## 2 Problem Description
### Mismatch between Plane Wave and Guided Modes  
A coordinate system is selected, as shown in Figure 1. In \(z<y_{0}\) half space, the plane wave propagates in free space and the E-field magnitude of the plane-wave stays to be a constant. The plane wave arrives at the dielectric bar at \(z=z_{0}\). Assume that all the excited modes under plane-wave excitation are guided modes (fields decay exponentially at \(x\) and  
Figure 4: Beat wave pattern under excitation of horn antenna (a) simulation model of a dielectric bar illuminated by a horn antenna (b) field distribution on \(x=0\) plane (c) field distribution on \(y=0\) plane. Figure 3: Beat wave pattern along a lossy dielectric bar. Figure 2: Schematic diagram of dielectric bar under plane-wave excitation (**a**), simulation model using CST Eigenmode Solver (**b**), simulation model using CST Time-Domain Solver (**c**) and Beat wave pattern along a dielectric bar simulated by CST Time-Domain Solver (**d**). \(z\) directions), i.e., radiation modes (approximated as leaky modes) and evanescent modes are negligible. The superposition of the guided modes should satisfy the boundary condition, which requires the continuity of the tangential fields at the interface. Without the loss of generality, let E represent the tangential component of the electric field of TM(\(x\)) mode. Then, a linear combination of the guided modes at \(z=z_{0}\) can be written as  
\[I=\sum_{m}\sum_{n}A_{nm}E_{nm}(x) \tag{1}\]  
Here, the variation of \(E\) versus \(z\) is not in consideration. There are infinite items in Equation (1). Assume that \(A_{mn}\) is the magnitude coefficient when the superposition of \(E_{mn}\) satisfies the continuity of the tangential field component. Let the decay factor of the electric field along \(x\) direction in cross-plane be denoted as decay factor \(\tau\). It is known that the decay factor becomes larger when the mode order increases. Therefore, the fundamental mode has the slowest rate of decay and the higher-order modes decay quicker. Thus, it can be derived from Equation (1) that  
\[I<I_{1}=\sum_{m}\sum_{n}A_{mn}E_{01}(x) \tag{2}\]  
Moreover, there is a maximum \(A\) because all the magnitude coefficients should be finite real numbers. Let \(A_{\text{max}}\) and \(\tau_{\text{min}}\) denote the maximum magnitude coefficient and the decay factor of the lowest mode, respectively. Equation (2) can be further derived as  
\[I<I_{1}<I_{2}=\sum_{i}A_{\text{max}}e^{-\tau_{\text{min}}x} \tag{3}\]  
where \(i\) denotes the mode number for convenience. Further consider the first \(n\)th items in Equation (3) and let \(n\) tend to infinity, then it can be obtained that  
\[I_{2}=\lim_{n,x\rightarrow\infty}nA_{\text{max}}e^{-\tau_{\text{min}}x}=\lim_ {n\rightarrow\infty}nA_{\text{max}}e^{-\tau_{\text{min}}n\delta_{x}} \tag{4}\]  
where \(\delta_{x}\) is any specific small quantity and \(\tau_{\text{min}}\) is a positive real number. It can be found that Equation (4) tends to be a geometric series when \(n\) increases to infinity, and the common ratio tends to be \(\exp(-\tau_{\text{min}}\delta_{x})\). It is proved that \(I_{2}\) tends to be zero at infinity, but the field of the plane wave stays a constant value at infinity. As a result, any a linear combination of the guided modes propagating along the dielectric bar cannot match the \(E\)-field magnitude of the incident plane wave at \(z=z_{0}\). A similar conclusion can be obtained when the dielectric bar is excited by the radiation from a horn antenna as shown in Figure 4. The incident field in transverse direction is polynomial, because the traditional directivity of the antenna has a form of \((\cos\theta)^{q}\). However, the guided modes under open boundary experience an exponential decay. The incident field experiences a slower decay in the same direction as that of the guided modes along the bar under open boundary. There is still a mismatch between the tangential electric field of the incident wave and the guided modes.