## 2 Statement of the Problem
\begin{array}{cc}-\underline{T}_{22}^{-1}\underline{T}_{21}&\underline{T}_{ 22}^{-1}\\ \underline{T}_{11}-\underline{T}_{12}\underline{T}_{22}^{-1}\underline{T}_{21} &\underline{T}_{12}\underline{T}_{22}^{-1}\end{array} \right] \tag{5}\]  
Then we determine the global transmission matrix by the product of successive partial transmission matrices. Thus, for the determination of the scattering matrices it is practically necessary also to determine the partial and global transmission matrices. Similarly, knowing the form of the scattering matrices \(\underline{S}^{(j)}\) of individual elements of the structure, the matrices \(T^{(j)}\) can be determined according to the formula:  
\[\left[ \begin{array}{c}T^{(j)}\\ =\end{array} \right]=\left[ \begin{array}{cc}\underline{T}^{(j)}_{-11}& \underline{T}^{(j)}_{-12}\\ \underline{T}^{(j)}_{-1}&\underline{T}^{(j)}_{-22}\\ \underline{T}^{(j)}_{-21}&\underline{T}^{(j)}_{-22}\end{array} \right]=\left[ \begin{array}{cc}\underline{S}^{(j)}_{-21}-\underline{S}^{(j)}_{-1}S^{(j)} _{-1}&\underline{S}^{(j)}_{-1}\\ -\underline{S}^{(j)}_{-1}S^{(j)}_{-1}&\underline{S}^{(j)}_{-1}\\ -\underline{S}^{(j)}_{-21}&\underline{-11}&\underline{-12}\end{array} \right] \tag{6}\]  
and hence we also obtain the global transmission matrix \(\underline{T}\) as a product of the matrix \(T^{(j)}\)[7]. In general, the cut-off numbers \(K_{j}\) of an infinite countable number of modes of a structure may change from one subsystem of the structure to another, i.e., they may depend on the number \(j\). For the simplicity of notation and further numerical implementation, however, assume that \(K_{j}=const.\) for any subsystem \(j=1,2,\ldots\)\(J\), which is also commonly used in numerical calculations. So we assume that the matrices \(\underline{S}^{(j)}\) and \(\underline{T}^{(j)}\) are square of the order \(2K\) and the matrices \(\underline{S}^{(j)}\)\({}_{kl}\), \(\underline{T}^{(j)}\)\({}_{kl}\), \(k\)\(l=1,2\) are of the order \(K\)[9]. Particular antenna subsystems, as well as the entire antenna system, can also be expressed by impedance matrices \(\underline{Z}^{(j)}\) or admittance matrices \(\underline{Y}^{(j)}\) according to the following formulas [4,5] (\([S]\equiv\underline{S}\)):  
\[[Z]=\frac{[1]+[S]}{[1]-[S]}[\zeta_{0}]=\left[Y^{-1}\right] \tag{7}\]  
\[[S]=\frac{[Z]\left[\zeta_{0}^{-1}\right]-[1]}{[Z]\left[\zeta_{0}^{-1}\right]+[ 1]}=\frac{[1]-[\zeta_{0}][Y]}{[1]+[\zeta_{0}][Y]} \tag{8}\]  
where: \([\zeta_{0}]\) diagonal matrix of characteristic impedances. The form of the matrix \(\underline{S}\), \(\underline{T}\), \(\underline{Z}\), or \(\underline{Y}\) depends on the geometry and parameters of the antenna structure medium, as well as on the adopted complete base of eigenvectors of the appropriate boundary problem. The electromagnetic field at any point in the structure is presented in the form of Fourier integrals or series with appropriate, generally complex, coefficients (amplitudes) of the distribution [10,11]. For infinite periodic structures, the equivalent of Fourier transforms is the field distributions to Floquet's harmonics. As a result, the transverse electric field \(\underline{E}_{t}\) is presented in the form of a series:  
\[ \begin{array}{l}\underline{E}_{t}(x,y,z)=\sum\limits_{k}v_{k}^{\pm} \underline{\Psi}_{k}exp(\pm ik_{zk}z)=\sum\limits_{k}v_{k}^{\pm}\underline{ \Psi}_{k}e_{k}(x,y)exp(\pm ik_{zk}z)\\ =\sum\limits_{k}v_{k}^{\pm}\underline{\Psi}_{k}\Big{[}ik_{zk}x+ik_{yk}y\Big{]} v_{k}^{\pm}\underline{\Psi}_{k}\Big{[}ik_{zk}x+ik_{yk}y\Big{]}\end{array}  \tag{9}\]  
where \(v_{k}^{\pm}\) denote the coefficients of the field distribution propagating in a positive (+) or negative (-) direction with a phase factor \(exp(\pm ik_{zk}z)\). The vectorial normalized to the unit power of [PERSON]'s harmonics \(\Psi_{k}\), defined in the next subsection, have a phase factor \(\epsilon_{k}(x,y)\) and vector amplitudes \(\psi_{k}\). The index \(k\) follows TE or TM polarization and the generally infinite Floquet harmonics indices in the \(x(m)\) and \(y(n)\) directions [12].