## 5 Results and Discussion
On the other hand, in the TE\({}_{11}\) mode, the electric field radiated locally in the radial direction from the continuous line slot antenna, and the radially radiated electric field was parallel to the tangential plane of the quartz window surface. In the radial direction, the electric field strength was maximal at the dielectric surface (alumina), which contacted a slot surface of the cylindrical waveguiding structure and it decreased with radial emission. In the standing waveform, the half-wavelength of the propagating quasi-coaxial line TEM mode through the coaxial line was approximately 48 mm. In the TE\({}_{11}\) mode, the half wavelength of the wave propagating through the dielectric was 22 mm, which was approximately 2.1 times smaller. The calculated wavelength by the simulation corresponded to the experimental results, which affected the electron density distribution along the \(Z\)-direction. The electric field attenuation in the axial direction had a considerable influence on the axial distribution of the inhomogeneous plasma. Figure 11 shows the simulation of the attenuation of the normalized E-field amplitude along the wave propagation direction. In general, the rate of attenuation of the wave propagating to the lossy media (X-direction) is expressed as a value proportional to \(\mathrm{e}^{-\alpha\mathrm{ex}}(\alpha\): attenuation coefficient) [1]. The attenuation rate of the electric field intensity (standing wave amplitude) along the wave propagation (Z-direction) could also be represented by the attenuation coefficient \(\alpha\). In the presence of the lossy media, the power absorption was concentrated at the beginning of the applied power [25]. The attenuation coefficient \(\alpha\) was affected by the wavelength of the standing wave. In this simulation, the value of the coefficient was represented by 5 in the quasi-coaxial line TEM and 10 in the circular TE\({}_{11}\). The value was inversely proportional to the wavelength. In the circular TE\({}_{11}\) wave mode with a short wavelength (which propagated in dielectric space), the power absorption was higher than the quasi-coaxial line TEM at the beginning of wave propagation. In the linear microwave source, from a macroscopic point of view, two factors affected the spatial distribution of the electron density significantly, namely, the attenuation of the field strength along the wave propagation direction (axial direction) and the coupled field profile at the window surface (radial direction). First, the electron density distribution along the wave propagation direction was proportional to the attenuation of the electric field strength. As mentioned in the previous section, power absorption mechanism in the plasma was related to the heat source from the electromagnetic wave power. The absorbed power was represented as a function of the variable E. Thus, the Re(J-E) of the power dissipation in the plasma chamber followed the \(E\) field profile. In both the cases of LMPS, power dissipation in the axial direction was concentrated at the surface of dielectric near field region. Figure 11: Attenuation of electric field norm \(|E|\) along Z-direction. However, the power dissipation in the XY plane appeared differently depending on the shape of the emitted field direction. In Figures 12 and 13, at a distance of 5 mm, 15 mm, and 25 mm from the quartz window surface (XZ plane at Y = 0), the electron density along the Z-axis is attenuated in proportion to the decrease in the electric field strength, as shown in Figure 11. In the circular TE\({}_{11}\) mode, the attenuation coefficient of the normalized electron density was five times higher than that of the quasi-coaxial line TEM mode, and the electron generation was concentrated at the initial 100 mm of the power application. Near the window surface, the same fluctuations in the electron density appeared as a change in the intensity of the amplitude with a standing wave effect.