions on characteristics of the microwave oscillations in a nonrelativistic electron beam with the virtual cathode formed in a decelerating field (low-voltage vircator). A numerical scheme allowing for ionization of a residual gas by an electron flow is proposed. It is shown that the residual-gas ionization in the operating chamber of a low-voltage vircator leads to a forcing of the virtual cathode out of the transit gap and to a cutoff of microwave oscillations. The obtained numerical data are confirmed by an experimental study using a low-voltage vircator model. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 49, No. 10, pp. 853–863, October 2006.     

Calculation of the characteristics of the radiation from a generator with a virtual cathode

A three-dimensional computer model, described by a system of Maxwell-Vlasov equations, for the interaction of a plasma with the electromagnetic field was used to calculate the excitation of a field by a relativistic electron beam with a virtual cathode oscillating in a resonance chamber. The characteristics of the generator radiation are investigated. Zh. Tekh. Fiz. 69, 87–92 (February 1999)     

Nonlinear dynamics of a generator on a virtual cathode with modulated emission

A numerical study of a low-voltage vircator with controlled emissions from a thermocathode is performed for when an external signal effects on the electron beam and modulates the emission. The strong influence of the modulation parameters on characteristics of oscillations of the beam with virtual cathode is noted. It is shown that when the modulation frequency is tuned to the one of harmonics of a virtual cathode’s free oscillations, there is a considerable increase in the power of high-frequency harmonics of the virtual cathode’s free oscillations in the output spectrum.     

Resonant cavity vircator driven by a thermionic cathode electronbeam gun

A resonant cavity vircator (virtual cathode oscillator) driven by an electron beam emitted from a broad-area thermionic cathode is tested. Narrow-bandwidth (1.0 MHz at the -3 dB level) excitation of the TM₀₂₃ mode of a cylindrical resonant cavity is observed at a frequency of 986 MHz with a pulselength of 1.2 μs. The single-cavity-mode excitation is attributed to the constant-voltage and -current electron beam emitted from the thermionic cathode     

Excitation of parametric oscillations in a triode with a virtual cathode with an external electromagnetic wave

The electromagnetic radiation of an electron flow in a triode with a virtual cathode is studied from the viewpoint of a parametric resonance. It is shown that the phase modulation of the electrons and the radiation saturation take place due to the oscillations of the virtual cathode as a result of large space charge. An expression for the coherent radiation intensity of the electron flow in a resonator is derived. It is shown that the phase modulation depth and the radiation efficiency increase with an external electromagnetic wave. Tomsk Polytechnic University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 76–81, February, 2000.     

Formation and dynamics of a virtual cathode in a tubular electron beam placed in a magnetic field

Mechanisms underlying the formation and dynamics of a virtual cathode (VC) in a tubular electron beam subjected to a magnetic field are studied using a numerical 2D model. Two qualitatively different competing types of space charge dynamics near the VC are discovered. Which of them predominates depends on the magnetic field strength. The beam current critical density at which a nonstationary VC forms in the system is also strongly dependent on the magnetic field. It is shown that the optimal strength of the magnetic field that minimizes the beam current critical density depends on the Brillouin magnetic field.     

Nonstationary processes in a Cerenkov VHF generator with dielectric loading

The dynamics of nonstationary processes in a VHF generator loaded with a dielectric and excited by a high current relativistic electron beam is investigated by means of the numerical solution of the self-consistent set of Maxwell-Vlasov equations. The results are compared with the linear theory of a stationary generator. The case of a beam current above the critical value, when a virtual cathode is formed in the drift space, is also included.     

行波管放大器的非线性混沌特性

以一组非线性方程为基础,从混沌动力学角度,通过计算电子的相空间轨道和辐射场的时空分布图谱,简要分析了行波管放大器的时空混沌特性,计算了行波管输出功率的演化。计算结果表明:行波管放大器辐射场与电子束是一个非线性相互作用的耗散系统,其参数影响整个系统的行为演化,当电流和失谐量等参数变化并超过一定阈值时,系统出现时空混沌特性,输出具有连续的宽频谱性。     

Using the extended Melnikov method to study the multi-pulse global bifurcations and chaos of a cantilever beam

