# Intuition and Symmetries in Electromagnetism: Eigenstates of Four Antennas
## 2 Problem Formulation
### Symmetries and Phases  
In this case, we consider only the case of two sources (a source and its image) having an even or odd symmetry relation (in the presence of magnetic or electric planes) [13; 16; 26; 27; 28; 29; 30; 31; 32; 33; 34]. According to the theorem of image explained in [16; 20; 21; 22; 23; 31; 32; 33; 34], we can simply count the phases established between two symmetrical or anti-symmetrical sources (in phase or in phase opposition), as indicated in Figure 2[13].  
In the case of even symmetry, the phase shift established between two sources was:  
\[\Phi_{S12}=0[2\pi]=2k\pi\Rightarrow 2\text{ sources are in phase} \tag{2}\]  
Inversely, in the case of odd symmetry, the phase shift was :  
\[\Phi_{S12}=(2k+1)[\pi]=(2k+1)\pi \tag{3}\]  
\(\Rightarrow\) 2 sources are in phase opposition  
According to Appendix A[17], \(S\) can be written as:  
\[[S]=\begin{pmatrix}S_{11}e^{-j\Phi 1}&S_{12}e^{-j(\Phi 2+\Phi 1)}\\ S_{21}e^{-j(\Phi 2+\Phi 1)}&S_{22}e^{-j\Phi 2}\end{pmatrix} \tag{4}\]  
\(\ast\)**Special cases:**  
\(\diamondsuit\) Case of even symmetry: if we fix \(\Phi 1=0\Rightarrow\Phi 2=0\).  
So,  
\[[S]=\begin{pmatrix}S_{11}&S_{12}\\ S_{21}&S_{22}\end{pmatrix} \tag{5}\]  
\(\diamondsuit\) Case of odd symmetry: if we fix \(\Phi 1=0\Rightarrow\Phi 2=\pi\) and vice versa if \(\Phi 1=\pi\Rightarrow\Phi 2=0\).  
\[[S]=\begin{pmatrix}S_{11}&S_{12}e^{-j\pi}\\ S_{21}e^{-j\pi}&S_{22}e^{-j\pi}\end{pmatrix} \tag{6}\]  
\(\diamondsuit\) In the case of an arbitrary phase shift and an odd symmetry: we define any phase \(\Phi 1\) will automatically add a phase shift of \(\pi\) to \(\Phi 2\) :  
Figure 2: Symmetry and antisymmetry between two sources [13].  
\(\Rightarrow\Phi 2=\Phi 1+\pi\Rightarrow\Phi 2-\Phi 1=\pi\) (Note that this case is verified under HFSS with the phase commands (e.g., Arg(\(S_{21}\))). Finally, the matrix S is written :  
\[[S]=\begin{pmatrix}S_{11}e^{-j\Phi 1}&S_{12}e^{-j(2\Phi 1+\pi)}\\ S_{21}e^{-j(2\Phi 1+\pi)}&S_{22}e^{-j(\Phi 1+\pi)}\end{pmatrix} \tag{7}\]  
We can generalize these different cases to study the configurations of the previous section (two by two between the four antennas). It is now possible to simulate this problem using commercial software (such as HFSS and CST) or other software.  
Finally, we can generalize the case of [S] parameters adapted to four-antennas structure in each symmetry state (in the presence of all combinations of magnetic and electrical walls), which is written as follows [13; 14; 17]:  
\[[S]=\begin{pmatrix}S_{11}e^{-j\Phi 1}&S_{12}e^{-j(\Phi 2-\Phi 1)}&S_{13}e^{-j(\Phi 3-\Phi 1)}&S_{14}e^{-j(\Phi 4-\Phi 1)}\\ S_{21}e^{-j(\Phi 2-\Phi 1)}&S_{22}e^{-j\Phi 2}&S_{23}e^{-j(\Phi 3-\Phi 2)}&S_{24}e^{-j( \Phi 4-\Phi 2)}\\ S_{31}e^{-j(\Phi 3-\Phi 1)}&S_{32}e^{-j(\Phi 3-\Phi 2)}&S_{33}e^{-j(\Phi 3)}&S_{34}e^{-j( \Phi 4-\Phi 3)}\\ S_{41}e^{-j(\Phi 4-\Phi 1)}&S_{42}e^{-j(\Phi 4-\Phi 2)}&S_{43}e^{-j(\Phi 4-\Phi 3)}&S_{44}e^{-j( \Phi 4)}\end{pmatrix} \tag{8}\]  
The symbols \(\Phi 1\), \(\Phi 2\), \(\Phi 3\) and \(\Phi 4\) are parameters that depend on the nature of the symmetry (odd\(\backslash\)even) used at each configuration. The particular cases of odd-even symmetries of Equation (8) will be treated in the same way as the case of two sources (see Equations (5)-(7)). Algebraically, the superposition of the S parameters is possible as the sources amplitudes, as proposed in the previous section.