# Modes in 1D cavity with loss

Let consider the simple problem of :

1D electromagnetic cavity with perfect conducting walls filled with a conducting medium

Let the length of the cavity be L, the conductivity be $\sigma$ and dielectric be $\epsilon$.

General solution

From Maxwell equations the solution of the wave equation inside this medium is :

$E_z=Ae^{-jk_{m}y}+Be^{+jk_{m}y}$

With $k_{m}$ given by the complex dispersion relation : $k_{m}=\frac{\omega}{c}\bar{n_{m}}=\beta-j\alpha$

Whose complex refractive index(which can easily be expanded) is : $\bar{n_{m}}=\sqrt{1+j\frac{\sigma}{\epsilon\omega}}$

Continuity at the PEC boundary $E_z(0,L)=0$

We consequently have the 2 sets of equations :

$A+B=0$

$Ae^{-j\bar{k_{m}L}}+Be^{+j\bar{k_{m}L}}=0$

Since $k_m$ is complex how do you manage to find the resonant frequencies along with their corresponding mode shapes???

I have tried manipulation but when you look at the amplitude of the complex sinh on a graph, it only passes by 0 once. What am I missing??

## 1 Answer

As you say, I think that your system of equations does not have a non trivial solution of the form $$E_z(y,t) = ( A e^{ik_m y} + B e^{-ik_m y} ) e^{i\omega t},$$ with $\omega\in\mathcal{R}$. These solutions would be "everlasting", in the sense that they are not decaying with time. Maybe the fact of having a complex refractive index (and therefore dissipation) forbids their existence.

If, on the other hand, you look for solutions decaying in time, you might find modes. I mean, suppose that you look for solutions of the form $$E_z(y,t) = ( A e^{ik_m y} + B e^{-ik_m y} ) e^{i(\omega+i\eta) t},$$ with $\omega,\eta\in\mathcal{R}$ and $\eta>0$. The condition to find solutions now (that is, the determinant of your system of equations for $A$ and $B$ being equal to 0) is then $$2\frac{\omega+i\eta}{c}\sqrt{1+i\frac{\sigma}{\epsilon(\omega+i\eta)}}=m\pi,$$ with $m\in\mathcal{Z}$. You can calculate $\omega$ and $\eta$ with this equation, finding then solutions damped with time, which may be the only ones existing.

Does this make sense to you?

• It makes sense indeed. The notion of modes may only stand for constant energy problems. The reason I looked at this problem was to get an intuitive feeling of the modes in plasmonic waveguides(which have complex refractive index). It seems then that the picture of these latter modes can not be explained in the same manner as for lossless waveguides. Jan 4, 2016 at 8:20