## 2 Multi-Mode MEICP Reactors Theoretical Background
This transition is often marked by the emergence of a centralized blue core (BC) that is influenced by the antenna geometry, magnetic confinement configuration, and the relative positioning of reactor components. The specific impulse is proportional to the square root of the internal plasma energy, thus requiring high plasma temperatures. Optimal thrust efficiency, on the other hand, requires effective wave-plasma energy conversion, significant plasma heating, and efficient transformation of internal energy into directed axial energy while minimizing plume divergence [28; 29]. From an operational perspective, a compact HPT utilizing an MEICP reactor in the 500 W power class predominantly transfers Radio Frequency (RF) energy via helicon and Trivelpiece-Gould (TG) waves throughout the plasma volume. This energy is subsequently converted into directed axial energy by an MN, enabling efficient thrust generation [6]. Helicon waves, which are low-frequency whistler waves in magnetized plasma, exhibit different characteristics in bounded environments compared to free space. While whistler waves in free space are right-circularly polarized electromagnetic waves, their behavior changes in a radially bounded axial reactor. In such configurations, helicon waves generate short-wavelength electrostatic components, known as TG-modes [37]. [PERSON] et al. [38; 39; 40; 41; 42] propose that the strong collisional damping of the TG wave plays a critical role in facilitating effective power absorption within the plasma. This damping mechanism enhances wave-plasma interactions, thereby improving energy transfer efficiency in the system. Between the lower hybrid and electron cyclotron frequencies (\(\omega_{LH}<\omega\ll\omega_{ce}\)), the wave refractive index may be expressed as follows [43]:  
\[N^{2}=\frac{k^{2}c^{2}}{\omega^{2}}=\frac{\omega_{p}^{2}}{\omega(\omega_{ce}k_ {z}/k-\omega)}\, \tag{1}\]  
where \(\omega,\omega_{ce},\) and \(\omega_{p}\) are the excited angular, cyclotron, and electron plasma frequencies, c is the speed of light, and \(k,k_{z},\) and \(k_{r}\) are the total, longitudinal, and transverse wave numbers, respecting the equality of \(k=\left(k_{r}^{2}+k_{z}^{2}\right)^{1/2}\). [PERSON] et al.and [PERSON] [41] propose that the dispersion relation for a magnetized inductively coupled plasma can be effectively represented by a biquadratic equation, enabling a comprehensive analysis of the wave propagation characteristics within such plasma environments. This formulation helps to better understand the complex interactions between the plasma and electromagnetic waves under varying conditions [44]:  
\[k_{r}^{2\pm}=k_{\parallel}^{2}\frac{1}{2\gamma^{2}\alpha^{2}\beta^{2}}\left(1- 2\gamma\alpha-2\gamma^{2}\alpha^{2}\beta^{2}\pm\sqrt{1-4\alpha}\right), \tag{2}\]  
where,  
\[\alpha=\frac{\omega_{p}^{2}}{\omega_{ce}^{2}N^{2}},\ \beta=\frac{\omega\omega_{ce}N^{2}}{ \omega_{p}^{2}},\gamma=1+i\Big{(}\frac{\
u}{\omega}\Big{)}, \tag{3}\]with \(\
u\) being the electron collision frequency between the neutral particles and ions and \(N_{z}\) being the longitudinal refractive index. Equation (2) underlines that there are two values of \(k_{r}\) related to the fast and slow waves branch; \(k_{r}^{-}\) corresponds to the FWB (helicon) perpendicular wave number and \(k_{r}^{+}\) corresponds to the slow waves branch (SWB) Trivelpiece--Gould perpendicular wave number. For helicon waves to propagate in a plasma, the condition for the bracketed term in Equation (2) must be positive, happening when \(\beta<1\) or \(\omega_{p}^{2}>\omega\omega_{ce}N_{z}^{2}\). Assuming, \(N_{z}=ck_{z}/\omega\), the parameters \(\alpha\) and \(\beta\) can be expressed as follows:  
\[\alpha=\frac{\omega_{p}^{2}\omega^{2}}{c^{2}\omega_{ce}^{2}k_{z}^{2}},\ \beta= \frac{\omega\omega_{ce}c^{2}k_{z}^{2}}{\omega_{p}^{2}\omega^{2}}, \tag{4}\]  
According to reference [45], \(\alpha\) is inversely proportional to the square of the applied magnetic field \(\left(\alpha\propto n_{0}/\left(B_{0}^{2}k_{z}^{2}\right)\right)\).