# The Helical Resonator: A Scheme for Radio Frequency Plasma Generation
## 1 Introduction  
The generation in the laboratory of an ionized gas, also called plasma, requires, in most instances, application to the originally neutral gas of an electric field that is large enough to start an avalanche process of ionization events driven by free electrons (also called breakdown). While the required conditions greatly vary, depending on many variables related to the chosen experimental layout and working conditions, in broad terms, the typical applied voltages are in the range of a few hundreds volts to several kV. Furthermore, the voltage can be stationary or varying in time with different rates: the rate determines both the technology to be used to generate and transmit the required voltage and the physics of the breakdown process and of the produced plasma. Among the different possibilities, the radio frequency (RF) voltage is a possible choice. This term broadly represents voltages varying with a frequency ranging from around 1 MHz to several hundred MHz. RF plasmas are characterized by a lower voltage required for breakdown with respect to lower frequency devices, and by the possibility of avoiding contact between electrodes and plasma, especially when the inductive coupling regime is achieved. In general terms, the production of a RF plasma requires a generator (possibly composed of an oscillator and an amplifier), and a matching network, which in this case, has the double role of matching the device input impedance to the output impedance of the generator, and to magnify the voltage to the required value for breakdown and subsequent plasma sustainment. The impedance matching feature is necessary to minimize the power reflected from the load to the generator. This is not an issue at lower frequencies, where the wavelength of the voltage disturbance is much larger than the apparatus size, so that instantaneous propagation can be assumed, and lumped element circuit treatments are appropriate. It becomes, however, important in the MHz range and beyond, where the wavelength becomes comparable to the system size. In this case, a finite time is required for voltage variation propagation, and transmission line theory comes into play. It is worthnoting that this brings many similarities between the field of plasma technology and that of radio transmission, although in the latter, voltage amplification is not a requirement, but rather an issue that should eventually be taken into account to avoid undesirable breakdown of air. Many possibilities exist for matching network construction, and sophisticated solutions are now commercially available. There is, however, a solution that has already been used for plasma generation, but has not encountered great diffusion despite its potential simplicity: the helical resonator. The helical resonator, as will be described in more detail in the following, is an open-ended air core coil that can be used both for voltage amplification, exploiting a resonance condition, and for impedance matching, by proper design of its layout. The absence of a ferrite core implies that very low dissipation levels are attainable if proper care is put in the design of the device, allowing the use of relatively low-power generators. It is worth making here the historical annotation that there is a strong connection between the helical resonator and the \"extra coil\" used by [PERSON] in his work at Colorado Springs to produce high voltage at high frequency [1]. Indeed, [PERSON] evolved from using a loosely coupled resonant transformer to a tightly coupled one, with the secondary circuit connected to an \"extra coil\", which was a capacitively loaded helical resonator. Keeping in mind that [PERSON] used frequencies of tens of kHz, so several issues related to dissipation are substantially different in our case (and the apparatus size as well), it is, however, important to remark that our present discussion is also useful for a better understanding of his work and of the modern day [PERSON] coil working principle.