Autoresonances of $m=2$ diocotron oscillations in non-neutral electron plasmas

The existence of autoresonances for $m=2$ diocotron oscillations of non-neutral electron plasmas in a uniform magnetic field was predicted by particle-in-cell simulations and it was confirmed in experiments. The obtained results show clear deviations from the standard threshold amplitude dependence on the sweep rate. The threshold amplitude approaches a constant at a lower sweep rate when there is a damping force. It was also found that the aspect ratio for the oval cross section of the confined plasma can be controlled by the frequency of the externally applied driving force.

Figure 1

(a) Schematic drawing of the experimental setup for $m=1$ and 2 diocotron autoresonances of non-neutral electron plasmas in a uniform magnetic field $B\sim 96$ G. Open squares are grounded ring electrodes. Five pairs of solid squares represent azimuthally segmented electrodes. Gray squares at both ends are ring electrodes where the potentials $V_1$ and $V_2$ are applied. The number of electrons is changed by the cathode potential and $V_1$ during the electron injection. (b) Experimental setup for autoresonances of the axial harmonic oscillations $B\sim 190$ G.

Figure 2

Radial profiles of a plasma with a large amplitude $m=2$ diocotron oscillation observed in (a) an experiment and (b) a simulation. The circular halo around the oval core is created when the $m=2$ diocotron oscillation is excited with large-amplitude rf bursts.

Figure 3

Autoresonance for $m=2$ diocotron oscillation with $f_0\sim 240$ kHz. (a) FFT spectra below a threshold amplitude $V_{th}$. The frequency of the excited oscillation stays constant. (b) FFT spectra above $V_{th}$. The frequency of the excited oscillation follows that of the external drive. (c) $V_{th}$ as a function of $\alpha /2\pi$. It deviates from the standard dependence of ${\alpha }^{3/4}$ at the lower sweep rate. (d) The decay time constant of the small amplitude $m=2$ diocotron oscillation is about 1.8 ms, which corresponds to the normalized decay constant ${\gamma }_n\sim 0.0023$.

Figure 4

(a) Threshold amplitude dependence on the sweep rate for $m=1$ diocotron autoresonances. (b) Threshold amplitude dependence on the sweep rate for autoresonances of the axial harmonic oscillations.

Figure 5

Red triangles and blue squares are $V_{th}$ determined by warp simulations with different numbers of macroparticles. Open circles, triangles, and squares are $V_{th}$ obtained from calculations with ${\gamma }_n=2\pi \times 0.001,2\pi \times 0.003$, and $2\pi \times 0.005$, respectively [18]. Solid circles and diamonds are the normalized experimental data for $f_0\sim 240$ and 404 kHz, respectively. Dotted and dashed lines are the fitting curves with Eq. (4), which gives ${\gamma }_n\sim 0.032$ and 0.045 for $f_0\sim 240$ and 404 kHz, respectively.

Figure 6

(a) Normalized frequency of $m=2$ diocotron oscillations as a function of the aspect ratio $\lambda$ calculated with Eq. (6). The thick solid line, dashed line, dotted line, dash-dotted line, and thin line correspond to $\sqrt{A_p/\pi R^2}\sim 0.29$, 0.43, 0.57, 0.71, and 0.86, in Eq. (6), respectively. Closed circles are from experiments and the red and blue lines are the simulation results. Also shown are phosphor screen images of plasmas extracted at the end of the external drive for the different stop frequencies of (b) 230, (c) 220, (d) 210, and (e) 200 kHz, respectively.