## 4 Vector Harmonics of Floquet--Electromagnetic Field Representation in the Plane (\(x\),\(y\))
### TE Polarization  
For the TE field, let us assume that the longitudinal component of the magnetic field \(H_{zTEmn}\) as a pseudo-periodic function \(f(x,y)\) with the appropriate phase factor depending on \(z\):  
\[H_{zTEmn}(x,y,z)=f(x,y)exp[ik_{zTEmn}z]. \tag{33}\]  
From [PERSON]'s equations, the electric transverse field \(E_{tTEmn}=(E_{xTEmn}\), \(E_{yTEmn})\) is expressed through the field \(H_{zTEmn}\)[16] as follows:  
\[\underline{E}_{tTEmn}=\frac{i\omega\mu}{k_{yTEmn}^{2}}\underline{u}_{z}\times \
abla_{t}H_{zTEmn}=\frac{i\omega\mu}{k_{xTEmn}^{2}+k_{yTEmn}^{2}}\underline{u}_ {z}\times i\left(\underline{u}_{x}k_{xTEmn}+\underline{u}_{y}k_{yTEmn}\right) H_{zTEmn} \tag{34}\]  
where: \(\underline{u}_{x},\underline{u}_{y},\underline{u}_{z}\) denote the unit vectors along the respective \(x,y\), and \(z\) axes. In further considerations, we will also denote the sequence of \(TEmn\) or \(TMmn\) pointers with one symbol from the set \(\{q\),\(p\}\). Let us introduce the orthogonal and normalized Floquet vector harmonics \(\Psi_{q}\) (\(q=\Psi_{q}\) (\(q=TEmn\))), i.e., satisfying the condition:  
\[\pm\underline{u}_{z}\circ\iint_{S}\underline{\psi}_{TEmnt}\times\underline{ \psi}_{TEmn}^{*}\ dxdy=\delta_{mmt}\,\delta_{mnt} \tag{35}\]  
where integration takes place over a single elementary cell of a periodic structure with the surface of \(S\). The normalized Floquet harmonics contain, in addition to the phase factor  
\[e_{TEmn}(x,y)=exp[i\left(k_{xTEmn}x+k_{yTEmn}y\right)] \tag{36}\]  
also normalizing vector amplitude coefficient:  
\[\underline{\psi}_{q}\left(k_{xq},k_{yq}\right)=\underline{u}_{x}\psi_{xTEmn}+ \underline{u}_{y}\psi_{xTEmn}=(d_{x}d_{y})^{-1/2}\left[\frac{\underline{u}_{x }k_{yq}-\underline{u}_{y}k_{xq}}{\left(k_{xq}^{2}+k_{yq}^{2}\right)^{1/2}} \right]. \tag{37}\]  
Thus, [PERSON]'s vector harmonics take the following form:  
\[\underline{\Psi}_{q}=\underline{\psi}_{q}\left(k_{xq},k_{yq}\right)e_{q}(x,y) = (d_{x}d_{y})^{-1/2}\left[\frac{\underline{u}_{x}k_{yq}-\underline{u}_{y}k_{q}}{ \left(k_{xq}^{2}+k_{yq}^{2}\right)^{1/2}}\right]exp[i\left(k_{xq}x+k_{yq}y \right)] \tag{38}\]  
with coordinates in the \(x\) and \(y\) directions, respectively:  
\[\Psi_{xq}\left(k_{xq},k_{yq}\right) = +\left(d_{x}d_{y}\right)^{-1/2}\left(k_{xq}^{2}+k_{yq}^{2}\right) ^{-1/2}k_{yq}exp[i\left(k_{xq}x+k_{yq}y\right)] \tag{39}\] \[\Psi_{yq}\left(k_{xq},k_{yq}\right) = -\left(d_{x}d_{y}\right)^{-1/2}\left(k_{xq}^{2}+k_{yq}^{2}\right) ^{-1/2}k_{xq}exp[i\left(k_{xq}x+k_{yq}y\right)] \tag{40}\]Finally, the transverse electric field with a polarization \(TE\)\(\underline{E}_{tq}\) (\(q=TEmn\)) is defined as proportional to the transverse vector harmonic \(\Psi_{tq}\):  
\[\underline{E}_{tq}(x,y)=(\eta_{q}^{*})^{+1/2}\underline{\Psi}_{q}(x,y) \tag{41}\]  
and the longitudinal component \(\underline{H}_{tq}\) of the magnetic field with a polarization \(TE\) is determined from the knowledge of the transverse electric field \(E_{tq}\)  
\[\underline{u}_{z}\times\underline{H}_{tq}=\pm\eta_{q}^{-1}\underline{E}_{tq} \tag{42}\]  
\[\underline{u}_{z}\times\underline{E}_{tq}=\pm\eta_{q}\underline{H}_{tq} \tag{43}\]  
and using the mode impedance \(\eta_{q}\) of a harmonic with a polarization \(TE\).  
The factor \((\eta_{q}^{*})^{1/2}\) standing at the normalized vector harmonic \(\Psi_{q}\) provides the normalization of the power of the harmonic field, which corresponds to the unit power flow conducted through this harmonic in the direction of the \(z\) axis:  
\[\pm\underline{u}_{z}\circ\iint_{S}\underline{E}_{tq}\times\underline{H}_{tq} ^{*}\,dxdy=\underline{u}_{z}\circ(\eta_{q}^{*})^{-1}\iint_{S}\underline{E}_{ tq}\times\left(\underline{u}_{z}\times\underline{E}_{tq}^{*}\right)dxdy= \delta_{qq} \tag{44}\]