## 2 Miniaturization of the Microwave System
### Microwave Transmission System Miniaturization Design  
To realize the miniaturization of the microwave system, the method we adopted was to directly match the microwave power source to the discharge chamber in the design, remove the circulator, and integrate the three-pin tuner function into the matching waveguide. The directional coupler is utilized chiefly for measuring microwave power, but it is canceled in the existing system. The system leaves just the magnetron, excitation cavity, high-voltage isolation waveguide, matching waveguide, and the source body. This method allows for the efficient transmission of high-power microwaves and decreases the total size of the microwave transmission system by a substantial amount. When combined with a compact discharge chamber, this allows for the ion source system to be located within the accelerator's high voltage terminal spherical shell. The microwave transmission system from power source to load includes: magnetron, excitation cavity, high voltage isolation waveguide, transition waveguide and source body. The source body is composed of a microwave window and a discharge chamber. The microwave window is composed of three pieces of 10 mm aluminum oxide and one piece of 2 mm boron nitride. The discharge chamber is made of stainless steel, and the inner cavity size is \(\Phi\)30 mm \(\times\) L40 mm. However, the discharge cells with the dimensions mentioned above have the following issues: In conventional electromagnetic wave theory, a cavity discharge chamber with a diameter of 30 mm is significantly smaller than the maximum discharge chamber diameter \(D=2\lambda/3.412\approx 72\) mm corresponding to the 2.45 GHz TE\({}_{11}\) wave in the classical electro  
Figure 3: Structural model of the ECR ion source experimental platform equipped with a conventional microwave transmission system and microwave transmission measurement results of the ion source: (**a**) is a three-dimensional structural model of the ECR ion source experimental platform equipped with a conventional microwave transmission system; (**b**) is a physical drawing of the experimental platform and microwave transmission measurement results of the ion source. magnetic wave theory [11]. The discharge chamber is so compact that electromagnetic waves cannot fit inside. The relative permittivity of the plasma \(\varepsilon_{\mathrm{P}}\) can be expressed as [14]:  
\[\varepsilon_{\mathrm{P}}=1-\frac{(\omega_{\mathrm{pe}}/\omega)^{2}}{1-\omega_{ \mathrm{ce}}/\omega}, \tag{1}\]  
where \(\omega\) represents the microwave frequencies, \(\omega_{\mathrm{ce}}\) is the electron cyclotron frequency, \(\omega_{\mathrm{pe}}\) is the plasma oscillation frequency given by \(\omega_{\mathrm{pe}}=\sqrt{n_{\mathrm{e}}e^{2}/m_{\mathrm{e}}\varepsilon_{0}}\), and \(\varepsilon_{0}\) is the dielectric constant in vacuum. The electron cyclotron frequency is denoted by \(\omega_{\mathrm{ce}}=eB/m_{\mathrm{e}}\), where \(e\) is the elementary charge, \(B\) is the axial magnetic induction, and \(m_{\mathrm{e}}\) is the electron mass. Additionally, the microwave frequency is represented by \(\omega=eB_{0}/m_{\mathrm{e}}\), with \(B_{0}\) corresponding to a magnetic flux density of 875 G for a microwave frequency of 2.45 GHz. By substituting these values into Equation (1), we obtain:  
\[\varepsilon_{\mathrm{P}}=1+\frac{n_{\mathrm{e}}m_{\mathrm{e}}}{\varepsilon_{ 0}B_{0}(B-B_{0})}, \tag{2}\]  
where \(n_{e}\) represents the electron density. According to the above equation, when \(B>B_{0}\), the dielectric constant will be larger than the vacuum dielectric constant, and the minimum diameter \(D_{\mathrm{P}}\) that can transmit a 2.45 GHz microwave can be changed to \(D_{\mathrm{P}}=\lambda/3.412\sqrt{\varepsilon_{\mathrm{P}}}\). When the plasma dielectric constant is large enough, the 2.45 GHz electromagnetic wave can enter the small size discharge chamber. To verify the miniaturization scheme, the numerical simulation of the microwave system was carried out.