# FDTD Formulation

## Maxwell's equation
$\nabla \times {\mathbf{E}(\mathbf{x},t)} = -\frac{\partial \mathbf{B}(\mathbf{x},t)}{\partial t}$
$\nabla \times {\mathbf{H}(\mathbf{x},t)} = \mathbf{J}(\mathbf{x},t) +\frac{\partial \mathbf{D}(\mathbf{x},t)}{\partial t}$

Consider inhomogenoues, isotropic material.

${\mathbf{D}(\mathbf{x},t)}={\epsilon(\mathbf{x})}{\mathbf{E}(\mathbf{x},t)}$
${\mathbf{B}(\mathbf{x},t)}={\mu(\mathbf{x})}{\mathbf{H}(\mathbf{x},t)}$

### 3-D
$\left\{
\begin{matrix}
\frac{\partial{H_x}}{\partial{t}} = \frac{-1}{\mu}\left(\frac{\partial{E_z}}{\partial{y}} - \frac{\partial{E_y}}{\partial{z}}\right) - \frac{1}{\mu}\left(M_{sx} + \sigma^{*}H_x \right) \\
\frac{\partial{H_y}}{\partial{t}} = \frac{-1}{\mu}\left(\frac{\partial{E_x}}{\partial{z}} - \frac{\partial{E_z}}{\partial{x}}\right) - \frac{1}{\mu}\left(M_{sy} + \sigma^{*}H_y \right) \\
\frac{\partial{H_z}}{\partial{t}} = \frac{-1}{\mu}\left(\frac{\partial{E_y}}{\partial{x}} - \frac{\partial{E_x}}{\partial{y}}\right) - \frac{1}{\mu}\left(M_{sz} + \sigma^{*}H_z \right) \\
\frac{\partial{E_x}}{\partial{t}} = \frac{1}{\epsilon}\left(\frac{\partial{H_z}}{\partial{y}} - \frac{\partial{H_y}}{\partial{z}}\right) - \frac{1}{\epsilon}\left(J_{sx} + \sigma E_x \right) \\
\frac{\partial{E_y}}{\partial{t}} = \frac{1}{\epsilon}\left(\frac{\partial{H_x}}{\partial{z}} - \frac{\partial{H_z}}{\partial{x}}\right) - \frac{1}{\epsilon}\left(J_{sy} + \sigma E_y \right) \\
\frac{\partial{E_z}}{\partial{t}} = \frac{1}{\epsilon}\left(\frac{\partial{H_y}}{\partial{x}} - \frac{\partial{H_x}}{\partial{y}}\right) - \frac{1}{\epsilon}\left(J_{sz} + \sigma E_z \right) \\
\end{matrix}
\right.$

### 2-D
Assuming z-direction directive as zero.
$\left\{
\begin{matrix}
\frac{\partial{H_x}}{\partial{t}} = \frac{-1}{\mu}\left(\frac{\partial{E_z}}{\partial{y}} \right) - \frac{1}{\mu}\left(M_{sx} + \sigma^{*}H_x \right) \\
\frac{\partial{H_y}}{\partial{t}} = \frac{-1}{\mu}\left( - \frac{\partial{E_z}}{\partial{x}}\right) - \frac{1}{\mu}\left(M_{sy} + \sigma^{*}H_y \right) \\
\frac{\partial{H_z}}{\partial{t}} = \frac{-1}{\mu}\left(\frac{\partial{E_y}}{\partial{x}} - \frac{\partial{E_x}}{\partial{y}}\right) - \frac{1}{\mu}\left(M_{sz} + \sigma^{*}H_z \right) \\
\frac{\partial{E_x}}{\partial{t}} = \frac{1}{\epsilon}\left(\frac{\partial{H_z}}{\partial{y}}\right) - \frac{1}{\epsilon}\left(J_{sx} + \sigma E_x \right) \\
\frac{\partial{E_y}}{\partial{t}} = \frac{1}{\epsilon}\left(- \frac{\partial{H_z}}{\partial{x}}\right) - \frac{1}{\epsilon}\left(J_{sy} + \sigma E_y \right) \\
\frac{\partial{E_z}}{\partial{t}} = \frac{1}{\epsilon}\left(\frac{\partial{H_y}}{\partial{x}} - \frac{\partial{H_x}}{\partial{y}}\right) - \frac{1}{\epsilon}\left(J_{sz} + \sigma E_z \right) \\
\end{matrix}
\right.$

