# 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 expressions below, the phase jumps at unit grating jumps, and the components of the wave vector in the \(x\) and \(y\) directions are, respectively, as follows:  
\[\underline{\Psi}_{k}(x,y)=\underline{\psi}_{k}exp[i\Big{(}k_{xk}x+k_{yk}y \Big{)}]=\Big{(}d_{x}d_{y})^{1/2}\underline{\psi}_{k}e_{k}(x,y) \tag{80}\]6  
\[\underline{\Psi}_{k}(x,y)=\underline{\psi}_{k}exp[i\Big{(}k_{xk}x+k_{yk}y \Big{)}]=\Big{(}d_{x}d_{y})^{1/2}\underline{\psi}_{k}e_{k}(x,y) \tag{80}\]7  
Similarly to the case of plane wave excitation, i.e., a single harmonic, we obtain the following expressions for the subsequent components of the electric field. The incident field:  
\[ \begin{array}{c}\underline{E}_{t}^{i}(x,y,0)=\sum\limits_{k}\underline{E}_{ tk}^{i}(x,y,0)=\sum\limits_{k}v_{k}^{i}\underline{\Psi}_{k}(x,y)=\sum\limits_{m.n}\sum\limits_{k}\underline{E}_{kkmn}^{i}(x,y,0)\\ =\sum\limits_{m.n}\sum\limits_{k}\underline{E}_{kt00}^{i}(x-x_{m},y-y_{n},0) exp\left(im\phi_{xk}+in\phi_{yk}\right)\end{array}  \tag{97}\]  
where  
\[\underline{E}_{ktmn}^{i}(x,y,0)=v_{k}^{i}h_{mn}(x,y)\underline{\Psi}_{k}(x,y), \tag{98}\]  
* The stimulating field:  
\[ \begin{array}{c}\underline{E}_{t}^{s}(x,y,0)=\sum\limits_{k}\underline{E}_{ tk}^{e}(x,y,0)=\sum\limits_{k}v_{k}^{e}\underline{\Psi}_{k}(x,y)=\sum_{m.n}\sum\limits_{k}\underline{E}_{ktmn}^{e}(x,y,0)=\\ -U\underline{\sum}_{m.n}\sum\limits_{k}\underline{E}_{kt00}^{i}(x-x_{m},y-y_ {n},0)exp\left(im\phi_{xk}+in\phi_{yk}\right)\end{array}  \tag{99}\]  
where  
\[\underline{E}_{ktmn}^{e}(x,y,0)=v_{k}^{e}h_{mn}(x,y)\underline{\Psi}_{k}(x,y) \tag{100}\]  
and by definition \(v_{k}^{e}=(1+\Gamma_{k})v_{k}^{i}=-Uv_{k}^{i}\), where \(\Gamma_{k}\) is the reflection coefficient of the harmonic with index \(k\) from the surface \(z=0\) without metal elements. For the scattered field:  
\[ \begin{array}{c}\underline{E}_{t}^{s}(x,y,0)=\sum\limits_{k} \underline{E}_{tk}^{s}(x,y,0)=\sum\limits_{k}v_{k}^{s}\underline{\Psi}_{k}(x, y)=\sum\limits_{m.n}\sum\limits_{k}\underline{E}_{ktmn}^{s}(x,y,0)\\ =\sum\limits_{m.n}\sum\limits_{k}\underline{E}_{kt00}^{s}(x-x_{m},y-y_{n},0) exp\left(im\phi_{xk}+in\phi_{yk}\right)\end{array}  \tag{101}\]  
where  
\[ \begin{array}{c}\underline{E}_{ktmn}^{s}(x,y,0)=v_{k}^{s}h_{mn}(x,y) \underline{\Psi}_{k}(x,y),\end{array}  \tag{102}\]  
For total area:  
\[ \begin{array}{c}\underline{E}_{t}(x,y,0)=\sum\limits_{k} \underline{E}_{tk}(x,y,0)=\sum\limits_{k}v_{k}\underline{\Psi}_{k}(x,y)=\sum \limits_{m.n}\sum\limits_{k}\underline{E}_{ktmn}(x,y,0)\\ =\sum\limits_{m.n}\sum\limits_{k}\underline{E}_{kt00}(x-x_{m},y-y_{n},0) exp\left(im\phi_{xk}+in\phi_{yk}\right)\end{array}  \tag{103}\]  
where  
\[ \begin{array}{c}\underline{E}_{ktmn}(x,y,0)=v_{k}h_{mn}(x,y)\underline{\Psi }_{k}(x,y)\end{array}  \tag{104}\]  
By definition of the total field:  
\[\underline{E}_{t}(x,y,z)=\underline{E}_{t}^{e}(x,y,z)+\underline{E}_{t}^{s}(x,y,z) \tag{105}\]  
we obtain the following:  
\[v_{k}=v_{k}^{e}+v_{k}^{s}=(1+\Gamma_{k})v_{k}^{i}+v_{k}^{s} \tag{106}\]The distribution coefficients \({v_{k}}^{s}\) and hence \({v_{k}}\) remain to be determined by applying the method of moments.