# The Helical Resonator: A Scheme for Radio Frequency Plasma Generation
## 3 Propagation of Electromagnetic Perturbations into the Resonator
This gives the following:  
\[E_{\theta 1}(b)\cos\psi+E_{z1}(b)\sin\psi=0 \tag{19}\]  
\[E_{\theta 2}(b)\cos\psi+E_{z2}(b)\sin\psi=0. \tag{20}\]  
The transverse component must be continuous, and since this is the only component present, this gives the following:  
\[B_{z1} = B_{1}I_{0}(\tau r)\]0  
which makes redundant one of the two previous ones. The longitudinal component of the magnetic field must be continuous since there is no current flowing on the surface perpendicular to this direction, so that the following holds:  
\[B_{z1} = B_{1}I_{0}(\tau r)\]1  
Finally, on the conducting shield (\(r=c\)) the electric field will be zero, that is,  
\[B_{z1} = B_{1}I_{0}(\tau r)\]2  
which complete the set of six conditions required to fix the six constants \(A_{1}\), \(B_{1}\), \(A_{2}\), \(A_{2}^{\prime}\), \(B_{2}\), \(B_{2}^{\prime}\). After some algebra, the following eigenvalue equation for \(\tau\) is obtained:  
\[B_{z1} = B_{1}I_{0}(\tau r)\]3This is the result that was obtained by [PERSON] and [PERSON] (2018), and also by [PERSON] (2018), which is incorrectly reported (likely due to a mere transcription error) in Equation (25) of [PERSON] et al. (2018). It has the form \((\tau b)^{2}g(\tau b,c/b)=(kb)^{2}\cot^{2}\psi\), which means that once the normalized frequency \(kb\) and the geometric factors \(c/b\) and \(\psi\) are specified, \(\tau b\) can be obtained, and subsequently, the normalized wave number \(\beta b\) using expression (9). The function \(g(\tau b,c/b)\) is plotted in Figure 2 for different values of \(c/b\). It can be seen that for large values of \(c/b\), this is a strongly decreasing function of \(\tau b\), whereas it tends to the constant value of 1 when the screen goes very near the coil. All the curves group together onto the unit value for large \(\tau b\) (indicatively, larger than 1). In Figure 2 are also reported the eigenvalues \(\tau b\) obtained from Equation (24), as a function of the parameter \(kb\cot\psi\). The curves display an increasing behavior of the eigenvalue with the frequency, and converge to the relation \(\tau b=kb\cot\psi\) as the shield approaches the coil. Once the eigenvalue equation is solved, one has the frequency as a function of the wave number, that is, the dispersion relation. It is then possible to evaluate the velocity factor, defined as the phase velocity \(v_{p}\) normalized to \(c_{0}\):  
\[B_{z1} = B_{1}I_{0}(\tau r)\]4  
We should now recall that we are dealing with axial propagation, and therefore the velocity factor refers to the axial velocity. However, this holds in the sheath helix approximation, while in reality, the voltage pulse propagation takes place longitudinally, along the helically shaped conductor. If we consider the longitudinal propagaton along the helical conductor, the wave vector is given, through a simple projection, by \(\beta^{L}=\beta\sin\psi\). Thus, the longitudinal velocity factor, given that the phase velocity is the ratio of the angular frequency to the wave number, is given by \(V_{f}^{L}=V_{f}/\sin\psi\). Such a longitudinal velocity factor is shown in Figure 3 for two different values of the pitch angle, corresponding to a tightly wound helix (\(\psi=1^{\circ}\)) and to a more loosely wound one (\(\psi=5^{\circ}\)). It can be seen that the longitudinal propagation takes place at a speed somehow larger than the speed of light in the vacuum, and that its frequency dependence is almost the same regardless of the pitch angle, with only a shift in the normalized frequency axis. We can identify three regions: at low frequency, the velocity factor is constant, with a value of 1 for the shield on the helix and increases to values larger than 3 when the shield is brought far away; a transition region; and a high frequency region where the propagation  
\[B_{z1} = B_{1}I_{0}(\tau r)\]5 **Left**: function \(g(\tau b,c/b)\) entering the eigenvalue equation, plotted as versus \(\tau b\) for different values of \(c/b\). **Right**: solutions of the eigenvalue equation, plotted versus \(kb\cot\psi\) for different values of \(c/b\).