## 2 Multi-Mode MEICP Reactors Theoretical Background
Helicon waves propagate radially outward from regions of higher plasma density to lower plasma density, with parameters \(\beta\) and \(\alpha\) playing critical roles in defining the wave's propagation characteristics. Specifically, for helicon waves, \(\beta\sim 1\) indicates that the plasma density is sufficiently high for efficient wave propagation and \(\alpha=1/4\) corresponds to the boundary conditions associated with the lower and higher cutoff frequencies for helicon waves. In the proposed HPT, radial non-uniformities are influenced by the cusped magnetic field configuration, leading to distinct propagation characteristics for helicon and TG waves within the plasma. These waves propagate in the interior of the plasma when \(\alpha<1/4\) and \(\beta>1\). The SWB (TG waves) propagates in the outer region of the plasma under similar conditions of \(\alpha<1/4,\ \beta>1\). As the external magnetic field strength increases, a clear separation between the helicon and TG waves becomes apparent. This separation allows for the distinct operational modes of the thruster. Notably, there exists a natural mode confluence at critical magnetic field strengths corresponding to the conditions where to \(\alpha=1/4\) and \(\beta\sim 1\)[46]. In high magnetic fields, helicon (H) and TG waves display distinct characteristics, but as the magnetic field strength decreases, their wave patterns converge, allowing for significant mode coupling between the two waves and facilitating energy transfers [45; 47]. The helicon waves, also referred to as the fast waves branch (FWB), can propagate within a specific density range, as outlined in the work of [PERSON] et.al and [PERSON] [41]. \[n_{Low}=\frac{m}{4\pi e^{2}}\omega\omega_{ce}N_{z}^{2}<n<n_{Up}=\frac{m}{16 \pi e^{2}}\omega_{ce}^{2}N_{z}^{2} \tag{5}\]  
\(n_{Low}\) represents the cutoff for the helicon wave; at \(n_{Up}\), the fast waves branch (FWB) merges with the SWB, known also as the TG-mode. According to references [40; 48], the propagating electromagnetic wave (FBW) transitions into an electrostatic resonance (SWB) when the two waves converge at the condition \(\alpha=1/4\), where \(k_{r}^{2-}=k_{r}^{2+}\)[35]. Unlike the helicon mode, which demonstrates maximum field strengths along the discharge axis, the electric field associated with the TG-mode is primarily localized near the plasma's radial boundaries. In ultra-compact helicon reactors, the generation of plasma is significantly enhanced through the interaction between the helicon wave and the dissipative TG-mode, which then transfers energy to the plasma electrons, promoting continuous ionization and sustaining the plasma state. The helicon and TG waves operate through distinct channels of RF input, facilitating power deposition in various regions of the plasma column. TG waves experience significant damping, primarily depositing energy in a narrow surface layer of the plasma. This surface power input channel is particularly evident when the plasma source operates outside the anti-resonance regime, thereby enhancing overall power transfer and plasma generation [49; 50]. At high magnetic fields, helicon (H) waves penetrate deeply into the plasma due to their low damping, while TG waves experience stronger damping and remain localized near the plasma edge, resulting in distinct propagation characteristics [45; 47]. The efficiency of converting helicon waves into TG waves is significantly determined by the plasma's density distribution and collisions rate. In high-density plasma, TG waves dominate power absorption at the plasma boundary, while Helicon waves primarily facilitate heating in the core. This complementary absorption pattern leads to a more balanced energy distribution within the plasma, enhancing overall heating performance. Mode conversion [35; 45; 51] between helicon and TG waves primarily occurs through two key mechanisms: surface mode conversion at the plasma boundary and bulk mode conversion within the plasma core.