# The Helical Resonator: A Scheme for Radio Frequency Plasma Generation
## 3 Propagation of Electromagnetic Perturbations into the Resonator  
In order to properly model the helical resonator, we first need to understand how electromagnetic disturbances propagate in it, deriving a dispersion relation that is useful for obtaining the basic properties, such as phase velocity and, in a limit case, the characteristic impedance of the resonator seen as a transmission line. To accomplish this task, it is customary to model the resonator as a \"sheath helix\". This name is used to indicate an idealized anisotropically conducting cylindrical surface of infinite length, with infinite conductivity along the helix and zero conductivity normal to it. This model is possibly originally due to [PERSON] (1983), and was treated by [PERSON] (1985) and by [PERSON] (2001). The starting point are Maxwell equations in vacuum as follows:  
\[\
abla\times\textbf{E}=-\frac{\partial\textbf{B}}{\partial t} \tag{5}\]  
\[\
abla\times\textbf{B}=\frac{1}{c_{0}^{2}}\frac{\partial\textbf{E}}{\partial t}. \tag{6}\]  
Combining them, we obtain the wave equation as follows:  
\[\frac{1}{c_{0}^{2}}\frac{\partial^{2}}{\partial t^{2}}\left\{ \begin{array}{ ll}\textbf{E}\\ \textbf{B}\end{array} \right\}-\
abla^{2}\left\{ \begin{array}{ll}\textbf{E}\\ \textbf{B}\end{array} \right\}=0 \tag{7}\]  
where \(c_{0}\) is the speed of light, which has as solutions in free space the electromagnetic plane waves. We now consider the cylindrical geometry of the resonator and seek solutions of the form \(\exp[i(\omega t-\beta z-m\theta)]\). Here, \(\beta\) is the axial wavevector, and \(m\) the azimuthal mode number. Furthermore, we restrict ourselves to \(m=0\) modes, because higher order modes are relevant only at very high frequencies. We thus deal with azimuthally symmetric perturbations of the form \(\exp[i(\omega t-\beta z)]\). The wave equation becomes the Helmholtz equation as follows:  
\[\left[\frac{1}{r}\frac{d}{dr}\bigg{(}r\frac{d}{dr}\bigg{)}-\beta^{2}+\frac{ \omega^{2}}{c_{0}^{2}}\right]\left\{ \begin{array}{ll}\textbf{E}\\ \textbf{B}\end{array} \right\}=0. \tag{8}\]  
Introducing \(k=\omega/c_{0}\) (this is the wavevector of electromagnetic plane waves in vacuum), and defining the radial eigenvalue,  
\[\tau^{2}=\beta^{2}-k^{2}, \tag{9}\]  
we have the following:  
\[\left(\frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d}{dr}-\tau^{2}\right)\left\{ \begin{array}{ll}\textbf{E}\\ \textbf{B}\end{array} \right\}=0. \tag{10}\]The general solution of this equation is of the form \(AI_{0}(\tau r)+BK_{0}(\tau r)\), where \(I_{0}\) and \(K_{0}\) are the modified Bessel functions of the first and second kind, respectively. It is important to remark, in order to avoid confusion, that \(k\) is here used as a normalized version of the frequency, whereas \(\beta\) is the actual wavenumber of the perturbations, and \(\tau\) defines the radial oscillation of the electromagnetic field perturbation. Normally, \(\beta\gg k\), so that \(\tau\approx\beta\). Let us now label with 1 the region inside the coil (\(0\leq r\leq b\)) and with 2, the region between the coil and the shield (\(b\leq r\leq c\)). Taking into account that \(K_{0}(x)\) diverges in \(x=0\), we have the following physically admissible solutions for the axial field components:  
\[E_{z1} = A_{1}I_{0}(\tau r) \tag{11}\] \[B_{z1} = B_{1}I_{0}(\tau r)\] (12) \[E_{z2} = A_{2}I_{0}(\tau r)+A_{2}^{\prime}K_{0}(\tau r)\] (13) \[B_{z2} = B_{2}I_{0}(\tau r)+B_{2}^{\prime}K_{0}(\tau r). \tag{14}\]  
The other components of the fields are deduced from Maxwell equations as follows:  
\[E_{r} = \frac{i\beta}{\tau^{2}}\frac{dE_{z}}{dr} \tag{15}\] \[E_{\theta} = -\frac{i\omega}{\tau^{2}}\frac{dB_{z}}{dr}\] (16) \[B_{r} = \frac{i\beta}{\tau^{2}}\frac{dB_{z}}{dr}\] (17) \[B_{\theta} = \frac{i\omega}{c_{0}^{2}\tau^{2}}\frac{dE_{z}}{dr} \tag{18}\]  
We now apply the boundary conditions relevant for the sheath helix. On the helix surface (\(r=b\)) the longitudinal electric field has to be zero, due to the hypothesis of infinite conductivity.