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.

Examples

2D Simulations

Scattering in $x-y$ plane

Waveguide modes $x-y$ plane