The aim of this paper is to investigate the multi-pulse global bifurcations and chaotic dynamics for the nonlinear non-planar oscillations of a cantilever beam subjected to a harmonic axial excitation and two transverse excitations at the free end by using an extended Melnikov method in the resonant case. First, the extended Melnikov method for studying the Shilnikov-type multi-pulse homoclinic orbits and chaos in high-dimensional nonlinear systems is briefly introduced in the theoretical frame. Then, this method is utilized to investigate the Shilnikov-type multi-pulse homoclinic bifurcations and chaotic dynamics for the nonlinear non-planar oscillations of the cantilever beam. How to employ this method to analyze the Shilnikov-type multi-pulse homoclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications is demonstrated through this example. Finally, the results of numerical simulation are given and also show that the Shilnikov-type multi-pulse chaotic motions can occur for the nonlinear non-planar oscillations of the cantilever beam, which verifies the analytical prediction.     

Effect of gas pressure on amplitude and duration of electron beam current in a gas-filled diode

The parameters of an electron beam generated in helium in the pressure range p = 10⁻⁴−12 atm are studied. Nanosecond high-voltage pulses are applied to a gap between a tubular cathode and planar anode, which is made of 45-μm-thick AlBe foil. Behind the anode, an electron beam is detected at a helium pressure of 12 atm. The pressure dependence of the beam current amplitude shows three peaks at p ≈ 0.01, ≈ 0.07, and ≈ 3 atm. The beam-induced glow of a luminescent film placed behind the foil and the discharge glow at different helium pressures in the gas-filled diode are photographed.     

Chaotization of the virtual cathode oscillations in the external magnetic field created by a ring magnet

The physical mechanisms leading to the chaotization of the virtual cathode oscillations in a low-voltage vircator system at an increase in the amplitude of the external inhomogeneous magnetic field created by a ring magnet have been studied within a two-dimensional numerical model. It has been established that the chaotization of the virtual cathode oscillations in a strongly inhomogeneous external magnetic field is due to the formation of the secondary electronic structure (electron beam) in the electron flow resulting from the magnetic trap in the outer layers of the electron beam.     

The effect of an external signal on output microwave power of a low-voltage vircator

This Letter is devoted to results of the both experimental and theoretical studies of electromagnetic radiation output power gain in the low-voltage microwave generator with a virtual cathode (vircator) under an external harmonic signal leading to the preliminary velocity modulation of the electron beam. The simple theoretical model of the electron beam with virtual cathode in a diode gap with retarding field under the external signal has been developed. The theoretical and numerical analysis have shown the possibility of power amplification in the vircator under the external influence. Obtained results of the theoretical consideration are proven by the experimental study.     

Influence of nonuniformities of a magnetic-mirror field on the space-time characteristics of a long-pulse relativistic electron beam

An experimental investigation is made of the influence of local nonuniformities of a mirror-configuration magnetic field on oscillations of the space charge and the structure of a long-pulse relativistic electron beam. It is found that the outcome depends on the axial configuration of the nonuniformity. A nonuniformity near the cathode can substantially reduce the amplitude of the oscillations and improve the beam transport. The creation of a nonuniformity far from the cathode leads to an accelerated increase in the oscillations and causes spreading of the transverse structure of the beam. A possible explanation is given for the mechanism responsible for the influence of these local magnetic field nonuniformities assuming reflection of the cathode plasma and electron flux from the magnetic mirror, and also allowing for a jump in the drift velocity. Zh. Tekh. Fiz. 67, 83–88 (August 1997)     

Numerical simulations of the reditron

The reflected-electrons discrimination microwave generator (reditron) is a high-power, narrow-band, and single-mode microwave generator that makes exclusive use of the oscillatory character of the virtual-cathode of a relativistic electron beam. The complex, nonlinear character of the virtual-cathode device necessitates particle-in-cell plasma simulation techniques. Investigations indicate two sources of the radiation: (1) the trapped electrons reflexing between the real and virtual cathodes, and (2) the oscillation of the virtual cathode. In the conventional design, the two mechanisms coexist and interfere with each other destructively, causing degradation of the efficiency of microwave generation. The authors have investigated a configuration with a slotted, thick anode and an external magnetic field, which effectively eliminates the reflexing electrons. Two-dimensional particle-in-cell simulations showed that such a configuration exploits the oscillation of the virtual cathode exclusively, and it generates single-mode, narrow bandwidth, and high-power microwave radiation with a potential efficiency over 10%. It was found that further optimization could be achieved by the use of a density (current) modulated electron beam at appropriate frequencies