Numerical Study of Nonstationary Phenomena in Gyro-Oscillators

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The possible applications of gyro-oscillators span a wide range of technologies [61 – 63]. In the field of magnetic confinement fusion studies, such applications as lower hybrid current drive, electron cyclotron resonance heating (ECRH) and current drive, plasma production for different processes, and active plasma diagnostic measurements have been demonstrated. For these applications, it is necessary to develop CW gyromonotrons that operate at both higher frequencies and higher output power. Hence single-mode 110-170 GHz CW gyromonotrons with a conventional cylindrical cavity, capable of high average power (0.5-1 MW per tube), and 2 MW coaxial cavity gyromonotrons are currently under development. Gyromonotrons are also successfully used in materials processing (e.g. in advanced ceramic sintering, surface hardening, or dielectric coating of metals and alloys) as well as in plasma chemistry. The use of gyromonotrons for such technological applications appears to be of interest if one can realize a relatively simple, low cost device which is easy in service. Hence gyromonotrons with low magnetic field (operating at the 2nd harmonic of the electron cyclotron frequency), low anode voltage, high efficiency, and long lifetime are under development. Potential applications of gyro-BWOs as a tunable millimeter-wave source including position-selective heating of fusion plasma, spectroscopy, materials processing, and drivers for ultra-high power amplifiers also motivate further theoretical and experimental investigations.

In many cases, the time dependence of the electromagnetic field in gyro-oscillators shows nonharmonic or multi-frequency behavior. This can be a transient process during a start-up phase (which generally causes th operating frequency to change with time), during a short-pulse operation, or during mode competition. In the latter case, the modes interacting at different cyclotron harmonics and/or in different regimes (gyromonotron or gyro-BWO) at the same time often show quite different operating frequencies. This can also be a nonstationary regime caused by operating the device at parameters significantly exceeding the start oscillation condition. Such operation regime can occur not only when increasing one of the operating parameters, for example, the beam current, but also when keeping the operating parameters constant and decreasing the start oscillation condition parameters, for example, by introducing reflections of the output power back from the load into the interaction space. Hence, in order to accurately simulate nonstationary phenomena in gyro-oscillators, a time-dependent multimode code which is capable to self-consistently model the electromagnetic field within broad spectral bandwidth is necessary.

In this work, the development of a self-consistent time-dependent analysis for gyro-oscillator simulation, which is believed to be as rigorous as the PIC code approach, but which is much more efficient especially in the case of an azimuthally non-homogeneous electromagnetic field has been presented. An accurate representation of the electromagnetic field is obtained by expanding the field components in terms of the solenoidal and the irrotational eigenfunctions of the equivalent completely closed cavity. The use of the eigenfunction expansion method reduces the boundary value problem for the field components to that of solving a linear system of ordinary differential equations (ODE) for the expansion coefficients. A convolution technique together with time domain analytic expressions for the characteristic admittances of the waveguide modes and for the reflection coefficients at the apertures of the cavity are applied to accurately formulate the time-dependent boundary conditions. The presented formulation is valid for describing a field having a broad frequency spectrum. The electron beam is represented by an ensemble of particles. The relativistic equations of motion associated with the particles are solved self-consistently with the ODE for the expansion coefficients by means of a multi-step integration scheme. After code validation, it has been used to attack a number of problems.

First, it has been demonstrated that the developed time-dependent self-consistent multimode code is capable to model the interaction between the modes operating at the fundamental and/or at higher harmonics of the cyclotron frequency. The numerical results for the fundamental modes agree well with the experimental results published in ¹⁴⁴. Moreover, it is shown that the use of the surface impedance model ¹³² for the calculation of modes operating at harmonics higher than the fundamental leads to wrong results in high-power coaxial cavity gyromonotrons. Hence the use of this model must always be checked if the radiation wavelength is less than about 5 times the width of the corrugation grooves.

The influence of reflections on the operation of an 1 MW, 140 GHz, TE22,6 mode gyromonotron developed at Forschungszentrum Karlsruhe has been studied by means of the described self-consistent time-dependent multimode analysis. It has been found that for the gyromonotron with operating parameters close to their optimum, the onset of an unstable operation regime occurs for reflection coefficients R ≥ 0.4, and the relative width of the spectrum of the output radiation increases up to 12%. The results of the numerical investigation on output power variation and frequency pulling due to a variation of the reflection parameters agree well with the available experimental data and demonstrate a strong dependence on the reflections. Moreover, our analysis of the gyromonotron operating in the presence of reflections has shown several new features of ist nonstationary behavior. First, it has been shown that the traditionally accepted scenario of the transition from the single-frequency stationary operation regime to the chaotic nonstationary regime when increasing the reflection coefficient is not correct. In contrast, an increase of the reflection coefficient above a certain threshold value (R ≥ 0.4) leads to the sequential excitation of a number of modes of the mismatched gyromonotron cavity. Then the gyromonotron shows a very complicated periodic or quasiperiodic behavior of a regular nature. No signs of chaotic behavior have been found. Second, for higher reflection coefficients (R ≥ 0.85), the main mode as well as most of the other modes of the mismatched cavity are suppressed by that mode which shows the highest eigenfrequency, and whose field profile shows its maximum in the vicinity of the left aperture. The eigenfrequency of this mode lies next to the cutoff frequency of the left aperture and limits the achievable bandwidth of the gyromonotron output radiation when one tries to increase it by means of reflections. Third, decreasing the distance to the reflecting load increases the frequency separation between the modes, what correspondingly reduces the number of excited modes and makes the spectrum of the gyromonotron output radiation less dense.

Furthermore, the obtained results reveal important details of the saturated behavior of the gyro-BWO which have not been shown in the nonstationary analysis performed before. Although on one side, they agree with results obtained by using a stationary code ⁵⁰ which employs a similar outgoing-wave boundary condition, and which also uses a self-consistent beam-field interaction model, on the other side, the results demonstrate the importance of timedependent calculations for a gyro-BWO stability analysis. Moreover, the results are in good agreement with recently available results obtained by another nonstationary code ⁵¹ as well as with the experiments ¹⁰. Furthermore, it has been shown that tuning the external magnetic field leads to the excitation of the undesired taper mode which reduces the magnetic tuning bandwidth. Moreover, in an injection-locked gyro-BWO, significant modifications of the locking bandwidth curve have been observed when the magnetic field is tuned. The locking bandwidth curve which has been found in the simulations as well as its asymmetry agree well with the experiments ¹⁰⁸.