1.6 Thesis Structure
2.1.1 Helicon Antenna
Helicon waves belong to a class of plasma waves known as whistler waves which were first identified as propagating through the ionosphere by telephone tech- nicians during the First World War[65, 66]. In nature, these whistlers are pro- duced from the broad-spectrum electromagnetic emission associated with lightning strikes and represent unbounded propagation modes of plasma waves. In labora- tory plasmas, we are typically more interested in the propagation of helicon waves in cylindrical plasma columns.
For the idealised case of an infinitely long uniform plasma column in a uniform magnetic field, the propagation of helicon waves can be described analytically
by a series of discrete wave modes with mode number m[53]. The details of this description are discussed in Appendix A, although an example of the axial evolution of the electric fields for the m = 0 and m = +1 helicon modes is reproduced in Figure 2.2.
Figure 2.2: The electric field direction in the transverse plane for (a) the
m = 0 and (b) the m = +1 helicon modes as their phase evolves along the z axis of an infinite uniform plasma column. See Figure 13.14 in Lieberman[52] after Chen[67]. Image removed due to copyright.
We note that the wave fields of helicon modes involve significant rotational or helical structure, which lends helicon waves their name. In order to excite these propagating modes it is therefore necessary for antenna currents to include specific overall geometric structure. This is in contrast to inductive coupling which does not depend on large scale geometric structure and typically operates purely based on uniform azimuthal antenna currents.
The utilisation of helicon waves in laboratory plasma was first achieved by Boswell in 1970, using a double-saddle antenna[68]. It was found that this antenna exclusively excites the m = +1 mode. Subsequent investigations have identified a broad array of antenna geometries which are efficient at coupling energy into
specific propagating modes[69–72], although the majority of these have focused on either them = 0 orm = +1 modes. Examples of these (Figure 2.3) include the double-saddle antenna which excites the m= +1 mode; the single-loop antenna, which is only capable of exciting the m = 0 mode due to its lack of geomet- ric structure; and the Nagoya III antenna with a left-handed 180◦ twist used in MAGPIE which excites the m = +1 mode but only in the direction antiparallel to the magnetic field (or parallel to the magnetic field for right handed antennae). The half turn Nagoya III antenna also excites an m = −1 mode in the opposite direction to the m = +1 mode, however, this is an evanescent mode which does not propagate[71].
double
saddle singleloop Nagoya III
m = +1
m = +1 m = 0
Figure 2.3: Antenna geometries of the double saddle, single loop, and left-handed half turn Nagoya III designs.
Since MAGPIE was developed to study plasma material interactions, the di- rectionality of the half turn Nagoya III antenna is particularly beneficial, as this means that coupled power is focused in the direction of target samples.
In addition to exciting idealised helicon modes along the axis of discharges, helicon antennae can also excite predominantly electrostatic slow wave modes known as Trivelpiece-Gould (TG) modes which originate directly under antenna straps and are often observed to propagate inwards along cyclotron resonance
cones[55,73–75]. These waves can heat the plasma resistively via collisional mech- anisms, but are also associated with direct ion heating via ion-Landau damping[69,
76,77]. Heating associated with TG modes is primarily edge-localised which can significantly affect the ionisation efficiency of helicon antenna. While the electro- magnetic fields of TG modes are not commonly detected directly[77], analysis of localised plasma heating from electron and ion temperature measurements is often used to identify the presence of TG modes[76,78,79]. In Chapter 4 we utilise this approach and observe that the discharge in MAGPIE under standard operating conditions is consistent with roughly a third of total coupled power going to each helicon coupling, TG coupling, and inductive coupling. This suggests that the half turn Nagoya III antenna utilised in MAGPIE is not fully optimised to launch helicon waves under the conditions that we investigate in this work.