### 1-D
Assuming z-direction and y-direction directives are both zero.
$\left\{
\begin{matrix}
\frac{\partial{H_x}}{\partial{t}} = - \frac{1}{\mu}\left(M_{sx} + \sigma^{*}H_x \right) \\
\frac{\partial{H_y}}{\partial{t}} = \frac{-1}{\mu}\left( - \frac{\partial{E_z}}{\partial{x}}\right) - \frac{1}{\mu}\left(M_{sy} + \sigma^{*}H_y \right) \\
\frac{\partial{H_z}}{\partial{t}} = \frac{-1}{\mu}\left(\frac{\partial{E_y}}{\partial{x}}\right) - \frac{1}{\mu}\left(M_{sz} + \sigma^{*}H_z \right) \\
\frac{\partial{E_x}}{\partial{t}} = - \frac{1}{\epsilon}\left(J_{sx} + \sigma E_x \right) \\
\frac{\partial{E_y}}{\partial{t}} = \frac{1}{\epsilon}\left(- \frac{\partial{H_z}}{\partial{x}}\right) - \frac{1}{\epsilon}\left(J_{sy} + \sigma E_y \right) \\
\frac{\partial{E_z}}{\partial{t}} = \frac{1}{\epsilon}\left(\frac{\partial{H_y}}{\partial{x}}\right) - \frac{1}{\epsilon}\left(J_{sz} + \sigma E_z \right) \\
\end{matrix}
\right.$

## Yee grid, FDTD
Yee grid from Meep document.
<img src='https://meep.readthedocs.io/en/latest/images/Yee-cube.png'></img>
Time step - Leap Frog

### Three dimensional FDTD formulation
$\left\{
\begin{matrix}
\mathbf{E} = (E_x|^{n+\frac{1}{2}}_{i, j+\frac{1}{2}, k+\frac{1}{2}}, E_y|^{n+\frac{1}{2}}_{i+\frac{1}{2}, j, k+\frac{1}{2}}, E_z|^{n+\frac{1}{2}}_{i+\frac{1}{2}, j+\frac{1}{2}, k}) \\
\mathbf{H} = (H_x|^{n}_{i+\frac{1}{2}, j, k}, H_y|^{n}_{i, j+\frac{1}{2}, k}, H_z|^{n}_{i, j, k+\frac{1}{2}}) \\
\end{matrix}
\right.$

$\left\{
\begin{matrix}
\frac{\partial{E_x}}{\partial{t}} = \frac{1}{\epsilon}\left(\frac{\partial{H_z}}{\partial{y}} - \frac{\partial{H_y}}{\partial{z}}\right) - \frac{1}{\epsilon}\left(J_{sx} + \sigma E_x \right) \\
\frac{\partial{E_x}}{\partial{t}} = \frac{\left (E_x|^{n+\frac{1}{2}}_{i, j+\frac{1}{2}, k+\frac{1}{2}} - E_x|^{n-\frac{1}{2}}_{i, j+\frac{1}{2}, k+\frac{1}{2}} \right )}{\Delta t}
\end{matrix}
\right.$

$\frac{\partial{E_x}}{\partial{t}} = \frac{1}{\epsilon}\left(\frac{\partial{H_z}}{\partial{y}} - \frac{\partial{H_y}}{\partial{z}}\right) - \frac{1}{\epsilon}\left(J_{sx} + \sigma E_x \right)$

$\frac{\left (E_x|^{n+\frac{1}{2}}_{i, j+\frac{1}{2}, k+\frac{1}{2}} - E_x|^{n-\frac{1}{2}}_{i, j+\frac{1}{2}, k+\frac{1}{2}} \right )}{\Delta t} = \frac{1}{\epsilon} \left(\frac{H_z|^{n}_{i, j+\frac{1}{2}, k+\frac{1}{2}} - H_z|^{n}_{i, j-\frac{1}{2}, k+\frac{1}{2}}}{\Delta y} - \frac{H_y|^{n}_{i+\frac{1}{2}, j, k+\frac{1}{2}} - H_y|^{n}_{i + +\frac{1}{2}, j, k-\frac{1}{2}}}{\Delta z}\right)$

## Numerical Dispersion and Courant Condition

## Reference

## Appendix