# The Helical Resonator: A Scheme for Radio Frequency Plasma Generation
## 8 Conclusions  
The production of RF plasmas requires both a good matching of the load to the power supply and a method to magnify the voltage so as to achieve the gas breakdown. The shielded helical resonator excited at a tap point near the grounded end is a concept that allows, in principle, to achieve both goals, allowing a very simple construction for the plasma source and a direct connection to the generator, without the need for a fault-prone matching network. However, in order to achieve both goals, a proper design is needed. In this paper, we derived formulas that allow a prediction of the resonator performance, thus enabling the plasma scientist to properly design his source according to the parameters required for his plasma, and we have tested them experimentally. Clearly, what is missing in this treatment is the effect of the plasma itself, once it is ignited. This puts an additional load on the resonator, and it is to be evaluated how much it can bring the operational conditions away from the optimal ones, especially in relation to the reflected power. This will be the topic of a subsequent study. Similarly, the results concerning the treatment of the resonator as a transmission line depend on the characteristic impedance, which, for the moment, can be explicitly calculated \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & **Shield** & **HV Probe** & \(\mathbf{Z_{c}(\Omega)}\) & \(\mathbf{\kappa H}\) & \(\mathbf{V_{f}}\) & \(\mathbf{\tau_{L}(ns)}\) & **C (pF)** \\ \hline Resonator 2 & No & No & 1003 & 0.119 & 0.025 & - & - \\ Resonator 2 & No & Yes & 1003 & 0.047 & 0.025 & 5.58 & 5.56 \\ Resonator 2 & Yes & No & 272.4 & 0.011 & 0.016 & - & - \\ Resonator 2 & Yes & Yes & 272.4 & 0.015 & 0.016 & 2.04 & 7.48 \\ Resonator 3 & No & No & 1064 & 0.173 & 0.055 & - & - \\ Resonator 3 & No & Yes & 1064 & 0.026 & 0.055 & 4.54 & 4.26 \\ Resonator 3 & Yes & No & 501.7 & 0.004 & 0.036 & - & - \\ Resonator 3 & Yes & Yes & 501.7 & 0.013 & 0.036 & 2.29 & 4.57 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters resulting from the fit of the input impedance curves for resonators 2 and 3. The load characteristic time \(\tau_{L}\) and the associated load capacitance \(C\) are also reported. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Shield** & **1/\(\mathbf{\kappa H}\)** & \(\mathbf{V_{out}/V_{in}}\) \\ \hline Resonator 2 & No & 21.4 & 41 \\ Resonator 2 & Yes & 61.8 & 80 \\ Resonator 3 & No & 38.0 & 30 \\ Resonator 3 & Yes & 77.6 & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the experimental amplification factor at resonance with the expected value \(1/\alpha H\) predicted from the fit outcome, for resonators 2 and 3, with and without shield. only for the shieldless case and needs to be measured experimentally in all other situations, particularly for the fully shielded resonator. Furthermore, even in the shieldless case, it was found that the formula gave a result different from the experimental one by a factor of two, something which needs to be better investigated. It is also to be remarked that to date, we have implicitly considered a capacitively coupled plasma source, where the amplified voltage gives rise to an electrostatic field, inducing breakdown. However, the very nature of the helical resonator makes it suitable also for the transition to an inductive regime [28]. This possibility and the effects on the load need also to be investigated. In conclusion, it is our belief that a renewed interest in the helical resonator concept could lead to new, simpler and more compact designs for RF plasma sources, increasing the efficiency and the reliability. Furthermore, the concept of a circuit element where the lumped circuit approximation fails and the pattern and speed of voltage propagation become important, is in itself a stimulating and non-trivial idea that should be more widely taught in plasma technology courses. Conceptualization, E.M., R.C., L.C. and M.Z. ; methodology, E.M.; software, E.M.; validation, R.C., L.C. and M.Z.