## 2 Theoretical Mode Analysis of 2-D Periodic Structure
Therefore, the lowest propagation mode is excited within the high-permittivity region, as shown in Figure 6, rather than the low-permittivity region, because the transverse size of the high-permittivity dielectric waveguide, denoted by \(l_{2}(\varepsilon_{h})^{0.5}\), is larger than that of the low-permittivity one and so is the cut-off wavelength. Figure 4: Propagation in the dielectric bar waveguide along the \(-x\) direction: (**a**) illustration of geometry and (**b**) simulation model using periodic boundary. \(w\) is the width of the bars in the \(y\) direction; \(l_{1}\) and \(l_{2}\) are the sizes of the high-permittivity and low-permittivity materials, respectively; and \(\varepsilon_{1}\) and \(\varepsilon_{h}\) are the permittivities of the materials and \(\varepsilon_{1}<\varepsilon_{h}\). The lowest guiding mode and the lowest leaky mode can be analyzed theoretically by [PERSON]'s method [11]. The method deals with the Helmholtz equation in a scalar form using the Borgnis function of the electric potential. Open boundaries in all directions are essential for applying this theoretical method. If other boundaries are applied, such as periodic boundaries or PEC boundaries, the obtained equations will be over-determined. The space inside and outside the central dielectric region can be separated into nine regions, as shown in Figure 6. Similar derivations have been described in Refs. [1, 11, 12, 13, 14, 15], and the equations to solve the wave numbers of the guiding mode can be expressed by  
\[k_{zi}^{2}+k_{yi}^{2}+\beta^{2}=k_{i}^{2}=\omega^{2}\mu_{0}\varepsilon_{i},\ i=1,\ 2,\ 3 \tag{1}\]  
\[k_{y1}=k_{y3} \tag{2}\]  
\[k_{z1}=k_{z2} \tag{3}\]  
\[\tanh(-jk_{y1}y+j\psi)=\frac{\varepsilon_{1}k_{y2}}{\varepsilon_{0}k_{y1}},y=a \tag{4}\]  
\[R(\theta)=e^{-2\eta\cos\theta\int_{0}^{d}\sigma_{x}(x)dx}\]0  
\[R(\theta)=e^{-2\eta\cos\theta\int_{0}^{d}\sigma_{x}(x)dx}\]1 Electric field distribution under plane wave excitation at different frequencies: (**a**) electric field at 2 GHz, (**b**) electric field at 4 GHz, (**c**) electric field at 6 GHz, (**d**) field strength of \(E_{y}\) component at 2 GHz, (**e**) field strength of \(E_{y}\) component at 4 GHz, and (**f**) field strength of \(E_{y}\) component at 6 GHz. \[R(\theta)=e^{-2\eta\cos\theta\int_{0}^{d}\sigma_{x}(x)dx}\]2 Region division for the theoretical analysis. However, for the leaky mode, the field grows in the corner region and decays in the third region denoted by \(\varepsilon_{3}\); thus, the equations for solving the leaky mode can be modified as  
\[R(\theta)=e^{-2\eta\cos\theta\int_{0}^{d}\sigma_{x}(x)dx}\]3  
\[R(\theta)=e^{-2\eta\cos\theta\int_{0}^{d}\sigma_{x}(x)dx}\]4  
\[R(\theta)=e^{-2\eta\cos\theta\int_{0}^{d}\sigma_{x}(x)dx}\]5  
\[R(\theta)=e^{-2\eta\cos\theta\int_{0}^{d}\sigma_{x}(x)dx}\]6  
\[R(\theta)=e^{-2\eta\cos\theta\int_{0}^{d}\sigma_{x}(x)dx}\]7  
where Equation (10) is assumed so that the field strength outside the high-permittivity region can be approximated by zero. Equations (6)-(10) consist of seven complex wave numbers that are unknowns to be solved and seven complex equations, as are Equations (1)-(5). Therefore, the guiding and leaky modes can be solved. The calculated period lengths of the beat pattern were compared to the simulation results in \[R(\theta)=e^{-2\eta\cos\theta\int_{0}^{d}\sigma_{x}(x)dx}\]8 In CST Microwave Studio, the frequency range was set to be [8.7 GHz, 10.7 GHz]. A single Gaussian pulse had a duration of \(\sim\)0.3 ns. The simulation time was no longer than 20 pulses. For good precision, the grid size in the dielectric bars in the _yoz_ plane was \(\sim\)1.3 mm and the grid size along \(x\) direction was \(\sim\)1.6 mm. The structure was simulated under periodic boundaries for transversal directions and the total experiment system was simulated under open boundaries, as will be shown in Section 4. The beat wave patterns are shown in Figure 7 using the simulation model shown in Figure 4b. Acceptable relative errors are obtained at the frequencies of 8.2 GHz, 8.7 GHz, and 9.2 GHz.