Some insight into the propagation of helicon and TG waves can be obtained by examining the dispersion relation which describes these waves. For an infinite, unbounded plasma in a magnetic field pointing in the z-direction, and for the frequency regime of ωci << ω < ωce << ωpe (where ωci, ωce, and ωpe are the ion
cyclotron, electron cyclotron, and electron plasma frequencies respectively), the dispersion can be described by the relation [52]
k =kz ωce 2ω " 1± s 1− 4ω 2 peω2 k2 zc2ωce2 # (2.1)
which can be simplified in terms of direct physical parameters to obtain
k=kz qeB0 2meω " 1± s 1− 4nω 2µ 0me k2 zB02 # (2.2)
Where k is the total wavenumber and kz its parallel component, B0 is the
amplitude of the magnetic field,nis the number density of the electrons, and ω is the frequency of the wave. We find that there are two solutions for an appropriate
value of kz due to the square root, with the positive branch corresponding to the
TG wave and the negative branch corresponding to the helicon wave. If we then consider the radial componentkr=
p
k2−k2
z, we can observe how the parallel and
perpendicular components of the wave modes depend on plasma conditions. We examine how these wave modes depend on each density and magnetic field strength over the range of conditions which are expected to be observed in MAGPIE in Figure 2.4.
Figure 2.4: Helicon and Trivelpiece-Gould wave number relations for a va- riety of typical plasma conditions observed in MAGPIE. Base parameters are set at ω= 2π×13.56×106s−1,B0 = 5mT, andn = 5×1018m−3. In
(a), density is varied by factors of 2, 5, and 10 between 1×1017m−3 and
2×1019m−3. In (b) the magnetic field is varied by factors of 2, 5, and 10
between 1 mT and 0.1 T. For each curve, the lower branch corresponds to helicon waves and the upper branch corresponds to TG waves.
For helicon waves that are launched by an antenna, kz is determined by the
axial length of the antenna, da. For the m = +1 mode which is launched by
MAGPIE’s antenna, an odd number of half wavelengths must correspond to the antenna length, which restricts kz to fixed longitudinal modes, described by the
relation [53]
kz = (2χ+ 1)
π da
where χ is the longitudinal mode number. For an antenna length of 22 cm, this corresponds to axial wave numbers of 14.3 m−1, 42.8 m−1, 71.4 m−1, and
so on for each successive longitudinal mode. This means that for a given set of plasma conditions, only a restricted set of longitudinal modes will have an axial wave number which corresponds to the helicon branch of the dispersion relation and can therefore propagate in the plasma as a helicon wave. In real systems, this is further complicated by the presence of boundary conditions associated with the plasma chamber and spatial gradients in both the density and magnetic field. However, from the dispersion relations for an unbounded uniform plasma displayed in Figure 2.4, we can note some general trends which we might expect to observe in the physical system. First we observe that if electron density is too low, or magnetic field strength too high, then the entire helicon branch will be restricted to axial wave numbers below that of the fundamental longitudinal mode. Under these conditions the antenna would not be able to excite propagating helicon waves and the discharge would be limited to evanescent inductive coupling, as discussed in Section 1.4. As electron density increases or magnetic field strength decreases, the range of axial wave numbers which correspond to the helicon branch moves to higher values ofkz. This restricts the longitudinal modes which can be excited to a
subset of higher order modes. In theory this could then be extended to arbitrarily high values of longitudinal wave number, however the efficiency with which a physical antenna can couple energy into each mode number will be limited by the specific antenna’s three-dimensional geometry, which would limit the antenna’s ability to excite higher order longitudinal modes. This then limits the maximum densities for which energy can be efficiently coupled into the system by a specific antenna.
For further discussions regarding the detailed physics of helicon wave propa- gation and coupling, see [52,53, 57, 81]
Impedance Matching
When antennas couple energy into a plasma, the antenna-plasma system acts as a load with a complex impedance, whose exact impedance depends both on the fixed antenna properties and the variable plasma properties which evolve both tempo- rally and with operating conditions. In order to ensure a matched impedance that maximises delivered power to the system with minimal reflections back to the power source, we must employ a tunable matching network. In MAGPIE, we utilise a standard π-type matching network described schematically in Figure 2.1. The network consists of a fixed 1.5 µH inductor (L1), between two grounded variable vacuum capacitors of 50-500 pF (C1) and 85-200 pF (C2). The network shown is intended for broad frequency operation, allowing MAGPIE to be oper- ated at frequencies from 7 to 28 MHz, although in this work MAGPIE is operated exclusively at 13.56 MHz.