## 2 Statement of the Problem  
We consider the composite periodic antenna structure shown in Figure 1. The system consists of many-layered elements bounded at the top and bottom by half-spaces (upper and lower) of homogeneous dielectric media. The third axis (\(y\)) of the orthogonal coordinate system (\(x\),\(y\),\(z\)) is perpendicular to the drawing plane. The basic components of the structure are: metallized, infinitely periodic in the plane (\(x\),\(y\)) antenna surface, the boundary between two dielectrics, and a homogeneous dielectric layer [5,6]. The periodic flat antenna structure is shown in Figure 1. The geometry of a single antenna element remains arbitrary. The distance between the individual antenna elements in the \(x\) and \(y\) directions is \(d_{x}\) and \(d_{y}\), respectively. The inclination angles of the symmetry axis of the antenna structure with respect to the \(x\) axis are respectively 0 and \(\Omega=90^{\circ}-\Theta\). Let us assume that the designations of the multi-layer antenna system are shown in Figure 2.  
The matrix \(\underline{\xi}^{(i)}\) or \(\underline{T}^{(j)}\) represents the planar discrete element of the antenna in the form of a flat metallized grid, a boundary between two dielectric media or a dielectric layer [7]. It was assumed that \(a_{j}^{\prime}=b_{j+1},\ a_{j}=b_{j+1}^{\prime}\). Vectors, \(a_{j}=\left[a_{j1},a_{j2},\ldots a_{j_{j}}\right]b_{j}=\left[b_{j1},b_{j2},b_{ j_{j}}\right]a_{j}^{\prime}=\left[a_{j1}^{\prime},a_{j1}^{\prime},\ldots a_{j_{j }}^{\prime}\right]\), \(b_{j}^{\prime}=\left[b_{j1}^{\prime},b_{j1}^{\prime},b_{j_{j}}^{\prime}\right]\) are the complex amplitudes of the normalized incoming \(\left(a_{j},a_{j}^{\prime}\right)\) and outgoing \(\left(b_{j},b_{j}^{\prime}\right)\) discrete modes of a single structure, represented by a scattering matrix \(\underline{\xi}^{(i)}\), or a transmission matrix \(\underline{T}^{(j)}\). The matrix \(\underline{\xi}^{(i)}\), binds the outgoing \(\left(b_{j},b_{j}^{\prime}\right)\) modes with falling modes \(\left(a_{j},a_{j}^{\prime}\right)\)  
\[\left[ \begin{array}{c}b_{j}\\ b_{j}^{\prime}\end{array} \right]=\left[ \begin{array}{c}S^{(j)}_{-}\\ -1\end{array} \right]\left[ \begin{array}{c}a_{j}\\ a_{j}^{\prime}\end{array} \right]=\left[ \begin{array}{c}S^{(j)}_{-1}\\ S^{(j)}_{-1}\\ S^{(j)}_{-1}\\ S^{(j)}_{-1}\\ -2\end{array} \right]\left[ \begin{array}{c}a_{j}\\ a_{j}^{\prime}\end{array} \right] \tag{1}\]and matrix \(\underline{T}\) connects the field on the \"right \(a_{j}^{(c)},\ b_{j}^{(c)},\ a_{N}^{(c)},\ b_{N}^{(c)}\)\", lower side of the structure with the field on the \"left\", upper side of the structure:  
\[\left[ \begin{array}{c}a_{j}^{(c)}\\ b_{j}^{(c)}\end{array} \right]=\left[\underline{T}^{(j)}\right]\left[ \begin{array} []{c}a_{j}\\ a_{j}^{(c)}\end{array} \right]=\left[ \begin{array}{cc}\underline{T}_{11}^{(j )}&\underline{T}_{21}^{(j)}\\ \underline{T}_{21}^{(j)}&\underline{T}_{22}^{(j)}\end{array} \right]\left[ \begin{array}{c}a_{j}\\ b_{j}\end{array} \right] \tag{2}\]  
Both types of matrices lead to a global description of the operation of the entire antenna structure [8]. Global transmission matrix \(\underline{T}\)  
\[\underline{T}\left[ \begin{array}{c}a_{1}\\ b_{1}\end{array} \right]=\left[ \begin{array}{c}a_{N}^{(c)}\\ b_{N}^{(c)}\end{array} \right] \tag{3}\]  
has the form of a cascade assembly of individual elements of the antenna structure, which corresponds to the product of the transmission matrix of individual structure elements:  
\[\underline{T}=\underline{T}^{(J)}\underline{T}^{(J-1)}\cdots\underline{T}^{( 2)}\underline{T}^{(1)}_{=} \tag{4}\]  
Figure 1: Flat periodic antenna structure infinite in the (\(x\),\(y\)) plane, multilayer structure. Figure 2: Block diagram of a multi-layer antenna. Knowing the matrices \(T^{(j)}\) and \(T\), it is possible to determine the matrices \(S^{(j)}\) and \(S\) according to the formula:  
\[[S]=\left[ \begin{array}{cc}\underline{S}_{11}^{(j)}&\underline{S}_{12}^{(j)} \\ \underline{S}_{21}^{(j)}&\underline{S}_{22}^{(j)}\end{array} \right]=\left[