## 3 Mode Analysis of Beat Wave Pattern
### Analysis of Propagation Characteristics of Guided Wave  
The theoretical analysis presented by [PERSON] [18] has been proved to be quite accurate when the frequency is bigger than the cut-off frequency of a guided mode. Field matching theory is applied for solving the propagation problem. The space inside and outside the wave-guide can be separated into 9 regions, as shown in Figure 6.  
Figure 5: Amplitude of electric field along \(y\) axis when the edge length of the losses dielectric bar is 14.4 mm and the incident plane wave is at 10 GHz.  
Figure 6: Region division for theoretical analysis.  
Considering the specific excitation in this problem, the propagating mode of concern, noted as TM\({}^{(y)}\) mode, consists mainly of \(E_{y}\) and \(H_{x}\) components. Under [PERSON]'s assumption, the Bergnis potential functions can be written as  
\[U_{1}^{(y)}=A_{1}\cosh(-jk_{x1}x+j\varphi)\cos(-jk_{y1}y+j\psi)e^{-j\beta z} \tag{5}\]  
\[U_{2}^{(y)}=A_{2}\cosh(jk_{x1}x+j\varphi)e^{-jk_{y2}y}e^{-j\beta z} \tag{6}\]  
\[U_{3}^{(y)}=A_{3}e^{-jk_{x3}x}\cosh(jk_{y1}y+j\psi)e^{-j\beta z} \tag{7}\]  
The Bergonis potential functions in the fourth and fifth regions are neglected due to the symmetry of the problem. The relations between Bergonis potential functions and field components are given as  
\[E_{xi}=\frac{\partial^{2}U_{i}^{(y)}}{\partial x\partial y} \tag{8}\]  
\[E_{yi}=\frac{\partial^{2}U_{i}^{(y)}}{\partial y^{2}}+k_{i}^{2}U_{i}^{(y)}= \Big{(}\beta^{2}+k_{xi}^{2}\Big{)}U_{i}^{(y)} \tag{9}\]  
\[E_{zi}=\frac{\partial^{2}U_{i}^{(y)}}{\partial z\partial y}=-j\beta\frac{ \partial U_{i}^{(y)}}{\partial y} \tag{10}\]  
\[H_{xi}=-j\omega\varepsilon_{i}\frac{\partial U_{i}^{(y)}}{\partial z}=- \omega\varepsilon_{i}\beta U_{i}^{(y)} \tag{11}\]  
\[H_{zi}=j\omega\varepsilon_{i}\frac{\partial U_{i}^{(y)}}{\partial x} \tag{12}\]  
From the wave equations in each region and the phase matching condition on the boundary, the relations between wave numbers in these regions are given by  
\[k_{xi}^{2}+k_{yi}^{2}+\beta^{2}=k_{i}^{2}=\omega^{2}\mu_{0}\varepsilon_{i}, \;i=1,2,3 \tag{13}\]  
\[k_{y1}=k_{y3} \tag{14}\]  
\[k_{x1}=k_{x2} \tag{15}\]  
where \(\varepsilon_{2}=\varepsilon_{3}=\varepsilon_{0}\). Using Equations (5)-(15) and applying the boundary conditions for tangential field components, the dispersion relationships of the guided modes can be observed by numerically solving  
\[\tanh(-jk_{y1}y+j\psi)=\frac{\varepsilon_{1}k_{y2}}{\varepsilon_{0}k_{y1}},y=a \tag{16}\]  
\[\tanh(-jk_{x1}x+j\varphi)=\frac{\varepsilon_{0}k_{x3}}{\varepsilon_{1}k_{x1} }\frac{\beta^{2}+k_{x1}^{2}}{\beta^{2}+k_{x3}^{2}},x=a \tag{17}\]  
Equations (13)\(-\)(17) consist of seven complex wave numbers that are unknowns to be solved and seven complex equations. Therefore, we can solve them numerically. The simulated and calculated real parts of the longitudinal phase constants are listed in Table 1. The edge length a is set to be 14.4 mm. Acceptable relative errors are obtained in the frequency range from 10\(-\)11 GHz. The field matching theory has enough precision for analyzing our problem.  
The comparison between the simulated and calculated field distribution is shown in Figure 7. It can be observed that the neglecting of the field in the corners in Figure 6 does not heavily influence the field pattern.