# Probing Collisional Plasmas with MCRS: Opportunities and Challenges
## Appendix B  
In this section, the resonance frequency \(f_{\text{res}}\) (\(f=\frac{\omega}{2\pi}\)) and the electromagnetic field distribution are derived for a cylindrical cavity with a height \(L\) and a radius \(r_{2}\) containing an homogenous coaxial plasma column with a radius \(r_{1}\). A schematic representation of the geometry is presented in Figure 13, where the azimuthal \(\hat{\phi}\), radial \(\hat{\rho}\), and axial \(\hat{z}\) unit vectors are indicated. The permittivity and permeability of region 1, i.e., the core, are \(\varepsilon_{1}\) and \(\mu_{1}\), respectively. The properties of the outer region, also referred to as region 2, are described by \(\varepsilon_{2}\) and \(\mu_{2}\). The cavity walls are assumed to be perfect conductors and, therefore, no fields can exist inside them. As the electric field \(\mathbf{E}\) parallel to the wall and the normal component of the magnetizing field \(\mathbf{H}\) need to be continuous at the interface of the wall and the cavity, the following equations must hold,  
\[\mathbf{\hat{n}}\times\mathbf{E} =0,\] (A6) \[\mathbf{\hat{n}}\cdot\mathbf{H} =0,\] (A7)  
where \(\mathbf{\hat{n}}\) is the normal vector of the wall of the cavity. Two types of resonant modes can be found for these boundary conditions: Transverse Magnetic (TM) and Transverse Electric (TE). For the former type, the magnetic field is perpendicular to the axis of the cylinder and for the latter, the electric field is perpendicular to that axis. For inhomogenously filled cavities, like in the coaxial geometry presented here, pure TM and TE modes only satisfy the boundary conditions if \(\varepsilon_{1}=\varepsilon_{2}\) and simultaneously \(\mu_{1}=\mu_{2}\) or one of the regions is a perfect electric or magnetic conductor. A superposition of a TM and TE mode can satisfy all the boundary conditions and this combination is called a hybrid mode [68]. Resonant modes are characterized by three integer indices \(m\), \(n\), and \(p\) which relate to the number of nodes in the \(\phi\)-, \(\rho\)-, and \(z\)-direction, respectively. The notation used to refer to a specific mode works as follows: TM\({}_{mnp}\) and TE\({}_{mnp}\) for pure modes and TM\({}_{mnp}\)* and TE\({}_{mnp}\)* for hybrid modes. Henceforward, the example of the 'two-dielectric' problem from ([33], pp. 219-221) will be expanded to allow for hybrid modes. More information relating to this problem can be found in [22]. The wave functions for the magnetizing field \(\psi^{\text{H,1}}\) and the electric field \(\psi^{\text{E,1}}\) for region 1 are,  
\[\psi^{\text{H,1}} =AB_{m}^{\text{H,1}}(k_{\rho,1}\rho)\cos(m\phi)\exp(-ik_{z}z),\] (A8) \[\psi^{\text{E,1}} =BB_{m}^{\text{E,1}}(k_{\rho,1}\rho)\sin(m\phi)\exp(-ik_{z}z),\] (A9)  
and for region 2,  
\[\psi^{\text{H,2}} =CB_{m}^{\text{H,2}}(k_{\rho,2}\rho)\cos(m\phi)\exp(-ik_{z}z),\] (A10) \[\psi^{\text{E,2}} =DB_{m}^{\text{E,2}}(k_{\rho,2}\rho)\sin(m\phi)\exp(-ik_{z}z),\] (A11)  
where \(A\), \(B\), \(C\), and \(D\) are constants, \(B_{m}^{\text{H,j}}\) is the Bessel function for the magnetic field and \(B_{m}^{\text{E,j}}\) the Bessel function for the electric field, \(m\) the order of the particular Bessel function and \(j\) the number of the region. [PERSON] ([69], pp. 342-346) explains in a very clear and intuitive manner the reason that these waves in a cylindrical geometry are described by Bessel functions. Later in this derivation, the Bessel functions \(B_{m}^{\text{H,2}}\) and \(B_{m}^{\text{E,2}}\) will be determined to be combinations of Bessel functions of the first kind \(J\) and second kind \(N\), also referred to as Neumann functions, and their derivatives indicated by \(J^{\prime}\) and \(N^{\prime}\).