## 4 Vector Harmonics of Floquet--Electromagnetic Field Representation in the Plane (\(x\),\(y\))
### TM Polarization  
For the TM field, let us take the longitudinal component of the electric field \(E_{zTMmn}\) as the functions \(f(x,y)\) with the appropriate phase factor depending on \(z\):  
\[E_{zTMmn}(x,y,z)=f(x,y)exp[ik_{zTMmn}z] \tag{45}\]  
Transverse fields: magnetic \(\underline{H}_{tTMmn}\) = (\(H_{xTMmn}\), \(H_{yTMmn}\)) and electric \(\underline{E}_{tTMmn}\) = (\(E_{xTMmn}\), \(E_{yTMmn}\)) are expressed through the \(E_{zTMmn}\) field as follows [6; 17]:  
\[\begin{array}{c}\underline{H}_{tTEmn}=\frac{iux}{k_{xTMmn}^{2}+k_{yTMmn}^{2} }\underline{u}_{z}\times\
abla_{t}H_{zTMmn}\\ =\frac{iux}{k_{xTMmn}^{2}+k_{yTMmn}^{2}}\underline{u}_{z}\times i\left( \underline{u}_{x}k_{xTMmn}+\underline{u}_{y}k_{yTMmn}\right)H_{zTMmn}\,,\end{array} \tag{46}\]  
\[\begin{array}{c}\underline{E}_{tTMmn}=-\frac{ik_{zTMmn}}{k_{xTMmn}^{2}+k_{yTMmn }^{2}}\
abla_{t}E_{zTMmn}.\end{array} \tag{47}\]  
Let us introduce the orthonormal vector harmonics of Floquet [18]: \(\Psi_{TMmn}\equiv\Psi_{p}\) (\(p=TMmn\)) about TM polarization  
\[\underline{\Psi}_{p}=(d_{x}d_{y})^{-1/2}\left[\frac{\underline{x}k_{xp}+ \underline{y}k_{yp}}{(k_{xp}^{2}+k_{yp}^{2})^{1/2}}\right]exp[i\left(k_{xp}x+k _{yp}y\right)], \tag{48}\]  
that is  
\[\psi_{xTMmn}\left(k_{x},k_{yk}\right)=+\left(d_{x}d_{y}\right)^{-1/2}\left(k_{ x}^{2}+k_{yk}^{2}\right)^{-1/2}k_{xTMmn}, \tag{49}\]  
\[\psi_{yTMmn}\left(k_{x},k_{yk}\right)=+\left(d_{x}d_{y}\right)^{-1/2}\left(k_{ x}^{2}+k_{yk}^{2}\right)^{-1/2}k_{yTMmn}. \tag{50}\]  
Then, analogically to the case of \(TE\) polarization, the orthogonality relationship takes place:  
\[\pm\underline{u}_{z}\circ\iint_{S}\underline{\Psi}_{Mmn}\times\underline{ \Psi}_{TMmn}^{*}\,dxdy=\delta_{mm}\delta_{nnr} \tag{51}\]  
Additionally, \(TE\) modes remain orthogonal to \(TM\) modes:  
\[\iint_{S}\underline{\Psi}_{Mmn}\times\underline{\Psi}_{TEmn}^{*}\,dxdy=0 \tag{52}\]which leads to the relationship of orthogonality in the general form (\(p\), \(q\) run for any values from the sets {_TE_, _TM_}, {_m_}, {_m_}):  
\[\pm\underline{\underline{u}_{z}}\,\circ\,\iint_{S}\underline{\underline{\Psi}}_ {p}\times\underline{\underline{\Psi}}_{d}^{*}\,dxdy=\delta_{pq}. \tag{53}\]  
Then, considering the mode impedance for the polarization we finally obtain the form of the transverse components [19] of the electric field \(\underline{E}_{tp}\) and magnetic \(\underline{H}_{tp}\) with _TM_ polarity (\(p=\textit{TM}\textit{nm}\)):  
\[\underline{E}_{tp}(x,y)=\left(\eta_{p}\right)^{+1/2}\underline{\underline{ \Psi}}_{p}(x,y) \tag{54}\]  
\[\pm\eta_{p}\underline{H}_{tp}=\underline{\underline{u}_{z}}\times\underline{ E}_{tp} \tag{55}\]  
\[\pm\eta_{p}^{-1}\underline{E}_{tp}=\underline{\underline{u}_{z}}\times \underline{H}_{tp}. \tag{56}\]  
Hence the transverse magnetic field \(\underline{H}_{tp}\) is expressed by vector harmonics \(\underline{H}_{tp}\) in the following way:  
\[\underline{H}_{tp}(x,y)=\pm\eta_{p}^{-1}\underline{\underline{u}_{z}}\times \underline{E}_{tp}(x,y)=\pm\eta_{p}^{-1/2}\underline{\underline{u}_{z}}\times \underline{\underline{\Psi}}_{p}(x,y) \tag{57}\]  
The coefficient (\(\eta_{p}\))\({}^{-1/2}\) at the normalized vector harmonic \(\underline{u}_{z}\times\Psi_{p}\) ensures the normalization of the harmonic field power with the polarization _TM_:  
\[\pm\underline{\underline{u}_{z}}\,\iint_{S}\underline{E}_{tp}\times\underline {H}_{tp}^{*}\,dxdy=\underline{\underline{u}_{z}}\,(\eta_{p})^{+1}\iint_{S} \left(\underline{\underline{u}_{z}}\times\underline{H}_{tp}\right)\times \underline{H}_{tp}^{*})\,dxdy=\underline{\underline{u}_{z}}\delta_{ppt} \tag{58}\]