# Radiation from a Cavity-Backed Circular Aperture Array Antenna Enclosed by an FSS Radome
## 2 Field Analysis  
Figure 1 shows the problem geometry. Assume that the \(z\)-oriented electric point source is located in a circular cavity with multiple circular apertures in a conducting plane. \(\epsilon_{n}\) and \(\delta_{n}\) are dielectric constant and loss tangent of each layer (\(n=1,2,3,4,5\)). Electromagnetic properties and size of each layer are shown in Table 1.  
Figure 2 shows an entire analysis procedure for the radiation from a cavity-backed circular aperture array antenna with the FSS radome based on the hybrid analysis method combining the mode matching method, the ray tracing technique, and HFSS, a commercial EM simulator, based on FEM. Firstly, for analysis of the antenna, we solved the electromagnetic boundary-value problem of the circular cavity with the circular aperture array antenna based on the mode matching method in our previous study [19]. The mode matching method provides an advantage in the fact that its solution is rigorous and theoretically robust. Moreover, it is time-efficient in an analysis of open-boundary problems compared to other numerical techniques such as FEM, FDTD, and MoM, to name a few. An analysis using the mode matching method takes the following steps. Above all, the whole region to be solved is divided into sub-regions on a basis of boundary. Electromagnetic fields are defined in each region. Afterwards, boundary conditions are enforced to obtain a set of simultaneous equations for modal coefficients. Matrix calculation enables an evaluation of the modal coefficients and electromagnetic fields in all regions can be calculated based on the obtained modal coefficients. Note that the mode matching method should be used in the separable coordinate systems where eigen-modes can be defined in all regions. By using the tangential electromagnetic field established from the mode matching method and the surface equivalence theorem [20], we can derive the equivalent magnetic surface currents on the apertures from the aperture fields, which will be used in a formation of rays. Meanwhile, the FSS layer is a cross-loop dipole FSS which has a passband resonant frequency at 10 GHz. A detailed configuration and size of the designed FSS can be found in the reference [10]. Reflection and transmission coefficients of the designed FSS layer are obtained via numerous simulations using HFSS, an electromagnetic full-wave simulator. Reflection and transmission coefficients between dielectric layers, not involved with FSS layer, can be determined on a polarization basis, as described in the reference [10]. Afterwards, to apply the ray tracing technique for the analysis of the radiation from the cavity-backed circular aperture antenna enclosed by the FSS \begin{table}
\begin{tabular}{c c c c} \hline
**Parameters** & **Value** & **Parameters** & **Value** \\ \hline D\({}_{1}\) & 284.8 mm & L\({}_{1}\) & 290.5 mm \\ D\({}_{2}\) & 286.4 mm & L\({}_{2}\) & 291.5 mm \\ D\({}_{3}\) & 292.3 mm & L\({}_{3}\) & 295.2 mm \\ D\({}_{4}\) & 292.4 mm & L\({}_{4}\) & 295.3 mm \\ D\({}_{5}\) & 298.4 mm & L\({}_{5}\) & 299.1 mm \\ D\({}_{6}\) & 300 mm & L\({}_{6}\) & 300 mm \\ \(\epsilon_{r1}\) & 4.35 & \(\delta_{1}\) & 0.0032 \\ \(\epsilon_{r2}\) & 1 & \(\delta_{2}\) & 0.0038 \\ \(\epsilon_{r3}\) & 4.4 & \(\delta_{3}\) & 0.02 \\ \(\epsilon_{r4}\) & 1 & \(\delta_{4}\) & 0.0038 \\ \(\epsilon_{r5}\) & 4.35 & \(\delta_{5}\) & 0.0032 \\ \hline \end{tabular}
\end{table} 
Table 1: Design parameters of the frequency selective surface (FSS) radome. Figure 1: Problem geometry. radome, we determined intercept points of the rays and the surfaces of the FSS radome using the iterative method [1]. Then, we can generate the rays from the equivalent magnetic surface currents on the apertures and trace the rays passing through the radome. When analyzing the multi-layer FSS radome, we used the ray tracing technique and the reflection and transmission coefficients at the FSS layer from the simulation of HFSS based on FEM [21].