## 2 Physical Model Analysis  
The high-frequency structure of the EIO consists of coupling slots, multiple gaps, and an electron beam channel. This study proposes an EIO dual-cavity coupled high-frequency structure with five gaps, and its electric field distribution is shown in Figure 1, consisting of a coupling cavity shared by one EIO operating at 94 GHz and 140 GHz. A sheet electron beam is used to minimize space charge effects and reduce the need for a focused magnetic field, while simultaneously expanding the interaction area between the electron beam and the electric field, thus enhancing circuit efficiency. The coupling slots on both sides of the high-frequency gap couple the longitudinal electric field of the gaps and effectively modulate the electron beam. The dimensions of the EIO are provided in Table 1.  
The electrons in the resonant cavity move along the electron beam channel towards the z-axis direction and are constrained by a uniform magnetic field of 0.5 T. The background material is set to copper with a conductivity of \(2.8\times 10^{7}\) S/m [16]. The operating mode of the EIO in this work is the \(2\pi\) mode. Compared to the \(\pi\) mode, the \(2\pi\) mode offers a higher characteristic impedance than the \(\pi\) mode, which contributes to enhancing electron efficiency and output power [17]. The TM11 mode is characterized by the strongest electric field component along the propagation direction of the electron beam at high frequencies. Therefore, the TM11-2\(\pi\) mode was chosen as the operating mode of the EIO in this study. Figure 1: Physical model of the resonant cavity. Additionally, the larger structure size associated with this mode makes it easier to manufacture. The period of the resonant cavity can be calculated by the following formula [18;19]:  
\[\beta_{e}p=m\pi \tag{1}\]  
while  
\[\beta_{e}=\frac{\omega}{v_{0}} \tag{2}\]  
\[v_{0}=c\sqrt{1-\frac{1}{\left(1+\frac{U}{5\Pi}\right)^{2}}}, \tag{3}\]  
where \(v_{0}\) is the electron beam velocity and \(U\) is operating voltage. The coupling coefficient \(M\) is used to characterize the degree of modulation experienced by the electron beam, which is defined as the ratio of the modulation voltage applied to the electron beam to the actual modulation voltage across the gap. The efficiency of the interaction between the electron beam and the electric field is described as characteristic impedance \(R/Q\)[20]. Their expressions are given as follows:  
\[M=\frac{\int_{-\infty}^{+\infty}f(z)e^{i\beta_{e}z}dz}{\int_{-\infty}^{+\infty }f(z)dz} \tag{4}\]  
\[R/Q=\frac{V_{m}}{2\omega_{0}W_{\mathrm{s}}}=\frac{\left[\int_{a}^{b}Edl\right] ^{2}}{\omega_{0}\int_{V}\left|E\right|^{2}dV} \tag{5}\]  
where \(E\) is the axial electric field, \(\omega\) is the resonant angular frequency, \(f(z)\) is the normalized field distribution function, \(V_{m}\) is the amplitude of the gap electric field, and \(W_{\mathrm{s}}\) is the energy stored in the gap. The effective characteristic impedance \(M^{2}R/Q\) is used to evaluate beam-wave interaction as the interaction strength between the characteristic impedance and coupling coefficient. The \(M^{2}R/Q\) variation versus voltage is shown in Figure 2. From this, it can be seen that 94 GHz and 140 GHz obtain the maximum effective characteristic impedance at operating voltages of 34 kV and 56 kV, respectively. For the \(2\pi\) mode, its cutoff frequency is mainly determined by \(gx\)[21]. As shown in Figure 3, with other parameters held constant, the resonant frequency of the gap corresponds to different \(gx\) variations, and with increasing \(gx\), the resonant frequency decreases, and the characteristic impedance gradually reaches a certain optimal value. Therefore, the operating frequency can be significantly adjusted during the design process by adjusting the gap narrow edge size \(gx\).