## 2 Theoretical Mode Analysis of 2-D Periodic Structure  
Under the experiment setup shown in Figure 1, the incident wave on the 2-D reflect array can be approximately considered as a plane wave toward the \(-x\) direction when the coated metal plate is rotated at an appropriate angle. Figure 3 shows the distribution of the microwave power reflected by the coated plate and the electric field distribution plotted as small arrows. Focusing antennas are suitable for mainly two reasons: (i) The power densities are enhanced significantly in the focal spots; so, high-power density is obtained for the microwave damage experiment. (ii) The microwave forms a beam with limited focusing spot so that it can be absorbed effectively in the absorbing coatings. In this work, a low-cost dielectric lens antenna was chosen. The design of this antenna followed the simple principle of light path compensation. The related theory and formula are presented in [10]. The material of the coating attached to the testing target is also modeled. An equivalent absorptive layer modeled in CST Microwave Studio can absorb  
Figure 3: Incident microwave after reflection: (**a**) \(E_{y}\) component on the _xoy_ plane, (**b**) power flow on the _xoy_ plane and (**c**) simulation model of the test target. simulations in this work were performed using the time-domain solver in CST Microwave Studio. The absorption rate is just as same as modern absorptive coatings on aircraft. Figure 3c shows the geometrical structure of the model of the coatings. The gray part is a conductive metal that can reflect any incident microwave; the green part is the equivalent artificial material. The required reflection coefficient is 50% when the direction of the incident wave is perpendicular to the coating surface. In previous studies on artificial absorptive layers, the reflection coefficient is formulated by  
\[R(\theta)=e^{-2\eta\cos\theta\int_{0}^{d}\sigma_{x}(x)dx}\]  
where \(\theta\) is the incident angle, \(\eta\) is the wave impedance of the absorptive material, and \(\sigma\) is the conductivity and is invariant as a function of \(x\) in this problem. The conductivity relates to the tangent loss, which was set to 34.5 in this model. The reflection coefficient of this model equals 50% when its thickness is 3 mm. The excited modes will propagate along the \(-x\) direction due to the equal-phase field distribution on the top surface, i.e., the perpendicular incidence. The propagation properties can be investigated by increasing the structure size in the \(-x\) direction and removing the PEC boundary, as shown in Figure 4. Guiding modes and leaky modes can be excited and propagate along the \(-x\) direction, as presented in Ref. [6], to match the boundary condition. The simulation results shown in Figure 5 verify the propagation of the modes and the beat wave pattern. In Figure 5, it can be found that the field pattern changes when the frequency of the incident plane wave increases. The arrows in Figure 5a-c show that \(E_{y}\) is the main component of the electric field. In Figure 5a,d, the electric field of an evanescent mode decays in the dielectric region for the lower frequency; however, in Figure 5c,f, it forms a sinusoidal distribution for the higher frequency, which means that the propagation mode was excited. It should be noticed that the sinusoidal field distributions cannot exist in both two dielectric regions for any propagation mode. Considering the boundary at \(y=b\), as shown in Figure 6, the \(E_{z}\) fields in the higher side for \(y>b\) will be continuous. The \(E_{z}\) field in the lower side for \(y<b\) is required to be continuous due to the boundary condition expressed by  
\[E_{z}\left|{}_{y=b+\Delta}=E_{z}\right|_{y=b-\Delta},\Delta\to 0\,,\]  
however, the \(E_{z}\) fields in the lower side for \(y<b\) will be discontinuous at \(z=a\) due to the different permittivities of the dielectric materials. This discrepancy limits the field pattern of the excited mode.