## 2 Wave Propagation through a Graded Layer
\begin{split} E_{y}=&-A\ \tau\frac{m\pi}{d_{x}}\sin \left(\frac{m\pi x}{d_{x}}\right)\cos\left(\frac{n\pi y}{d_{y}}\right)\exp\left( 2p\ \frac{z}{z_{0}}\right)\\ &\cdot\left[1+\exp\left(2\ \frac{z}{z_{0}}\right)\right]^{-2p} \ _{2}F_{\rm t}\left[a,b,c;\frac{1}{1+\exp(2\ z/z_{0})}\right]\end{split}  \tag{8}\]  
where \(A\) is a constant proportional to the incident electric field amplitude \(E_{0}\), and \(\tau\) is the transmission coefficient which will be given below. Furthermore, \(\ _{2}F_{\rm t}(a,b,c;u)=F(a,b,c;u)\) is the ordinary Gaussian hypergeometric function [24]. In (7) and (8), several short-hand notations are introduced, as also seen in [20; 21], and are defined as follows  
\[ \begin{split} a&=2p+\frac{1}{2}+\sqrt{r^{2}+\frac{1 }{4}}\\ b&=2p+\frac{1}{2}-\sqrt{r^{2}+\frac{1}{4}}\\ c&=2p+1\end{split}  \tag{9}\]  
with  
\[ \begin{split} p&=\mathrm{i}\frac{z_{0}}{2}\sqrt{k^{2} \epsilon_{\rm G}(\omega)-k_{\rm t}^{2}}=\frac{\mathrm{i}k_{z_{\rm G}}(\omega) z_{0}}{2}\\ r&=k_{2}\sqrt{\epsilon_{\rm L}(\omega)-\epsilon_{ \rm G}(\omega)}\end{split}  \tag{10}\]  
where \(k_{z_{\rm G}}(\omega)=\sqrt{k^{2}\epsilon_{\rm G}(\omega)-k_{\rm t}^{2}}\) is the \(z\)-component of the wave vector of the asymptotic waves for \(z\rightarrow\pm\infty\), and the transverse wavenumber \(k_{\rm t}\) is given by  
\[k_{\rm t}^{2}=\left(\frac{m\pi}{d_{x}}\right)^{2}+\left(\frac{n\pi}{d_{y}} \right)^{2} \tag{11}\]  
for a rectangular waveguide. Based on the above solutions, from [20], the general expressions for the transmission coefficient (\(\tau\)) and the reflection coefficient (\(\rho\)), are  
\[ \begin{split}\tau&=\frac{\Gamma(a)\Gamma(b)}{\Gamma( c)\Gamma(a+b-c)}\,\\ \rho&=\frac{\Gamma(a)\Gamma(b)}{\Gamma(c-a)\Gamma(c-b )}\frac{\Gamma(c-a-b)}{\Gamma(a+b-c)}\.\end{split}  \tag{12}\]  
where \(\Gamma(z)\) is the Gamma function [24]. In lossless media, the power reflection coefficient is given by \(R_{P}=|\rho|^{2}\) and the power transmission coefficient is given by \(T_{P}=|\tau|^{2}\). These coefficients satisfy \(C_{\rm abs}=1-T_{P}-R_{P}=0\), since there is no absorption of energy in the two media. Power Considerations with Losses  
For lossy media, \(C_{\rm abs}=1-T_{P}-R_{P}\) is nonzero, such that \(C_{\rm abs}\ (0<C_{\rm abs}<1)\), quantifies the absorption (electromagnetic energy loss) in the entire waveguide. Plotting this quantity, one can investigate the dependence of the absorption on frequency and on layer size \(z_{0}\). In order to study the absorption in the dielectric layer only, we need to calculate the power reflection coefficient (\(R_{P}\)) and power transmission coefficient (\(T_{P}\)) over the layer (\(-z_{0}\leq z\leq+z_{0}\)). In order to exclude the losses in the surrounding medium, we use the following exact analytical formulas [22]  
\[ \begin{split} T_{P}&=|\tau|^{2}\cdot\left(1+\text{e} ^{2}\right)^{-4\Re\{p\}}\cdot\chi(a,b,c)\\ R_{P}&=|\rho|^{2}\cdot\left(1+\text{e}^{2}\right)^{-4 \Re\{p\}}\cdot\chi(a,b,c)\end{split}  \tag{13}\]  
where we have defined an auxiliary scale factor \(\chi(a,b,c)\) by  
\[\chi(a,b,c)=\left|\frac{F(a,b,c;\frac{1}{1+\text{e}^{2}})}{F(c-a,c-b,c-a-b+1; \frac{1}{1+\text{e}^{2}})}\right|^{2} \tag{14}\]  
where \(F(a,b,c;u)\) is the Gauss hypergeometric function and Equation (10) can be used to relate \(p\) to \(k_{z\text{c}}z_{0}\). If we want to study the losses, i.e., the absorption, in any finite part of the waveguide, we need to calculate the power reflection coefficient (\(R_{Pm}\)) and power transmission coefficient (\(T_{Pm}\)) over some arbitrary layer (\(-m\cdot z_{0}\leq z\leq+m\cdot z_{0}\)), where \(m\) is a number that defines how large is the studied part in the units of \(z_{0}\). It is not necessarily an integer, but can be any fraction. We then obtain the generalized fomulas  
\[ \begin{split} T_{Pm}&=|\tau|^{2}\cdot\left(1+\text{e }^{2m}\right)^{-4\Re\{p\}}\cdot\chi_{m}(a,b,c)\\ R_{Pm}&=|\rho|^{2}\cdot\left(1+\text{e}^{2m}\right)^ {-4\Re\{p\}}\cdot\chi_{m}(a,b,c)\end{split}