FDTD in OghmaNano

1. Introduction

Theoretical background

References for this section are . This section of the manual aims to describe the FDTD code in full with verbose derivations to help understanding/pick up errors.

Ampere’s law is given as \[\sigma \boldsymbol{E} + \epsilon \frac{\partial \boldsymbol{E}}{\partial t} = \nabla \times \boldsymbol{H} = \begin{vmatrix} \hat{\boldsymbol{x}} & \hat{\boldsymbol{y}} & \hat{\boldsymbol{z}} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ H_{x} & H_{y} & H_{z} \end{vmatrix}\]

which can be expanded as

\[\sigma E_{x} + \epsilon \frac{\partial E_{x}}{\partial t} = \frac{\partial H_{z}}{\partial y}-\frac{\partial H_{y}}{\partial z}\]

\[\sigma E_{y} + \epsilon \frac{\partial E_{y}}{\partial t} = -\frac{\partial H_{z}}{\partial x}+\frac{\partial H_{x}}{\partial z}\]

\[\sigma E_{z} + \epsilon \frac{\partial E_{z}}{\partial t} = \frac{\partial H_{y}}{\partial x}-\frac{\partial H_{x}}{\partial y}\]

For the case \(\frac{\partial}{\partial y}=0\)

\[\begin{split} &\sigma E_{x} + \epsilon \frac{\partial E_{x}}{\partial t} =-\frac{\partial H_{y}}{\partial z}\\ &\sigma E_{y} + \epsilon \frac{\partial E_{y}}{\partial t} = -\frac{\partial H_{z}}{\partial x}+\frac{\partial H_{x}}{\partial z}\\ &\sigma E_{z} + \epsilon \frac{\partial E_{z}}{\partial t} = \frac{\partial H_{y}}{\partial x} \end{split}\]

for \(E_{x}\) \[\begin{split} &\sigma E_{x} + \epsilon \frac{\partial E_{x}}{\partial t} =-\frac{\partial H_{y}}{\partial z}\\ &\sigma \frac{E_{x}^{t+1}[]+E_{x}^{t}[]}{2} + \epsilon \frac{E_{x}^{t+1}[]-E_{x}^{t}[]}{\Delta t} = -\frac{H_{y}^{t+\frac{1}{2}}[\frac{1}{2}]-H_{y}^{t+\frac{1}{2}}[-\frac{1}{2}]}{\Delta z}\\ &\sigma \frac{E_{x}^{t+1}[]}{2} + \epsilon \frac{E_{x}^{t+1}[]}{\Delta t} = -\frac{H_{y}^{t+\frac{1}{2}}[\frac{1}{2}]-H_{y}^{t+\frac{1}{2}}[-\frac{1}{2}]}{\Delta z}-\sigma \frac{E_{x}^{t}[]}{2}+\epsilon \frac{E_{x}^{t}[]}{\Delta t}\\ &\sigma \frac{E_{x}^{t+1}[]}{2} + \epsilon \frac{E_{x}^{t+1}[]}{\Delta t} = -\frac{H_{y}^{t+\frac{1}{2}}[\frac{1}{2}]-H_{y}^{t+\frac{1}{2}}[-\frac{1}{2}]}{\Delta z}-\sigma \frac{E_{x}^{t}[]}{2}+\epsilon \frac{E_{x}^{t}[]}{\Delta t}\\ & \frac{\sigma \Delta t + 2 \epsilon }{ 2 \Delta t}E_{x}^{t+1}[] = -\frac{H_{y}^{t+\frac{1}{2}}[\frac{1}{2}]-H_{y}^{t+\frac{1}{2}}[-\frac{1}{2}]}{\Delta z}-\sigma \frac{E_{x}^{t}[]}{2}+\epsilon \frac{E_{x}^{t}[]}{\Delta t}\\ & E_{x}^{t+1}[] = \left ( -\frac{H_{y}^{t+\frac{1}{2}}[\frac{1}{2}]-H_{y}^{t+\frac{1}{2}}[-\frac{1}{2}]}{\Delta z}-\sigma \frac{E_{x}^{t}[]}{2}+\epsilon \frac{E_{x}^{t}[]}{\Delta t} \right ) \frac{2 \Delta t}{\sigma \Delta t + 2 \epsilon} \end{split}\]

for \(E_{y}\) \[\begin{split} &\sigma E_{y} + \epsilon \frac{\partial E_{y}}{\partial t} = -\frac{\partial H_{z}}{\partial x}+\frac{\partial H_{x}}{\partial z}\\ &\sigma \frac{E_{y}^{t+1}[]+E_{y}^{t}[]}{2} + \epsilon \frac{E_{y}^{t+1}[]-E_{y}^{t}[]}{\Delta t} = -\frac{H_{z}^{t+\frac{1}{2}}[\frac{1}{2}]-H_{z}^{t+\frac{1}{2}}[-\frac{1}{2}]}{\Delta x}+\frac{H_{x}^{t+\frac{1}{2}}[\frac{1}{2}]-H_{x}^{t+\frac{1}{2}}[-\frac{1}{2}]}{\Delta z}\\ &\sigma \frac{E_{y}^{t+1}[]}{2} + \epsilon \frac{E_{y}^{t+1}[]}{\Delta t} = -\frac{H_{z}^{t+\frac{1}{2}}[\frac{1}{2}]-H_{z}^{t+\frac{1}{2}}[-\frac{1}{2}]}{\Delta x}+\frac{H_{x}^{t+\frac{1}{2}}[\frac{1}{2}]-H_{x}^{t+\frac{1}{2}}[-\frac{1}{2}]}{\Delta z}-\sigma \frac{E_{y}^{t}[]}{2} + \epsilon \frac{E_{y}^{t}[]}{\Delta t}\\ &E_{y}^{t+1}[] = \left ( -\frac{H_{z}^{t+\frac{1}{2}}[\frac{1}{2}]-H_{z}^{t+\frac{1}{2}}[-\frac{1}{2}]}{\Delta x}+\frac{H_{x}^{t+\frac{1}{2}}[\frac{1}{2}]-H_{x}^{t+\frac{1}{2}}[-\frac{1}{2}]}{\Delta z}-\sigma \frac{E_{y}^{t}[]}{2} + \epsilon \frac{E_{y}^{t}[]}{\Delta t} \right ) \frac{2 \Delta t}{\sigma \Delta t + 2 \epsilon} \end{split}\]

