Oblique Bernstein Waves
Electron cyclotron harmonic waves, also called Bernstein waves, are electrostatic waves propagating predominantly perpendicular to an ambient magnetic field. For oblique propagation the wave is usually strongly Landau damped. This regime is explored in the present experiment. It is performed in a uniform dc discharge plasma which is magnetized and well diagnosed. As shown in Fig. 1 the waves are excited with a gridded plane electrode and received with a smaller similar grid, all connected via shielded coaxial lines. In order to minimized the transmission of an electromagnetic signal the grids are shielded by adjacent grounded grids. The exciter grid orientation can be varied so as to produce oblique Bernstein waves. The waves are analyzed with an interferometer setup.
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Fig. 1. Schematic picture of the exciter grid to launch plane Bernstein waves oblique to B0. |
Typical wave measurements are shown in Fig. 2. Of main interest is the wave propagation which is obtained from the measurement of the phase fronts of the waves. The grid surface forms an equipotential surface just like the phase fronts of electrostatic waves. Thus oblique waves are emitted with k-vector normal to the grid. However, Bernstein waves are backward waves, i.e. phase and group velocities point in opposite directions. By causality the energy flows away from the exciter, hence the group velocity points away from the grid while the phase velocity points toward the exciter.
. Group and phase velocity directions are indicated by arrows.',
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Fig. 2. Phase fronts of oblique Bernstein waves (fc=48 MHz, fp=82 MHz, theta=70 deg). Group and phase velocity directions are indicated by arrows. |
By measuring the wavelength from the phase measurements at different frequencies the wave dispersion relation is obtained. Figure 3a shows the dispersion curves of Bernstein waves propagating perpendicular to B0 where damping is negligible. The cutoffs (k=0) arise at the electron cyclotron frequency and its harmonics. The lower branches have a negative slope or group velocity (vgroup=d omega/dk) for positive phase velocities, vphase=omega/k. The upper branches have a positive slope, hence are forward waves.
 Dispersion relation for electron cyclotron waves (Bernstein waves) for perpendicular propagation (f<sub>c</sub>=48 MHz, f<sub>p</sub>=94 MHz). (b) Polar plot of the real k vs angle of propagation for different frequencies. Arrows indicate group velocity. f<sub>c</sub>=48 MHz, f<sub>p</sub>=82 MHz.',
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Fig. 3. (a) Dispersion relation for electron cyclotron waves (Bernstein waves) for perpendicular propagation (fc=48 MHz, fp=94 MHz). (b) Polar plot of the real k vs angle of propagation for different frequencies. Arrows indicate group velocity. fc=48 MHz, fp=82 MHz. |
For oblique propagation the magnitude of the k-vector changes with propagation angle as shown in Fig. 3b. The normal to the refractive index curves points in the direction of the group velocity. The imaginary value of k is indicated by the numbers near the curves for the real value of k. Damping rapidly increases with propagation angle. The damping is due to the Landau resonance and the fact that obliques waves have a parallel electric field to interact with resonant electrons.
References
- Observations of obliquely propagating electron Bernstein waves, R. J. Armstrong, J. Juul Rasmussen, R. L. Stenzel, and J. Trulsen, Phys. Lett. A 85A, 281-284 (1981). [Link to original publication.]