Wave Guides
#Physics
Conductive tubes that guide light waves
Topics
$$
\begin{cases}
\vec{E}^{\parallel }=\vec{0} \\B^{\perp}=0
\end{cases}
$$
- Boundary conditions for a hollow wave guide
$$\displaystyle \begin{cases}
\partial_{x}E_{y}-\partial_{y}E_{x}=i\omega B_{z} \\\partial_{y}E_{z}-ikE_{y}=i\omega B_{x} \\ikE_{x}-\partial_{x}E_{z}=i\omega B_{y} \\\partial_{x}B_{y}-\partial_{y}B_{x}=-\frac{i\omega}{c^{2}}E_{z} \\\partial_{y}B_{z}-ikB_{y}=-\frac{i\omega}{c^{2}}E_{x} \\ikB_{x}-\partial_{x}B_{z}=-\frac{i\omega}{c^{2}}E_{y}
\end{cases}
$$
- Equations of electric fields obtained by Maxwell's equations (Griffith's pg. 426)
$$
\begin{cases}
E_{x}= \frac{i}{\left( \frac{\omega}{c} \right)^{2}-k^{2}}(k \partial_{x}E_{z}+\omega \partial_{y}B_{z}) \\E_{y}= \frac{i}{\left( \frac{\omega}{c} \right)^{2}-k^{2}}(k \partial_{y}E_{z}-\omega \partial_{x}B_{z}) \\B_{x}= \frac{i}{\left( \frac{\omega}{c} \right)^{2}-k^{2}}\left( k \partial_{x}B_{z}-\frac{\omega}{c^{2}}\partial_{x}E_{z} \right) \\B_{y}= \frac{i}{\left( \frac{\omega}{c} \right)^{2}-k^{2}}\left( k \partial_{y}B_{z}-\frac{\omega}{c^{2}}\partial_{x}E_{x} \right)
\end{cases}
$$
- Transverse components of EM waves in wave guides
$$
\begin{cases}
\left[ \partial_{x}^{2}+\partial_{y}^{2}+\left( \frac{\omega}{c} \right)^{2}-k^{2} \right]E_{z}=0 \\\left[ \partial_{x}^{2}+\partial_{y}^{2}+\left( \frac{\omega}{c} \right)^{2}-k^{2} \right]B_{z}=0
\end{cases}
$$
- Longitudinal components of waves in wave guides
$\displaystyle !(E_{z}=B_{z}=0)$
- You cannot have transverse EM waves in hollow wave guides