## 2 Wave Propagation through a Graded Layer  
The geometry of the problem is presented in Figure 2. The surrounding non-magnetic lossy waveguide medium is described by the complex relative permittivity \(\varepsilon_{\mathrm{G}}(\omega)\). The thin lossy non-magnetic layer, inserted about \(z=0\), is described by the complex relative permittivity \(\varepsilon_{\mathrm{L}}(\omega)\). In a previous study by one of the present authors [20], it was found that a homogeneous straight waveguide medium with a single thin layer can be described as a stratified medium with frequency-dependent permittivity \(\varepsilon(\omega,z)\) being a function of the \(z\)-coordinate (waveguide axis direction). The two different media in the waveguide, as shown in Figure 2, can be modeled as a single stratified medium with a spatially dependent permittivity \(\varepsilon(\omega,z)=\varepsilon_{0}\varepsilon_{r}(\omega,z)\), where \(\varepsilon_{r}(\omega,z)\) denotes the relative permittivity, given as a function of the waveguide axis direction  
\[\varepsilon_{r}(\omega,z)=\varepsilon_{\mathrm{L}}(\omega)-[\varepsilon_{ \mathrm{L}}(\omega)-\varepsilon_{\mathrm{G}}(\omega)]\tanh^{2}\biggl{(}\frac{ z}{z_{0}}\biggr{)} \tag{1}\]  
where \(2z_{0}\) represents the thickness of the inserted dielectric layer around the plane \(z=0\). Far away from the thin dielectric layer (\(z\to\pm\infty\)) the function (1) behaves as follows  
\[\tanh^{2}\biggl{(}\frac{z}{z_{0}}\biggr{)}\to 1\ \Rightarrow\ \varepsilon(\omega,\pm\infty)= \varepsilon_{0}\varepsilon_{\mathrm{G}}(\omega)\, \tag{2}\]  
while at the center position of the thin dielectric layer (\(z\to 0\)) the relative permittivity (1) becomes  
\[\tanh^{2}\biggl{(}\frac{z}{z_{0}}\biggr{)}\to 0\ \Rightarrow\ \varepsilon(\omega,0)= \varepsilon_{0}\varepsilon_{\mathrm{L}}(\omega). \tag{3}\]  
This asymptotic behavior ((2) and (3)) is in agreement with the geometry of the problem shown in Figure 2. A geometry with a very thin layer and rapid smooth transition from \(\varepsilon_{\mathrm{G}}(\omega)\) to \(\varepsilon_{\mathrm{L}}(\omega)\) and back to \(\varepsilon_{\mathrm{G}}(\omega)\), is obtained in the limit \(z_{0}\to 0\). The waveguide description presented here is a general theory, where any permittivity model can be employed as long as it satisfies the Kramer-Kronig relations. Two examples of permittivity models [18; 19; 23] for the two dispersive media, \(\varepsilon_{\mathrm{G}}(\omega)\) and \(\varepsilon_{\mathrm{L}}(\omega)\), are given in the next sections of this paper. To describe the wave propagation in a waveguide, without field sources inside (\(\rho=0\), \(\mathbf{J}=0\)), we use the following wave equations [20] for the electric and magnetic fields for TE-waves (with \(E_{z}=0\))  
\[\
abla^{2}\mathbf{E}+k^{2}\varepsilon_{r}(z)\mathbf{E}=0\,\
abla^{2}H_{z}+k^{2} \varepsilon_{r}(z)H_{z}=0. \tag{4}\]  
Here we solve the first of Equations (4) for the electric field \(\mathbf{E}\), whereby the magnetic field \(\mathbf{H}\) is readily obtained from the first of [PERSON]'s equations, i.e., using  
\[\mathbf{H}=\frac{\mathrm{i}}{\omega\mu_{0}}\
abla\times\mathbf{E}. \tag{5}\]  
Thus, the first of Equations (4) with \(E_{z}=0\) is equivalent to two scalar equations, both of which are of the form  
\[\
abla^{2}E_{j}+k^{2}\varepsilon_{r}(z)E_{j}=0\,\ j\in\{x,y\}. \tag{6}\]In the special case of a rectangular waveguide with cross-sectional lengths \(d_{x}\) and \(d_{y}\), we then obtain the overall expressions for the two electric field components in the form  
\[ \begin{split} E_{x}=& A\ \tau\frac{n\pi}{d_{y}}\cos \left(\frac{m\pi x}{d_{x}}\right)\sin\left(\frac{n\pi y}{d_{y}}\right)\exp\left( 2p\ \frac{z}{z_{0}}\right)\\ &\cdot\left[1+\exp\left(2\ \frac{z}{z_{0}}\right)\right]^{-2p} \ _{2}F_{\rm t}\left[a,b,c;\frac{1}{1+\exp(2\ z/z_{0})}\right]\end{split}  \tag{7}\]  
\[