# Galerkin's Spectral Method in the Analysis of Antenna Wall Operation
## 2 Radiation Field of Unconfined Antenna Array Placed on Multilayer Dielectric  
The geometry of the analyzed antenna array and the adopted coordinate system are shown in Figure 1. It shows an antenna located in free space, and the case of the antenna location on the boundary of two different dielectric media is specified only by the form of Green's function. In this work, the special case of a rectangular antenna grid is considered, i.e., \(\Omega=\pi\Omega\), although generalization of the analysis to the case of a skew antenna grid \(\Omega\
eq\pi\Omega\) does not change the presented procedure for solving the scattering problem. The relationship between the scattered fields in the observation plane (\(x\),\(y\),\(z\))  
\[\underline{E}_{t}^{s}(x,y,z)=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} \underline{E}(\underline{E}_{t}^{s})\big{(}k_{x},k_{y},z)exp\big{\{}i[k_{x}x +k_{y}y]\big{\}}dk_{x}dk_{y} \tag{1}\]  
and the scattered field in the antenna plane (\(x\),\(y\),\(0\))  
\[\underline{E}_{t}^{s}(x,y,0)=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} \underline{E}(\underline{E}_{t}^{s})\big{(}k_{x},k_{y},0)exp\big{\{}i[k_{x}x +k_{y}y]\big{\}}dk_{x}dk_{y} \tag{2}\]  
is expressed in spectral space by the phase propagation factor \(exp(ik_{z}z)\) relating their Fourier transforms as follows:  
\[ \begin{array}{c}\underline{E}(\underline{E}_{t}^{s})\big{(}k_{x},k_{y},z)= \underline{E}(\underline{E}_{t}^{s})\big{(}k_{x},k_{y},0\big{)}exp(ik_{z}z)\\ =(2\pi)^{-2}\!\!\int\int\underline{E}_{t}^{s}(x^{\prime},y^{\prime},0)exp\big{\{ -i[k_{x}x^{\prime}+k_{y}y^{\prime}+k_{z}z]\big{\}}dx^{\prime}dy^{\prime}\end{array}  \tag{3}\]  
The dispersion relationship in these equations is as follows:  
\[k_{z}=\sqrt{k^{2}-k_{x}^{2}-k_{y}^{2}};\quad Re(k_{z})\geq 0;\quad Im(k_{z})\geq 0 \tag{4}\]  
Figure 1: Periodic two-dimensional antenna system. This results from the Helmholtz equation in the free (homogeneous isotropic) half-space over the antenna structure:  
\[\left[\partial_{z}^{2}+\partial_{x}^{2}+\partial_{z}^{2}+k^{2}\right]\underline {E}_{t}^{s}(x,y,z)=0 \tag{5}\]  
This is fulfilled in the free (homogeneous, isotropic) half-space above the antenna structure by the scattered field. Thus, the scattered field in the observation plane \(z=const.\) is expressed by the appropriate integral transform from the distribution of the scattered field in the antenna plane \(z=0\) :  
\[ \begin{array}{c}\underline{E}_{t}^{s}(x,y,z)=\int_{-\infty}^{+\infty}\int_{ -\infty}^{+\infty}\underline{E}(\underline{E}_{t}^{s})\big{(}k_{x},k_{y},0 \big{)}\,exp\big{(}ik_{z}z\big{)}exp\big{\{}i\big{[}k_{x}x+k_{y}y\big{]}\, \big{\}}dk_{x}dk_{y}\\ =(2\pi)^{-2}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\int_{-\infty}^{+ \infty}\int_{-\infty}^{+\infty}\underline{E}_{t}^{s}(x^{\prime},y^{\prime},0 )exp\big{\{}i\big{[}k_{x}(x-x^{\prime})+k_{y}(y-y^{\prime})+k_{z}z\big{]} \,\big{\}}dx^{\prime}dy^{\prime}dk_{x}dk_{y}\end{array}  \tag{6}\]  
The above formula is valid for any distribution of the scattered field in the plane \(z=0\). Then, we introduce the division of the antenna surface in the plane \(z=0\) into identical cells numbered by integers (\(m\),\(n\)), where the central point (\(x_{0}\),\(y_{0}\)) of the central cell (0,0) is located at the origin of the coordinate system \(x=0\), \(y=0\), and the points \(x=x_{m}\), \(y=y_{n}\) are the central points of the subsequent cells (\(m\),\(n\)).