## 2 Multi-Mode MEICP Reactors Theoretical Background
Bulk conversion happens near the mode conversion surface, where the properties of helicon and TG waves intersect, significantly influencing plasma behavior, shaping its density profile, and enhancing the efficiency of energy absorption from external sources [39; 42]. Effective plasma generation in an MEICP reactor is achieved through volume oscillations that are resonantly excited by antennas configured for an azimuthal mode of m = +1. This mode enables optimal coupling of the RF power into the plasma, enhancing the overall plasma density and stability. Considering [PERSON]'s law and the waves expressed in the form of \(\sim exp(i(kz-\omega t+m\theta))\), the magnetic components of the bounded helicon waves \(B\) can be represented using Bessel functions [52]. When the wave is confined within a reactor of radius \(R_{r}\), the boundary conditions for the wave, assuming \(k_{z}\ll 1\), can be derived from the following equation:  
\[\frac{\omega\omega_{p}^{2}}{\omega_{ce}k_{z}^{2}c}\frac{m}{k_{r}R_{r}}J_{m}(k _{r}R_{r})+I_{m}^{\prime}(k_{r}R_{r})=0, \tag{6}\]  
where \(\theta\) is the angle of the helicon wave propagation relative to the magnetic field, \(J_{m}\) and \(J_{m}^{\prime}\) are the Bessel function and its derivative with respect to argument, and \(m\) represents the helicon wave azimuthal mode number. In the case of the m = +1 operating mode, the lowest Bessel root is \(k_{r}R_{r}=3.83\), imposing that \(k_{r}\ll k_{z}\), and Equation (5) becomes [44]:  
\[\frac{3.83}{R_{r}}=\frac{\omega}{k_{z}}\frac{\mu_{0}en_{0}}{B_{0}}\ \propto\frac{\omega}{k_{z}}\frac{n_{0}}{B_{0}} \tag{7}\]  
with \(\mu_{0}\) and \(e\) being the magnetic permeability of the vacuum and the electric charge. Equation (7) highlights a linear relationship between the plasma density (\(n_{0}\)) and the external magnetic field (\(B_{0}\)) for a fixed mode, where both the angular frequency \(\omega\) and the axial wave vector \(k_{z}\) are held constant. This suggests that plasma density is approximately proportional to magnetic field strength (\(n_{0}\propto B_{0}\)) [43]. As the magnetic field increases, the plasma density tends to rise, indicating that the magnetic field significantly influences the ionization and overall characteristics of the plasma. This correlation is essential for optimizing plasma conditions in various applications, including EP systems, where precise control over plasma density is crucial for performance. One peculiar characteristic of the MEICP reactor dedicated to a HPT is the operating mode transition from the inductive (H-mode) to the helicon wave (W) mode [53; 54]. Generally, the H-W transition is associated with sharp changes in plasma impedance and a jump in the plasma density. Research by [PERSON] et al. [55] has demonstrated that, under specific operational conditions, the helicon mode exhibits pronounced ionization effects, particularly near the center of the plasma. This strong ionization is closely linked to the presence of helicon waves, which facilitate enhanced energy transfer and electron heating within the plasma. As a result, the H-W transition not only alters the operational dynamics of the plasma but also plays a crucial role in optimizing the performance of the HPT, ensuring efficient propulsion capabilities. Achieving a significant increase in plasma density--especially following mode transitions--relies heavily on the effective design of RF antennas in a helicon reactor. Various antenna configurations have been proposed to optimize this process, including loop antennas, double saddle coils, Nagoya Type III antennas, spiral antennas, birdcage antennas, and half-helix antennas [56]. As electromagnetic waves are expressed as a combination of the Bessel function, the choice of antenna geometry plays a critical role in determining the azimuthal mode numbers and the associated electromagnetic field distributions, which are essential for maximizing energy transfer to the plasma.