## 2 Miniaturization of the Microwave System
### Microwave Transmission System Miniaturization Design
#### 2.2.2 Conventional Length Waveguide Plus Miniaturized Ridged Waveguide  
The ridge waveguide has a low cut-off frequency, low impedance, and high bandwidth. It is commonly used to connect the front-end waveguide system and the ion source body. The impedance of the whole ridge waveguide depends on several factors, such as the impedance of each ridge waveguide (\(Z_{n}\)), the number of ridge steps (\(N\)), the length of the ridge segment (\(L_{n}\)), the gap between the ridge segments (\(d_{n}\)), and the width of the ridge segment (\(s_{n}\)). The commonly used ridge waveguide has 3-4 segments, and the length of each part is \(\lambda_{g}/4\), where \(\lambda_{g}\) is the waveguide wavelength for transmitting microwaves. The relationship of each segment of the ridge waveguide is as follows [16,17]:  
\[Z_{n+1}=\exp(\ln(Z_{n})+2^{-N}C_{n}^{N}\ln\frac{Z_{L}}{Z_{IV}}), \tag{3}\]  
where \(n\) is the current number of ridge step, \(N\) is the total number of ridge steps, \(Z_{IV}\) is the input impedance, and \(Z_{L}\) is the load impedance. The impedance of a ridge waveguide is described differently, with the characteristic impedance being determined by the voltage-to-power ratio as follows [18,19]:  
\[Z_{PV}=\frac{V_{0}{}^{2}}{2P}, \tag{4}\]  
where \(V_{0}\) is the peak voltage across the ridge gap, and \(P\) is the power transmitted by the ridge waveguide. Assuming that the medium in the waveguide is air, at frequency \(f=\infty\), the characteristic impedance \(Z_{PV}\) of the ridge waveguide is expressed as follows:  
\[Z_{PV}^{\infty}=\frac{1}{\frac{k}{120\pi^{2}\beta}\left\{m\frac{2\beta}{k} \cos^{2}\frac{\pi\gamma}{k}\ln\csc\frac{\pi d}{2b}+\frac{\pi\gamma}{2k}+\frac {1}{4}\sin\frac{2\pi\gamma}{2k}+\frac{\phi}{b}\frac{\cos^{2}\frac{\pi\gamma}{ 2k}}{\frac{\pi\delta}{k}}\left[\frac{\pi\delta}{k}-\frac{1}{4}\sin\frac{4\pi \delta}{k}\right]\right\}}, \tag{5}\]  
\[\mathrm{k=}\frac{\lambda_{c}}{a};\;\beta=\frac{d}{a};\;\gamma=\frac{s}{a};\; \delta=\frac{(1-s/a)}{2}, \tag{6}\]  
where \(\lambda_{c}\) represents the cut-off wavelength of the relevant frequency, \(a\) represents the length of the rectangular surface of the ridge waveguide, and \(b\) represents its breadth. According to the formula mentioned above, the impedance of each conventional ridge waveguide segment may be determined. However, the length of each part of a standard ridge waveguide is typically a quarter of the wavelength \(\lambda_{g}\) of the waveguide. For 2.45 GHz, each section of the ridge waveguide is 30.6 mm, so the length of the conventional waveguide used should be around 91.8 mm or 122.4 mm. The dimensions of the waveguide do not suit our requirements. Therefore, based on the traditional ridge waveguide, referring to the design of the transition waveguide, the cavity part of the ridge waveguide is designed as a transition waveguide with three sections of openings gradually reduced. Three equal-width steps are arranged along the transmission direction at the center of both sides of the long side of the inner surface of the cavity, and the gaps between the three steps gradually decrease. By adjusting the width of the ridge and the relative size of the steps, the impedance-matching of the entire microwave system can be realized. The overall length of the miniaturized ridge waveguide is about half of the conventional ridge waveguide. Figure 4b shows the simulation results of the miniaturized ridge waveguide instead of the transition waveguide. Using the miniaturized ridge waveguide as a matching waveguide can boost the electric field strength inside the discharge chamber. The maximum  
field strength inside the discharge chamber is increased to 6027 V/m, roughly four times that of using the BJ26 to BJ32 transition waveguide. The curves (plan 2) in Figure 5 show that the system has an experience with a scattering parameter S11 of \(-\)21.44 dB at 2.45 GHz, and the microwave transmission efficiency into the discharge chamber is greater than 90%.