## 1 Introduction  
High-power microwaves (HPMs) are frequently used for disturbing or attacking various aircrafts. Absorptive coatings on aircraft surfaces can be heated at particular frequencies by high-power microwaves radiated by antenna arrays on the ground. The heated coatings can be damaged and even removed under bombardment from high-speed air molecules. To investigate such an effect experimentally, high-speed airflows and high-power density microwaves in a wind tunnel are required. Figure 1 shows the HPM radiation effect experiment system [1]. The wall of the wind tunnel is considered as a PEC metal. The high-speed airflow moves along the longitudinal z-direction. The high-power microwave can be typically generated by vacuum electronic sources, such as klystrons or traveling wave tubes. The continuous-wave (CW) microwave power moves into a wind tunnel as a beam formed by a focusing antenna. In the high-power density microwave damaging experiment, the microwave source can be a klystron, delivering 700 kW of CW waves [2]. If the microwave is again reflected by the coated metal plate and the metal wall, the microwave power will return back to the radiation source, which could be harmful to the operation of the source. In practice, it is acceptable that the returned power is \(\sim\)1% of the output power of the HPM source, and the less the better. Separation devices for blocking the reflected microwave power have to face the high-power effect, which can result in over-heating and life-shortening. Usual absorbing materials and microwave meta-materials can hardly be applied due to insufficient robustness to stay firmly under the strike of high-speed airflows [3; 4]. The airflows in the wind tunnel have supersonic speed. The temperature in the wind tunnel varies from \(-20\)\({}^{\circ}\)C to \(30\)\({}^{\circ}\)C and the pressure will not exceed the range of [50 kPa, 400 kPa]. Two-dimensional dielectric bar arrays have been investigated as a microwave scatter [5; 6; 7]. They can be attached to the metal wall of the wind tunnel and work well. Previously, [PERSON] et al. published a method to analyze the problem of electromagnetic wave scattering by a double-periodic grating, in which the reflected power as functions of frequency and direction angle can be obtained [5]. However, more physical insights such as the excited modes cannot be solved. [PERSON] et al. presented a semi-numerical method to analyze the modes of multi-layer 2-D periodic structures [6]. The thin metal patterns can be arbitrarily chosen, but the dielectric layers are assumed to be isotropically uniform. [PERSON] and [PERSON] studied the electromagnetic wave scattering by a 1-D array of axially magnetized plasma columns [7]. The resonance scattering mechanism was analyzed and discussed. The presented method was not extended to 2-D periodic problems. The 2-D dielectric bar array under concern consists of parallel bars that have two dielectric blocks in each period, as shown in Figure 2. The incident wave travels approximately toward the \(-x\) direction, as shown in Figure 1. The periodicity of the bars can provide directional reflection, which cannot be obtained using uniform bars. Theoretically, the reflection is not related to the azimuth angle when only one mode is excited because of the uniform phase distribution on the surface plane. The periodic bar as shown in Figure 2 is expected to support at least two modes so that the reflection properties can be controlled by adjusting the geometric sizes. Figure 1: Illustration of HPM radiation effect experiment system [1]. Figure 2: Illustration of the 2-D dielectric bar reflect array. To the best of our knowledge, it is unknown how well such a 2-D periodic dielectric reflect array can control the reflected wave. Although the geometry of the 2-D array is simple, the difficulty and complexity in the theoretical mode analysis of arrays of lossy bars are unavoidable [1]. To reveal the mechanism of directional reflection, mode analysis is helpful in finding the physical insight.