## 3 Optimization of Discharge Chamber Size and Magnetic Field  
Optimization is based on the simulation calculations with consideration that the microwave and plasma parameter evolution are decoupled to each other, because the time scale is at the ns level. Considering the time scale of microwaves, the motion of ions is assumed to be negligible in comparison to the motion of electrons, the model ignoring the plasma's specific evolution [20; 21]. This means that we can only focus on whether the microwave is effectively absorbed and simplify the consideration of the type of ionized gas and the input power value. Therefore, we will examine the ionization of the gas at 10 watts of microwave. The influence of parameters such as the magnetic field and the size of the discharge chamber on the process of absorbing microwave energy by plasma is evaluated based on the state of the plasma. In microwave plasma, the electric field (\(E\)) distribution satisfies the wave equation [20; 22]:  
\[\
abla\times{\mu_{0}}^{-1}(\
abla\times E)-{k_{0}}^{2}(\varepsilon_{\rm r}- \frac{i\sigma}{\omega\varepsilon_{0}})E=0, \tag{7}\]  
where \(\mu_{0}\) is vacuum permeability, \(k_{0}\) is the vacuum wavenumber, \(\varepsilon_{\rm r}\) is relative permittivity, \(\varepsilon_{0}\) is the vacuum permittivity, and \(\sigma\) is the plasma conductivity, which is a full tensor determined by the electron density collision frequency and the static magnetic flux density. Define the following parameters:  
\[\alpha=\frac{q}{m_{\rm e}(v_{\rm e}+i\omega)}, \tag{8}\]where \(q\) is the electron charge, \(m_{\rm e}\) is the electron mass, and \(v_{\rm e}\) is the collision frequency between electrons and neutral particles. The inverse matrix of plasma conductivity can be expressed as:  
\[qn_{\rm e}\sigma^{-1}= \begin{bmatrix}-1&-\alpha B_{\rm z}&\alpha B_{\rm y}\\ \alpha B_{\rm z}&1&-\alpha B_{\rm x}\\ -\alpha B_{\rm y}&\alpha B_{\rm x}&1\end{bmatrix} , \tag{9}\]  
The drift-diffusion equation can be used to determine the electron density and electron energy density if the electron density is constant in the resonance region, and the Debye length is much smaller than the microwave interaction length [21; 23]:  
\[\frac{\partial}{\partial t}(n_{\rm e})+\
abla\cdot[-n_{\rm e}(\mu_{\rm e}\cdot E )-D_{\rm e}\cdot\
abla n_{\rm e}]=R_{\rm e}, \tag{10}\]  
\[\frac{\partial}{\partial t}(n_{\rm e})+\
abla\cdot[-n_{\rm e}(\mu_{\rm e}\cdot E )-D_{\rm e}\cdot\
abla n_{\rm e}]+E\cdot\Gamma_{\rm e}=R_{\rm e}, \tag{11}\]  
where \(E\cdot\Gamma_{\rm e}\) is the heating term, which comes from microwave heating and the motion of electrons in the bipolar field. \(R_{\rm e}\) is the electron source term, \(R_{\rm e}\) is the energy loss due to inelastic collisions, \(D_{\rm e}\) is the electron diffusion coefficient, \(D_{\rm e}\) is the electron energy diffusion coefficient, \(\mu_{\rm e}\) is the electron energy mobility, and \(\mu_{\rm e}\) is the electron mobility, which are full tensors affected by the static magnetic flux density; the specific expressions are as follows:  
\[D_{\rm e}=\mu_{\rm e}T_{\rm e};\ D_{\rm e}=\mu_{\rm e}T_{\rm e};\ \mu_{\rm e}=(5/3)\mu_{\rm e}. \tag{12}\]  
According to the above formula, different boundary conditions and the static magnetic field distribution inside the discharge chamber will change the conductivity, the mobility, and the diffusion coefficient of the plasma, resulting in ionization characteristics. The factors affecting the ionization characteristics can be studied by analyzing the electron density and temperature of the plasma. The structure of the physical model used to perform the plasma calculations is shown in Figure 2b. It consists of a microwave window, a mounting barrel, a discharge chamber, and permanent magnet rings. The microwave window is composed of three pieces of ceramics with a thickness of 10 mm and a piece of BN with a thickness of 2 mm. This configuration effectively couples the microwave into the discharge chamber and can be easily replaced at a later stage.