## 4 Simulation  
### Simulation Methods  
Conventionally, analyses of long linear microwave sources reveal limitations to solving the transmission model self-consistently [25; 26; 27]. Self-consistency requires that the wave field and continuity equations be solved simultaneously to account for the effects of the plasma radial and axial inhomogeneities. The plasma spatial inhomogeneity can be calculated by using numerical methods [28; 29; 30; 31; 32; 33; 34].  
Based on numerical methods, in this study, a fluid plasma simulation model(using COMSOL Multiphysics) in the 3-D domain was presented to solve the plasma and wave equations self-consistently [35]. The spatially self-consistent description of the transmis  
Figure 4: Front (**a**) and top (**b**) views of the experimental and diagnostic setup diagram.  
Figure 5: Diagnostic setup in the deposition region.  
sion line could be achieved because the plasma fluid simulation could solve the wave and plasma equations simultaneously in 3-D space. The correlation between the plasma generation, longitudinal power absorption, and sustaining the electric field generated from electromagnetic waves could be calculated.  
In Figure 6, the simulation geometry was designed in the same way as in the Figure 2 (experiment). The height and the diameter of the chamber were designed the same as the experiment. The parts of the power applicator and structures in the simulation were designed, as shown in the upper parts or panels of the figure. In the case of the quasi-coaxial TEM LMPS, shown in Figure 6a, wave excitation was applied at one side (power feeding port) of the WR340 waveguide structure as the rectangular TE\({}_{10}\) mode. The rectangular TE\({}_{10}\) wave was converted to a quasi-coaxial TEM mode through a rod-shaped antenna at the location \(3/4\times\lambda_{up340}\) (\(\lambda_{up340}\): WR340 guide wavelength, 14.36 cm in 2.45 GHz) away from the power input port where the electric field was maximized. On the other hand, in the circular TE\({}_{11}\) mode, shown in Figure 6b, the WR340 waveguide and the converter of the rectangular to circular waveguing structure was omitted to reduce the computational loads. As the wave mode was conserved from the rectangular TE\({}_{10}\) to the circular TE\({}_{11}\)[1], the power applicator was designed to be simplified by applying a cylindrical TE\({}_{11}\) wave power directly to one side of the cylindrical alumina surface (power feeding port), as shown in Figure 6b. In the electromagnetic wave model in COMSOL Multiphysics [35], power application method could be selected so that the input power of the active port is fully deposited on the conductive material [36]. In both the cases of this study, this power application model was selected to reduce the calculation time. 500 mTorr Ar plasma with 2.45 GHz and 0.6 kW microwave power was applied. The background gas temperature was set to 300 K, and the neutral gas density was assumed to be uniform and was calculated under the assumption of an ideal gas. The velocity of the electron was assumed to have a Maxwellian distribution.  
Table 1 lists the Ar chemical reaction set used in the simulation. In this simulation, electrons, Ar\({}^{+}\) ions, and Ar\({}^{*}\) (first Ar metastable species) were considered [37; 38; 39; 40]. In particular, the reaction set containing the Ar excited states and step-ionization was selected to increase the accuracy of the calculations in the pressure regime [36; 37; 38; 39; 40; 41; 42]. The cross sections are given for the charged particle reaction sets, and the rate coefficients are given for the reactions of quenching and ionization by atomic collisions.  
Figure 6: Geometry of MWP 3-D simulation (**a**) quasi-coaxial TEM linear MWP and (**b**) circular TE\({}_{11}\) waveguing structure.