# Galerkin's Spectral Method in the Analysis of Antenna Wall Operation
## 3 Radiation Field Analysis of Unrestricted Antenna Array
### Spectral Representation of the Field--The Case of Arbitrary Excitation  
In the general case, for any excitation, the incident field \(\underline{E}_{t}^{i}\), the excitation field \(\underline{E}_{t}^{e}\), the scattered field \(\underline{E}_{t}^{s}\) and the total field \(\underline{E}_{t}^{s}\) are presented as scalar decompositions:  
\[e_{k}(x,y)=\Big{(}d_{x}d_{y})^{-1/2}exp[i\Big{(}k_{xk}x+k_{yk}y\Big{)}\Big{]} \tag{79}\]  
and Floquet vector harmonics:  
\[\underline{\Psi}_{k}(x,y)=\underline{\psi}_{k}exp[i\Big{(}k_{xk}x+k_{yk}y \Big{)}]=\Big{(}d_{x}d_{y})^{1/2}\underline{\psi}_{k}e_{k}(x,y) \tag{80}\]The values of the wave vector components are determined by the periodicity of the structure and the assumed direction of incidence of the incident wave, i.e.,  
\[k_{xk}=2\pi m^{\prime}/d_{x}+ksin\theta_{k}cos\phi_{k} \tag{81}\]  
\[k_{yk}=2\pi n^{\prime}/d_{y}+ksin\theta_{k}sin\phi_{k} \tag{82}\]  
with the distribution coefficients \(v_{k}^{i},v_{k}^{i},v_{k}^{s},v_{k}\), respectively, where \(k=\{l,\,m^{\prime},\,n^{\prime}\}\), \(l=1\) for TE-polarization and \(l=2\) for TM-polarization. The values of the vector amplitudes \(\underline{y}_{k}\) of the Floquet harmonics are as follows:  
\[\underline{\psi}_{k}=\underline{u}_{x}\psi_{xk}+\underline{u}_{y}\psi_{yk}=( -1)^{l}(d_{x}d_{y})^{-1/2}\left[\frac{\underline{u}_{x}k_{yk}-\underline{u}_{ y}k_{xk}}{(k_{x}^{2}+k_{yk}^{2})^{1/2}}\right] \tag{83}\]  
They are ensured by the orthonormality condition of scalar harmonics:  
\[\iint\limits_{S_{00}}e_{k}(x,y)e_{k}^{*}(x,y)dxdy=\delta_{kk}. \tag{84}\]  
They also fulfill the condition of vector orthonormality of harmonics:  
\[\iint\limits_{S_{00}}\underline{\psi}_{k}^{\circ}\underline{\psi}_{k}^{*}= \delta_{kk}. \tag{85}\]  
where the symbol \"\(\circ\)\" denotes the product of two tensor quantities, and the integration is performed over the surface of any periodic cell of the antenna (here, the surface of the central cell \(S_{00}\)). The generating functions (\(H_{zk}\) for TE-polarization, \(E_{zk}\) for TM-polarization) and the remaining components of the fields for vector harmonics are, respectively, as follows:  
TE-polarization:  
\[H_{zk}(x,y)=(\omega\mu)^{-1}\sqrt{k_{x}^{2}+k_{y}^{2}}e_{k}(x,y) \tag{86}\]  
\[H_{yk}=\pm\eta_{k}^{-1}\Psi_{xk} \tag{87}\]  
\[H_{xk}=\mp\eta_{k}^{-1}\Psi_{yk} \tag{88}\]  
TM-polarization:  
\[\underline{\Psi}_{k}(x,y)=\underline{\psi}_{k}exp[i\Big{(}k_{xk}x+k_{yk}y \Big{)}]=\Big{(}d_{x}d_{y})^{1/2}\underline{\psi}_{k}e_{k}(x,y) \tag{80}\]0  
\[\underline{\Psi}_{k}(x,y)=\underline{\psi}_{k}exp[i\Big{(}k_{xk}x+k_{yk}y \Big{)}]=\Big{(}d_{x}d_{y})^{1/2}\underline{\psi}_{k}e_{k}(x,y) \tag{80}\]1  
\[\underline{\Psi}_{k}(x,y)=\underline{\psi}_{k}exp[i\Big{(}k_{xk}x+k_{yk}y \Big{)}]=\Big{(}d_{x}d_{y})^{1/2}\underline{\psi}_{k}e_{k}(x,y) \tag{80}\]2  
The total power flux \(P_{z}\) of the field in the direction of the \(z\)-axis is given by  
\[\underline{\Psi}_{k}(x,y)=\underline{\psi}_{k}exp[i\Big{(}k_{xk}x+k_{yk}y \Big{)}]=\Big{(}d_{x}d_{y})^{1/2}\underline{\psi}_{k}e_{k}(x,y) \tag{80}\]3  
where the sign \(\pm\) corresponds to the directions of propagation of the harmonics, respectively, in line with or against the direction of the \(z\)-axis, and the coefficient \(\Gamma_{k}\) is determined by the mode impedance of the harmonic:  
\[\underline{\Psi}_{k}(x,y)=\underline{\psi}_{k}exp[i\Big{(}k_{xk}x+k_{yk}y \Big{)}]=\Big{(}d_{x}d_{y})^{1/2}\underline{\psi}_{k}e_{k}(x,y) \tag{80}\]4  
\[\underline{\Psi}_{k}(x,y)=\underline{\psi}_{k}exp[i\Big{(}k_{xk}x+k_{yk}y \Big{)}]=\Big{(}d_{x}d_{y})^{1/2}\underline{\psi}_{k}e_{k}(x,y) \tag{80}\]5Summation over the index \(k\) covers both propagating modes \([Im(k\mathrm{z}k)]=0\) and decaying modes \(Im(k_{zk})\
eq 0\).