Real Electromagnetic Waveguide Modes in loss media

Suppose a waveguide with 2 Perfect electric conductor at both boundaries. The waveguide is filled with a lossy media modelled with a conductivity $\sigma$.

Solving for the following Maxwell's equations (TE mode) : \begin{eqnarray} \mu \partial_t\tilde H_x &=& - \partial_y\tilde E_z \\ \mu \partial_t\tilde H_y &=& + \partial_x \tilde E_z \\ \epsilon \partial_t\tilde E_z + \sigma \tilde E_z &=& + \partial_x \tilde H_y - \partial_y\tilde H_x \end{eqnarray}

one can find that a valid solution adopting a complex notation is: \begin{eqnarray}\label{eq:ShapeComplexSol} H_x &=& +jE_0\frac{k_y}{\omega\mu} \cos(k_yy) e^{j(\omega t - k_x x)}\\ H_y &=& - E_0\frac{k_x}{\omega\mu} \sin(k_yy) e^{j(\omega t - k_x x)}\\ E_z &=& + E_0 \sin(k_yy) e^{j(\omega t - k_x x)} \end{eqnarray}

provided the dispersion relation : \begin{equation}\label{eq:PECdispersion} k_0^2n^2 = k_x^2 + k_y^2 ~~\mbox{with}~~ k_y =\frac{m\pi}{a} ~~\mbox{and}~~ n^2 = \mu\epsilon\left(1 - j\frac{\sigma}{\omega\epsilon}\right) ~~\mbox{and}~~ \end{equation}

where $k_y$ is imposed by the PEC boundary conditions.

Note that since the refractive index n is a complex number, $k_x=\beta-j\alpha=|k_x|e^{j\phi}$ is a complex number too.Consequently, the real part of the solution is :

\begin{eqnarray}\label{eq:RealSol} \Re(H_x) &=& -E_0\frac{ k_y }{\omega\mu}\cos(k_yy)e^{-\alpha x}sin(\omega t - \beta x) \\ \Re(H_y) &=& -E_0\frac{|k_x|}{\omega\mu}\sin(k_yy)e^{-\alpha x}cos(\omega t - \beta x + \phi) \\ \Re(E_z) &=& +E_0 sin(k_yy)e^{-\alpha x}\cos(\omega t - \beta x) \end{eqnarray}

Which does not satisfy the 3rd equation of Maxwell's equations. Indeed, \begin{eqnarray} \partial_x H_y -\partial_y H_x &=& \frac{E_0}{\omega\mu}sin(k_yy)e^{-\alpha x}(|k_x|(\alpha cos(\omega t - \beta x + \phi) - \beta sin(\omega t - \beta x + \phi)) +k_y^2sin(\omega t - \beta x)) \\ &= &\frac{E_0}{\omega\mu}sin(k_yy)e^{-\alpha x}sin(\omega t - \beta x) (|k_x|^2 + k_y^2) \\ &\neq&E_0sin(k_yy)e^{-\alpha x}(-\omega\epsilon sin(\omega t - \beta x) + \sigma cos(\omega t - \beta x)) \label{eq:modulus}\\ &= &\epsilon \partial_t \tilde E_z + \sigma \tilde E_z \end{eqnarray}
My question is the following :

Since taking the real part of the fields is not a valid solution of the presented Maxwell's equations anymore, how can one find the corresponding real modes that would actually be found if one were to carry an experiment?

• You may wish to explain what waveguide you are considering and what coordinates you are using. If this is a rectangular waveguide, why does $E_z$ only depend on 2 coordinates? – akhmeteli Nov 24 '16 at 13:46
• It is a planar waveguide so no dependence on z. Indeed, the Maxwell's equations are written in 2D. $k_x$ is the propagation axis and is the transverse axis hence the quantization of the component $k_y$. – Ronan Tarik Drevon Nov 24 '16 at 16:59

I believe the real part of the fields is still a valid solution of the Maxwell equations. The modes will be attenuated in the direction of propagation $x$.
I don't quite understand though why the component of the magnetic field in the direction of propagation $H_x$ does not vanish for a TM mode.
• @RonanTarikDrevon: In particular, be careful when you calculate the real part of $e^{j(\omega t- k_x x)}$ and $j e^{j(\omega t- k_x x)}$, where $k_x$ is not real. – akhmeteli Nov 24 '16 at 17:48
• @RonanTarikDrevon: 1. Did you check that the complex solution satisfies the Maxwell equations? 2. With all due respect, I don't like your calculations: you should choose one form for complex $k_x$, rather than two forms: one using $\alpha$ and $\beta$ and the other using $|k_x|$ and $\phi$. As of now, I am not sure the Maxwell equations are not satisfied for the real part. – akhmeteli Nov 24 '16 at 18:08