for \(E_{z}\) \[\begin{split} &\sigma E_{z} + \epsilon \frac{\partial E_{z}}{\partial t} = \frac{\partial H_{y}}{\partial x}\\ &\sigma \frac{E_{z}^{t+1}[]+E_{z}^{t}[]}{2} + \epsilon \frac{E_{z}^{t+1}[]-E_{z}^{t}[]}{\Delta t} = \frac{H_{y}^{t+\frac{1}{2}}[\frac{1}{2}]-H_{y}^{t+\frac{1}{2}}[-\frac{1}{2}]}{\Delta x}\\ &\sigma \frac{E_{z}^{t+1}[]}{2} + \epsilon \frac{E_{z}^{t+1}[]}{\Delta t} = \frac{H_{y}^{t+\frac{1}{2}}[\frac{1}{2}]-H_{y}^{t+\frac{1}{2}}[-\frac{1}{2}]}{\Delta x}-\sigma \frac{E_{z}^{t}[]}{2} + \epsilon \frac{E_{z}^{t}[]}{\Delta t}\\ &E_{z}^{t+1}[]= \left ( \frac{H_{y}^{t+\frac{1}{2}}[\frac{1}{2}]-H_{y}^{t+\frac{1}{2}}[-\frac{1}{2}]}{\Delta x}-\sigma \frac{E_{z}^{t}[]}{2} + \epsilon \frac{E_{z}^{t}[]}{\Delta t} \right ) \frac{2 \Delta t}{\sigma \Delta t + 2 \epsilon}\\ \end{split}\]

Faraday’s law is given as \[-\sigma_{m} \boldsymbol{H} - \mu \frac{\partial \boldsymbol{H}}{\partial t} = \nabla \times \boldsymbol{E} = \begin{vmatrix} \hat{\boldsymbol{x}} & \hat{\boldsymbol{y}} & \hat{\boldsymbol{z}} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ E_{x} & E_{y} & E_{z} \end{vmatrix}\]

which can be expanded to give:

\[-\sigma_{m} H_{x} - \mu \frac{\partial H_{x}}{\partial t} = \frac{\partial E_{z}}{\partial y}-\frac{\partial E_{y}}{\partial z}\]

\[-\sigma_{m} H_{y} - \mu \frac{\partial H_{y}}{\partial t} = -\frac{\partial E_{z}}{\partial x}+\frac{\partial E_{x}}{\partial z}\]

\[-\sigma_{m} H_{z} - \mu \frac{\partial H_{z}}{\partial t} = \frac{\partial E_{y}}{\partial x}-\frac{\partial E_{x}}{\partial y}\]

With \(\sigma_m=0\) and \(\frac{\partial}{\partial y}=0\)

\[\begin{split} &\frac{\partial H_{x}}{\partial t} = \frac{1}{\mu} \left ( \frac{\partial E_{y}}{\partial z} \right )\\ &\frac{\partial H_{y}}{\partial t} = \frac{1}{\mu} \left ( \frac{\partial E_{z}}{\partial x}-\frac{\partial E_{x}}{\partial z} \right )\\ &\frac{\partial H_{z}}{\partial t} = - \frac{1}{\mu} \left ( \frac{\partial E_{y}}{\partial x} \right ) \end{split}\]

which discretizing gives

\[\begin{split} & H_{x}^{t+1} = \frac{1}{\mu} \left ( \frac{E_{y}^{t+\frac{1}{2}}[\frac{1}{2}]-E_{y}^{t+\frac{1}{2}}[-\frac{1}{2}]}{\Delta z} \right ) \Delta t + H_{x}^{t}[]\\ & H_{y}^{t+1} = \frac{1}{\mu} \left ( \frac{E_{z}^{t+\frac{1}{2}}[\frac{1}{2}]-E_{z}^{t+\frac{1}{2}}[-\frac{1}{2}]}{\Delta x}-\frac{E_{x}^{t+\frac{1}{2}}[\frac{1}{2}]-E_{x}^{t+\frac{1}{2}}[-\frac{1}{2}]}{\Delta z} \right ) \Delta t+ H_{y}^{t}[]\\ & H_{z}^{t+1} = \frac{1}{\mu} \left ( - \frac{E_{y}^{t+\frac{1}{2}}[\frac{1}{2}]-E_{y}^{t+\frac{1}{2}}[-\frac{1}{2}]}{\Delta x} \right ) \Delta t + H_{x}^{z}[] \end{split}\]