# Galerkin's Spectral Method in the Analysis of Antenna Wall Operation
## 2 Radiation Field of Unconfined Antenna Array Placed on Multilayer Dielectric
The maximum of the array factors is obtained from the following condition:  
\[y^{\prime}-y_{m}=y^{\prime} \tag{31}\]0which corresponds to the propagation angle \(\theta=\theta_{Max}\), where  
\[y^{\prime}-y_{m}=y^{\prime} \tag{31}\]1  
\(m=0\), 1, 2..., zeros of the array factor occur when the condition is met:  
\[y^{\prime}-y_{m}=y^{\prime} \tag{31}\]2  
This corresponds to the propagation angle \(\Theta\)\(=\Theta_{0}\), where  
\[y^{\prime}-y_{m}=y^{\prime} \tag{31}\]3  
\(m^{\prime}=0\), 1, 2,..., excluding the value \(m^{\prime}=m(2M+1)\). For perpendicular excitation, i.e., without phase modulation, for sufficiently small values of the array constantly satisfying the condition,  
\[y^{\prime}-y_{m}=y^{\prime} \tag{31}\]4  
The array factor consists solely of the main beam propagating perpendicularly to the antenna surface. Thus, with the increase in the number of array elements or the increase in the field frequency, the directional selectivity of the antenna increases. Hence, the name _Frequency Selective Surfaces_ is assigned to the periodic antenna structures considered here. In the limiting case of an unlimited antenna wall, we obtain the following:  
\[ \begin{array}{c}(2M+1)(2N+1)Af(k_{x},k_{y})\frac{}{2M+1\rightarrow\infty\ 2N+ 1\rightarrow\infty}\pi^{2}\delta(\phi_{x}/2)\delta(\phi_{y}/2)=\\ =\pi^{2}\delta\left(\frac{(k_{x}-k_{x}^{i})d_{x}}{2}\right)\delta\left(\frac{( k_{y}-k_{y}^{i})d_{y}}{2}\right)\end{array}  \tag{54}\]  
Then,  
\[E_{t\Lambda MN}^{s}\frac{}{2M+1\rightarrow\infty\ 2N+1\rightarrow\infty}0 \tag{55}\]  
and the scattered field in the observation plane is expressed by a reflected plane wave and appropriately modified by the Fourier transform of the scattered field distribution on the central cell:  
\[ \begin{array}{c}E_{t}^{s}(x,y,z)=\frac{4\pi^{2}}{d_{x}d_{y}}F(E_{t00}^{s})(k_ {x}^{i},k_{y}^{i},0)exp[i(k_{x}^{i}x+k_{y}^{i}y+k_{z}^{i}z)]\\ =\frac{4\pi^{2}}{d_{x}d_{y}}F(E_{t00}^{s})(ksin\phi^{i}sin\phi^{i},ksin\phi^{i }cos\phi^{i},0)exp[iksin\phi^{i}(xsin\phi^{i}+ycos\phi^{i})]\\ \times exp[ikz\sqrt{1-sin^{2}\theta^{i}(sin^{2}\phi^{i}+cos^{2}\phi^{i})}]. \end{array}  \tag{56}\]  
It follows from this that the amplitude of the field propagating in a given direction depends on the Fourier transform of the field distribution scattered on the central cell and is inversely proportional to the cell area. At the same time, the direction of propagation is determined by the phasing of the excitation of individual cells determined by the angles \(\theta^{i}\) i \(\phi^{i}\).i. From the above formula, it follows that for an unconstrained grid composed of point sources,  
\[E_{t}^{s}(x,y,0)=d_{x}d_{y}\sum_{m,n}\delta(x-x_{m})\delta(y-y_{n}) \tag{57}\]  
The scattered field is a spectral reflection of the incident field:  
\[E_{t}^{s}(x,y,z)=exp[i\left(k_{x}^{i}x+k_{y}^{i}y+k_{z}^{i}z\right)] \tag{58}\]  
\[F(E_{t00}^{s})=\left(2\pi\right)^{-2}d_{x}d_{y} \tag{59}\]In another limiting case of an antenna grid of finite dimensions, but with a concentrated field spectrum (different from zero) in the vicinity of the geometrically reflected field spectrum, i.e., when the following conditions are met:  
\[\phi_{x}\cong 0 \tag{60}\]  
\[\phi_{y}\cong 0 \tag{61}\]  
the array factor is approximately equal to unity:  
\[Af\big{(}k_{x},k_{y}\big{)}\cong 1 \tag{62}\]  
and the spectrum of the scattered field is expressed by the spectral distribution of the field in the area of the central cell, appropriately modified (multiplied) by the number of cells:  
\[\underline{E}_{t}^{s}(x,y,z)\ \cong(2M+1)(2N+1)\int_{-\infty}^{+\infty}\int_{- \infty}^{+\infty}F(\underline{E}_{t00}^{s})(k_{x},k_{y},0)e^{ik_{z}z}e^{i(k_{ x}x+kyy)}dk_{x}dk_{y}. \tag{63}\]