### Physics Model

From Maxwell Equations: $\nabla \times \vec{H} = \vec{j}+\frac{\partial \vec{D}}{\partial t}$

$\nabla \times \vec{E} =- \frac{\partial \vec{B}}{\partial t}$

The material equations are the following:

$\vec{B} = \mu \vec{H}$

$\vec{j} = \sigma \vec{E}$

$\vec{D} = \epsilon \vec{E}$

Combined with the material equations:

$\nabla \times \vec{H} = \sigma \vec{E} + \epsilon \frac{\partial \vec{E}}{\partial t}$

$\nabla \times \vec{E} = - \mu \frac{\partial \vec{H}}{\partial t}$

are resulting two coupled equations, with two unknowns.

In order to make the equations solvable, try to make a differential equation with one unknown.

$-\nabla \times (\mu^{-1} \nabla \times \vec{E}) = \sigma \frac{\partial \vec{E}}{\partial t} + \epsilon \frac{\partial^2 \vec{E}}{\partial t^2}$

And if $\epsilon^{-1} \sigma$ is isotropic:

$-\nabla \times (\epsilon^{-1} \nabla \times \vec{H}) = \epsilon^{-1} \sigma \mu \frac{\partial \vec{H}}{\partial t} + \mu \frac{\partial^2 \vec{H}}{\partial t^2}$

In 3 dimensions, whether we solve (1) for (E), or (2) for (H) they will give the same results, but in lower dimensions one should make constrictions.

The propagation direction lies in the $x-y$ plane:

TE Mode: $-\nabla \times (\mu^{-1} \nabla \times E_z) = \sigma \frac{\partial E_z}{\partial t} + \epsilon \frac{E_z}{\partial t^2}$

TM mode $-\nabla \times (\epsilon^{-1} \nabla \times H_z) = \epsilon^{-1} \sigma \mu \frac{\partial H_z}{\partial t} + \mu \frac{\partial^2 H_z}{\partial t^2}$

The propagation direction lies in the $z$ axis:

TE Mode, with isotropic $\mu$ tensor: $- \mu \Delta_{xy} \vec{E} = \sigma \frac{\partial \vec{E}}{\partial t} + \epsilon \frac{\vec{E}}{\partial t^2}$

TM mode, with isotropic $\epsilon$ tensor: $- \epsilon \Delta_{xy} \vec{H} = \epsilon^{-1} \sigma \mu \frac{\partial \vec{H}}{\partial t} + \mu \frac{\partial^2 \vec{H}}{\partial t^2}$

If in the media exists a source, which generates the radiation, then an additional $\vec{S}$ source temp is added to the right side.